Daily Papers

We analyze the stability of a class of eigensolvers that target interior eigenvalues with rational filters. We show that subspace iteration with a rational filter is robust even when an eigenvalue is near a filter's pole. These dangerous eigenvalues contribute to large round-off errors in the first iteration, but are self-correcting in later iterations. For matrices with orthogonal eigenvectors (e.g., real-symmetric or complex Hermitian), two iterations is enough to reduce round-off errors to the order of the unit-round off. In contrast, Krylov methods accelerated by rational filters with fixed poles typically fail to converge to unit round-off accuracy when an eigenvalue is close to a pole. In the context of Arnoldi with shift-and-invert enhancement, we demonstrate a simple restart strategy that recovers full precision in the target eigenpairs.

Krylov subspace, which is generated by multiplying a given vector by the matrix of a linear transformation and its successive powers, has been extensively studied in classical optimization literature to design algorithms that converge quickly for large linear inverse problems. For example, the conjugate gradient method (CG), one of the most popular Krylov subspace methods, is based on the idea of minimizing the residual error in the Krylov subspace. However, with the recent advancement of high-performance diffusion solvers for inverse problems, it is not clear how classical wisdom can be synergistically combined with modern diffusion models. In this study, we propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if the tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG initialized with the denoised data ensures the data consistency update to remain in the tangent space. This negates the need to compute the manifold-constrained gradient (MCG), leading to a more efficient diffusion sampling method. Our method is applicable regardless of the parametrization and setting (i.e., VE, VP). Notably, we achieve state-of-the-art reconstruction quality on challenging real-world medical inverse imaging problems, including multi-coil MRI reconstruction and 3D CT reconstruction. Moreover, our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method. Code is available at https://github.com/HJ-harry/DDS

Learning neural operators for solving partial differential equations (PDEs) has attracted great attention due to its high inference efficiency. However, training such operators requires generating a substantial amount of labeled data, i.e., PDE problems together with their solutions. The data generation process is exceptionally time-consuming, as it involves solving numerous systems of linear equations to obtain numerical solutions to the PDEs. Many existing methods solve these systems independently without considering their inherent similarities, resulting in extremely redundant computations. To tackle this problem, we propose a novel method, namely Sorting Krylov Recycling (SKR), to boost the efficiency of solving these systems, thus significantly accelerating data generation for neural operators training. To the best of our knowledge, SKR is the first attempt to address the time-consuming nature of data generation for learning neural operators. The working horse of SKR is Krylov subspace recycling, a powerful technique for solving a series of interrelated systems by leveraging their inherent similarities. Specifically, SKR employs a sorting algorithm to arrange these systems in a sequence, where adjacent systems exhibit high similarities. Then it equips a solver with Krylov subspace recycling to solve the systems sequentially instead of independently, thus effectively enhancing the solving efficiency. Both theoretical analysis and extensive experiments demonstrate that SKR can significantly accelerate neural operator data generation, achieving a remarkable speedup of up to 13.9 times.

Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground and excited-state energies of a quantum many-body system. While QKSD algorithms typically rely on using the Hadamard test for estimating Krylov subspace matrix elements of the form, langle ϕ_i|e^{-iHτ}|ϕ_j rangle, the associated quantum circuits require an ancilla qubit with controlled multi-qubit gates that can be quite costly for near-term quantum hardware. In this work, we show that a wide class of Hamiltonians relevant to condensed matter physics and quantum chemistry contain symmetries that can be exploited to avoid the use of the Hadamard test. We propose a multi-fidelity estimation protocol that can be used to compute such quantities showing that our approach, when combined with efficient single-fidelity estimation protocols, provides a substantial reduction in circuit depth. In addition, we develop a unified theory of quantum Krylov subspace algorithms and present three new quantum-classical algorithms for the ground and excited-state energy estimation problems, where each new algorithm provides various advantages and disadvantages in terms of total number of calls to the quantum computer, gate depth, classical complexity, and stability of the generalized eigenvalue problem within the Krylov subspace.

We study first-order methods with preconditioning for solving structured nonlinear convex optimization problems. We propose a new family of preconditioners generated by symmetric polynomials. They provide first-order optimization methods with a provable improvement of the condition number, cutting the gaps between highest eigenvalues, without explicit knowledge of the actual spectrum. We give a stochastic interpretation of this preconditioning in terms of coordinate volume sampling and compare it with other classical approaches, including the Chebyshev polynomials. We show how to incorporate a polynomial preconditioning into the Gradient and Fast Gradient Methods and establish the corresponding global complexity bounds. Finally, we propose a simple adaptive search procedure that automatically chooses the best possible polynomial preconditioning for the Gradient Method, minimizing the objective along a low-dimensional Krylov subspace. Numerical experiments confirm the efficiency of our preconditioning strategies for solving various machine learning problems.

We made a comparative analysis of numerical methods for multidimensional optimization. The main parameter is a number of computations of the test function to reach necessary accuracy, as it is computationally "slow". For complex functions, analytic differentiation by many parameters can cause problems associated with a significant complication of the program and thus slowing its operation. For comparison, we used the methods: "brute force" (or minimization on a regular grid), Monte Carlo, steepest descent, conjugate gradients, Brent's method (golden section search), parabolic interpolation etc. The Monte-Carlo method was applied to the eclipsing binary system AM Leo.

Within the realm of early fault-tolerant quantum computing (EFTQC), quantum Krylov subspace diagonalization (QKSD) has emerged as a promising quantum algorithm for the approximate Hamiltonian diagonalization via projection onto the quantum Krylov subspace. However, the algorithm often requires solving an ill-conditioned generalized eigenvalue problem (GEVP) involving erroneous matrix pairs, which can significantly distort the solution. Since EFTQC assumes limited-scale error correction, finite sampling error becomes a dominant source of error in these matrices. This work focuses on quantifying sampling errors during the measurement of matrix element in the projected Hamiltonian examining two measurement approaches based on the Hamiltonian decompositions: the linear combination of unitaries and diagonalizable fragments. To reduce sampling error within a fixed budget of quantum circuit repetitions, we propose two measurement strategies: the shifting technique and coefficient splitting. The shifting technique eliminates redundant Hamiltonian components that annihilate either the bra or ket states, while coefficient splitting optimizes the measurement of common terms across different circuits. Numerical experiments with electronic structures of small molecules demonstrate the effectiveness of these strategies, reducing sampling costs by a factor of 20-500.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

This work provides a nonasymptotic error analysis of quantum Krylov algorithms based on real-time evolutions, subject to generic errors in the outputs of the quantum circuits. We prove upper and lower bounds on the resulting ground state energy estimates, and the error associated to the upper bound is linear in the input error rates. This resolves a misalignment between known numerics, which exhibit approximately linear error scaling, and prior theoretical analysis, which only provably obtained scaling with the error rate to the power 2{3}. Our main technique is to express generic errors in terms of an effective target Hamiltonian studied in an effective Krylov space. These results provide a theoretical framework for understanding the main features of quantum Krylov errors.

We introduce a deep neural network-based numerical method for solving kinetic Fokker Planck equations, including both linear and nonlinear cases. Building upon the conservative dissipative structure of Vlasov-type equations, we formulate a class of generalized minimizing movement schemes as iterative constrained minimization problems: the conservative part determines the constraint set, while the dissipative part defines the objective functional. This leads to an analog of the classical Jordan-Kinderlehrer-Otto (JKO) scheme for Wasserstein gradient flows, and we refer to it as the kinetic JKO scheme. To compute each step of the kinetic JKO iteration, we introduce a particle-based approximation in which the velocity field is parameterized by deep neural networks. The resulting algorithm can be interpreted as a kinetic-oriented neural differential equation that enables the representation of high-dimensional kinetic dynamics while preserving the essential variational and structural properties of the underlying PDE. We validate the method with extensive numerical experiments and demonstrate that the proposed kinetic JKO-neural ODE framework is effective for high-dimensional numerical simulations.

Plasma modeling is central to the design of nuclear fusion reactors, yet simulating collisional plasma kinetics from first principles remains a formidable computational challenge: the Vlasov-Maxwell-Landau (VML) system describes six-dimensional phase-space transport under self-consistent electromagnetic fields together with the nonlinear, nonlocal Landau collision operator. A recent deterministic particle method for the full VML system estimates the velocity score function via the blob method, a kernel-based approximation with O(n^2) cost. In this work, we replace the blob score estimator with score-based transport modeling (SBTM), in which a neural network is trained on-the-fly via implicit score matching at O(n) cost. We prove that the approximated collision operator preserves momentum and kinetic energy, and dissipates an estimated entropy. We also characterize the unique global steady state of the VML system and its electrostatic reduction, providing the ground truth for numerical validation. On three canonical benchmarks -- Landau damping, two-stream instability, and Weibel instability -- SBTM is more accurate than the blob method, achieves correct long-time relaxation to Maxwellian equilibrium where the blob method fails, and delivers 50% faster runtime with 4times lower peak memory.

We revisit the problem of large-scale bundle adjustment and propose a technique called Multidirectional Conjugate Gradients that accelerates the solution of the normal equation by up to 61%. The key idea is that we enlarge the search space of classical preconditioned conjugate gradients to include multiple search directions. As a consequence, the resulting algorithm requires fewer iterations, leading to a significant speedup of large-scale reconstruction, in particular for denser problems where traditional approaches notoriously struggle. We provide a number of experimental ablation studies revealing the robustness to variations in the hyper-parameters and the speedup as a function of problem density.

Context. Radiation plays a significant role in solar and astrophysical environments as it may constitute a sizeable fraction of the energy density, momentum flux, and the total pressure. Modelling the dynamic interaction between radiation and magnetized plasmas in such environments is an intricate and computationally costly task. Aims. The goal of this work is to demonstrate the capabilities of the open-source parallel, block-adaptive computational framework MPI-AMRVAC, in solving equations of radiation-magnetohydrodynamics (RMHD), and to present benchmark test cases relevant for radiation-dominated magnetized plasmas. Methods. The existing magnetohydrodynamics (MHD) and flux-limited diffusion (FLD) radiative-hydrodynamics physics modules are combined to solve the equations of radiation-magnetohydrodynamics (RMHD) on block-adaptive finite volume Cartesian meshes in any dimensionality. Results. We introduce and validate several benchmark test cases such as steady radiative MHD shocks, radiation-damped linear MHD waves, radiation-modified Riemann problems and a multi-dimensional radiative magnetoconvection case. We recall the basic governing Rankine-Hugoniot relations for shocks and the dispersion relation for linear MHD waves in the presence of optically thick radiation fields where the diffusion limit is reached. The RMHD system allows for 8 linear wave types, where the classical 7-wave MHD picture (entropy and three wave pairs for slow, Alfven and fast) is augmented with a radiative diffusion mode. Conclusions. The MPI-AMRVAC code now has the capability to perform multidimensional RMHD simulations with mesh adaptation making it well-suited for larger scientific applications to study magnetized matter-radiation interactions in solar and stellar interiors and atmospheres.

The design of fusion devices is typically based on computationally expensive simulations. This can be alleviated using high aspect ratio models that employ a reduced number of free parameters, especially in the case of stellarator optimization where non-axisymmetric magnetic fields with a large parameter space are optimized to satisfy certain performance criteria. However, optimization is still required to find configurations with properties such as low elongation, high rotational transform, finite plasma beta, and good fast particle confinement. In this work, we train a machine learning model to construct configurations with favorable confinement properties by finding a solution to the inverse design problem, that is, obtaining a set of model input parameters for given desired properties. Since the solution of the inverse problem is non-unique, a probabilistic approach, based on mixture density networks, is used. It is shown that optimized configurations can be generated reliably using this method.

The Numerical Flow Iteration (NuFI) method has recently been proposed as a memory-slim solution method for the Vlasov--Poisson system. It stores the temporal evolution of the electric field, instead of the distribution functions, and reconstructs the solution in each time step by following the characteristics backwards in time and reconstructing the solution from the initial distribution. NuFI has been shown to be more accurate than other state-of-the-art Vlasov solvers given the same amount of degrees of freedom as well as interpolation order, essentially making NuFI a high-fidelity but low-memory cost scheme. In this paper, we build on the Hamiltonian structure of the full Vlasov--Maxwell system to extend NuFI to handle electro-magnetic kinetic plasma dynamics. We show that the advanced structure-preserving properties of NuFI are preserved when extending to the electro-magnetic case.

Hyperparameter tuning almost always means search: fit the model at every value on a grid, score each by cross-validation, and keep the winner. For spline regression that search is unnecessary. The optimal resolution can be solved for in closed form, to the accuracy an exhaustive search reaches, at a fraction of the compute. Three ingredients make this possible: classical approximation theory pins the squared bias to a known power of the resolution G, exactly the Kolmogorov n-width of the smoothness class; the basis dimension is an explicit polynomial in G; and leave-one-out error follows from a single fit via the PRESS identity. Balancing the two known curves gives the minimizer analytically. We extend this calculus to many coordinates by replacing ambient input dimension with interaction order, the number of active low-order components in an ANOVA decomposition, yielding a scaling law in which the optimal resolution and error are power functions of the effective density (sample size per active component), with input dimension absent from the exponent. The law becomes an algorithm. KORE (Kolmogorov-optimal Order-aware Resolution Estimation) fits two pilot resolutions, solves a leverage-calibrated 2x2 system for the bias and noise scales, and evaluates the closed-form plug-in resolution with a tiny leave-one-out certificate: about a dozen fits instead of a full grid sweep, with a consistency guarantee as the sample grows. Across additive and sparse pairwise targets up to 80 input dimensions, KORE matches exhaustive 3-fold cross-validation and the full classical ladder (GCV, Mallows' Cp, AIC, BIC) while fitting roughly 8x fewer models; on 36 real tabular datasets it ranks first among 21 methods in accuracy per unit of compute, ahead of tuned boosters and kernel machines. When complexity lives in low interaction order, solving for the resolution beats searching for it.

In this paper, we develop and implement an efficient asymptotic-preserving (AP) scheme to solve the gas mixture of Boltzmann equations under the disparate mass scaling relevant to the so-called "epochal relaxation" phenomenon. The disparity in molecular masses, ranging across several orders of magnitude, leads to significant challenges in both the evaluation of collision operators and the designing of time-stepping schemes to capture the multi-scale nature of the dynamics. A direct implementation of the spectral method faces prohibitive computational costs as the mass ratio increases due to the need to resolve vastly different thermal velocities. Unlike [I. M. Gamba, S. Jin, and L. Liu, Commun. Math. Sci., 17 (2019), pp. 1257-1289], we propose an alternative approach based on proper truncation of asymptotic expansions of the collision operators, which significantly reduces the computational complexity and works well for small varepsilon. By incorporating the separation of three time scales in the model's relaxation process [P. Degond and B. Lucquin-Desreux, Math. Models Methods Appl. Sci., 6 (1996), pp. 405-436], we design an AP scheme that captures the specific dynamics of the disparate mass model while maintaining computational efficiency. Numerical experiments demonstrate the effectiveness of the proposed scheme in handling large mass ratios of heavy and light species, as well as capturing the epochal relaxation phenomenon.

We present a novel formulation for mesh-free, reduced-order simulation of deformable hyperelastic objects. Existing work in reduced-order elastodynamic simulation represents the input geometry by either meshes, which can be difficult to obtain due to challenges in scanning and triangulating complex shapes, or by neural fields that require per-shape optimization. We propose to adopt a Reproducing Kernel Particle Method (RKPM) representation, which enables the construction of reduced-order skinning weights by solving a generalized eigensystem on the Hessian matrix of the elastic energy. We demonstrate that this formulation not only leads to a 40x training speedup compared with the per-shape optimization of neural fields, but also achieves lower simulation error when evaluated against the converged results of finite element method. We show our simulation results on a wide variety of objects in different representations including meshes and Gaussian splats, as well as the application of our method in the downstream task of robot simulation.

Quantum computers provide new avenues to access ground and excited state properties of systems otherwise difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues on quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, which applies to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

The study demonstrates the capabilities of a vector-based approach for calculating stoichiometric coefficients in chemical equations, using black powder as an illustrative example. A method is proposed for selecting and constraining intermediate interactions between reactants, as well as for identifying final products. It is shown that even a small number of components can lead to a large number of final and intermediate products. Through concrete calculations, a correlation is established between the number of possible chemical equations and the number of reactants. A methodology is proposed for computing all possible chemical equations within a reaction system for arbitrary component ratios, enabling the derivation of all feasible chemical reactions. Additionally, a method is developed for calculating the chemical composition for a fixed set of reactants, allowing for the evaluation of the set of products resulting from all possible chemical interactions given a specified initial composition.

Contemporary models of high dimensional physical systems are constrained by the curse of dimensionality and a reliance on dense data. We introduce KHRONOS (Kernel Expansion Hierarchy for Reduced Order, Neural Optimized Surrogates), an AI framework for model based, model free and model inversion tasks. KHRONOS constructs continuously differentiable target fields with a hierarchical composition of per-dimension kernel expansions, which are tensorized into modes and then superposed. We evaluate KHRONOS on a canonical 2D, Poisson equation benchmark: across 16 to 512 degrees of freedom (DoFs), it obtained L_2-square errors of 5e-4 down to 6e-11. This represents a greater than 100-fold gain over Kolmogorov Arnold Networks (which itself reports a 100 times improvement on MLPs/PINNs with 100 times fewer parameters) when controlling for the number of parameters. This also represents a 1e6-fold improvement in L_2-square error compared to standard linear FEM at comparable DoFs. Inference complexity is dominated by inner products, yielding sub-millisecond full-field predictions that scale to an arbitrary resolution. For inverse problems, KHRONOS facilitates rapid, iterative level set recovery in only a few forward evaluations, with sub-microsecond per sample latency. KHRONOS's scalability, expressivity, and interpretability open new avenues in constrained edge computing, online control, computer vision, and beyond.

We propose a physics-informed neural particle method (PINN--PM) for the spatially homogeneous Landau equation. The method adopts a Lagrangian interacting-particle formulation and jointly parameterizes the time-dependent score and the characteristic flow map with neural networks. Instead of advancing particles through explicit time stepping, the Landau dynamics is enforced via a continuous-time residual defined along particle trajectories. This design removes time-discretization error and yields a mesh-free solver that can be queried at arbitrary times without sequential integration. We establish a rigorous stability analysis in an L^2_v framework. The deviation between learned and exact characteristics is controlled by three interpretable sources: (i) score approximation error, (ii) empirical particle approximation error, and (iii) the physics residual of the neural flow. This trajectory estimate propagates to density reconstruction, where we derive an L^2_v error bound for kernel density estimators combining classical bias--variance terms with a trajectory-induced contribution. Using Hyvarinen's identity, we further relate the oracle score-matching gap to the L^2_v score error and show that the empirical loss concentrates at the Monte Carlo rate, yielding computable a posteriori accuracy certificates. Numerical experiments on analytical benchmarks, including the two- and three-dimensional BKW solutions, as well as reference-free configurations, demonstrate stable transport, preservation of macroscopic invariants, and competitive or improved accuracy compared with time-stepping score-based particle and blob methods while using significantly fewer particles.

The relationship between material properties and independent variables such as temperature, external field or time, is usually represented by a curve or surface in a multi-dimensional space. Determining such a curve or surface requires a series of experiments or calculations which are often time and cost consuming. A general strategy uses an appropriate utility function to sample the space to recommend the next optimal experiment or calculation within an active learning loop. However, knowing what the optimal sampling strategy to use to minimize the number of experiments is an outstanding problem. We compare a number of strategies based on directed exploration on several materials problems of varying complexity using a Kriging based model. These include one dimensional curves such as the fatigue life curve for 304L stainless steel and the Liquidus line of the Fe-C phase diagram, surfaces such as the Hartmann 3 function in 3D space and the fitted intermolecular potential for Ar-SH, and a four dimensional data set of experimental measurements for BaTiO3 based ceramics. We also consider the effects of experimental noise on the Hartmann 3 function. We find that directed exploration guided by maximum variance provides better performance overall, converging faster across several data sets. However, for certain problems, the trade-off methods incorporating exploitation can perform at least as well, if not better than maximum variance. Thus, we discuss how the choice of the utility function depends on the distribution of the data, the model performance and uncertainties, additive noise as well as the budget.

Every Cartesian three-dimensional PDE solver hides a structural secret that production CFD codes have used for half a century and that graduate-level textbooks rarely state plainly. The derivative matrices, the compact Padé line solves, the Galerkin mass inversions, the alternating-direction-implicit substeps, and even the fast Poisson and Helmholtz diagonalization transforms all factor along the coordinate axes and collapse into repeated one-dimensional banded kernels executed along the grid lines. The three-dimensional operator exists only on paper; it is never assembled, factored, or stored. This paper is the manual for that collapse. We derive the Kronecker-product algebra that makes it exact, carry it cleanly through central differences, compact schemes, tensor-product Galerkin, B-spline and isogeometric methods, collocation, ADI time stepping, and direct Poisson and Helmholtz solves, and bring into the open the three production tricks that turn the reduction into hardware-conscious floating-point throughput on real machines: the multi-right-hand-side reshape that exposes a sweep as one batched line kernel (a dense BLAS-3 GEMM when the line factor is dense or element-local, a banded or stencil kernel when it is not), the sum factorization that rescues high-order Galerkin from the O(p^{2d}) quadrature trap, and the pencil decomposition that keeps every direction contiguous across an MPI cluster. For fixed stencil width or fixed polynomial degree, the compute cost stays O(N) in the total number of unknowns N = N_x N_y N_z; the operator storage drops to O(N_x + N_y + N_z) up to bandwidth constants; direct separable Poisson and Helmholtz solvers add the expected transform cost; the line kernels are embarrassingly parallel. These facts are familiar to practitioners but rarely assembled in one place; this paper collects them and shows how to use them.

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as ``the simple method'') optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi--Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

Common moment-based radiative transfer methods, such as flux-limited diffusion (FLD) and the M1 closure, suffer from artificial interactions between crossing beams. In protoplanetary disks, this leads to an overestimation of the midplane temperature due to the merging of inward and outward vertical fluxes. Methods that avoid these artifacts typically require angular discretization, which can be computationally expensive. In the spirit of the two-stream approximation, we introduced a half-moment (HM) closure that integrates the radiative intensity over hemispheres, thereby suppressing beam interactions along a fixed spatial direction. We derived a multidimensional HM closure via entropy maximization and replaced this closure with an approximate expression that closely matches it, coinciding with it in the diffusion and free-streaming regimes while remaining expressible through simple operations. We implemented HM and M1 closures via implicit-explicit schemes, including multiple frequency groups. We tested these methods in numerical benchmarks such as computing the temperature in an irradiated disk around a T Tauri star, comparing our results with Monte Carlo (MC) radiative transfer simulations. The HM closure correctly reproduces the diffusion limit and prevents crossing flux interactions in a chosen spatial direction. In disk simulations, our multigroup HM method closely matches midplane temperature distributions obtained with classical MC methods. While the M1 closure produces midplane temperatures 44% higher than MC with one frequency group and 21% higher with 22 groups, HM reduces this discrepancy to 6% with 22 groups. Even with just three groups, HM significantly outperforms M1, with maximum departures of 8% compared to M1's 23%. Our results show that combining HM with a multigroup treatment yields more realistic disk temperatures than M1, particularly in optically thick regions.

We discuss methods for modeling eclipsing binary stars using the "physical", "simplified" and "phenomenological" models. There are few realizations of the "physical" Wilson-Devinney (1971) code and its improvements, e.g. Binary Maker, Phoebe. A parameter search using the Monte-Carlo method was realized by Zola et al. (2010), which is efficient in expense of too many evaluations of the test function. We compare existing algorithms of minimization of multi-parametric functions and propose to use a "combined" algorithm, depending on if the Hessian matrix is positively determined. To study methods, a simply fast-computed function resembling the "complete" test function for the physical model. Also we adopt a simplified model of an eclipsing binary at a circular orbit assuming spherical components with an uniform brightness distribution. This model resembles more advanced models in a sense of correlated parameter estimates due to a similar topology of the test function. Such a model may be applied to detached Algol-type systems, where the tidal distortion of components is negligible.

This study uses plasma-profile data from the EUROfusion pedestal database, focusing on the electron-temperature and electron-density profiles in the edge region of H-mode ELMy JET ITER-Like-Wall (ILW) pulses. We make systematic predictions of the electron-temperature pedestal, using the density profiles and engineering parameters of the pulses as inputs. We first present a machine-learning algorithm that, given more inputs than theory-based modelling and 80\% of the database as training data, can reconstruct the remaining 20\% of temperature profiles within 20\% of the experimental values, including accurate estimates of the pedestal width and location. The most important engineering parameters for these predictions are magnetic field strength, particle fuelling rate, plasma current, and strike-point configuration. This confirms the potential of accurate pedestal prediction using large databases. Next, we take a simple theoretical approach assuming a local power-law relationship between the gradients of density (R/L_{n_e}) and temperature (R/L_{T_e}): R/L_{T_e}=Aleft(R/L_{n_e}right)^α with αapprox 0.4 fits well in the steep-gradient region. When A and α are fit independently for each pedestal, a one-to-one correlation emerges, also valid for JET-C data. For α= 1, A equiv η_e, a known control parameter for turbulence in slab-ETG theory. Measured values of η_e in the steep-gradient region lie well above the slab-ETG stability threshold, suggesting a nonlinear threshold shift or a supercritical turbulent state. Finally, we test heat-flux scalings motivated by gyrokinetic simulations, and we provide best-fit parameters for reconstructing JET-ILW pedestals. These models require additional experimental inputs to reach the accuracy of the machine-learning reconstructions.

Many problems in astrophysics cover multiple orders of magnitude in spatial and temporal scales. While simulating systems that experience rapid changes in these conditions, it is essential to adapt the (time-) step size to capture the behavior of the system during those rapid changes and use a less accurate time step at other, less demanding, moments. We encounter three problems with traditional methods. Firstly, making such changes requires expert knowledge of the astrophysics as well as of the details of the numerical implementation. Secondly, some parameters that determine the time-step size are fixed throughout the simulation, which means that they do not adapt to the rapidly changing conditions of the problem. Lastly, we would like the choice of time-step size to balance accuracy and computation effort. We address these challenges with Reinforcement Learning by training it to select the time-step size dynamically. We use the integration of a system of three equal-mass bodies that move due to their mutual gravity as an example of its application. With our method, the selected integration parameter adapts to the specific requirements of the problem, both in terms of computation time and accuracy while eliminating the expert knowledge needed to set up these simulations. Our method produces results competitive to existing methods and improve the results found with the most commonly-used values of time-step parameter. This method can be applied to other integrators without further retraining. We show that this extrapolation works for variable time-step integrators but does not perform to the desired accuracy for fixed time-step integrators.

When solving inverse problems, it is increasingly popular to use pre-trained diffusion models as plug-and-play priors. This framework can accommodate different forward models without re-training while preserving the generative capability of diffusion models. Despite their success in many imaging inverse problems, most existing methods rely on privileged information such as derivative, pseudo-inverse, or full knowledge about the forward model. This reliance poses a substantial limitation that restricts their use in a wide range of problems where such information is unavailable, such as in many scientific applications. To address this issue, we propose Ensemble Kalman Diffusion Guidance (EnKG) for diffusion models, a derivative-free approach that can solve inverse problems by only accessing forward model evaluations and a pre-trained diffusion model prior. We study the empirical effectiveness of our method across various inverse problems, including scientific settings such as inferring fluid flows and astronomical objects, which are highly non-linear inverse problems that often only permit black-box access to the forward model.

Hall-effect thrusters (HETs) are widely used for modern near-earth spacecraft propulsion and are vital for future deep-space missions. Methods of modeling HETs are developing rapidly. However, such methods are not yet precise enough and cannot reliably predict the parameters of a newly designed thruster, mostly due to the enormous computational cost of a HET plasma simulation. Another approach is to use scaling techniques based on available experimental data. This paper proposes an approach for scaling HETs using neural networks and other modern machine learning methods. The new scaling model was built with information from an extensive database of HET parameters collected from published papers. Predictions of the new scaling model are valid for the operating parameters domain covered by the database. During the design, this model can help HET developers estimate the performance of a newly-designed thruster. At the stage of experimental research, the model can be used to compare the achieved characteristics of the studied thruster with the level obtained by other developers. A comparison with the state-of-the-art HET scaling model is also presented.

We here propose a machine learning approach for monitoring particle detectors in real-time. The goal is to assess the compatibility of incoming experimental data with a reference dataset, characterising the data behaviour under normal circumstances, via a likelihood-ratio hypothesis test. The model is based on a modern implementation of kernel methods, nonparametric algorithms that can learn any continuous function given enough data. The resulting approach is efficient and agnostic to the type of anomaly that may be present in the data. Our study demonstrates the effectiveness of this strategy on multivariate data from drift tube chamber muon detectors.

Fermi operator expansion (FOE) methods are powerful alternatives to diagonalization type methods for solving Kohn-Sham density functional theory (KSDFT). One example is the pole expansion and selected inversion (PEXSI) method, which approximates the Fermi operator by rational matrix functions and reduces the computational complexity to at most quadratic scaling for solving KSDFT. Unlike diagonalization type methods, the chemical potential often cannot be directly read off from the result of a single step of evaluation of the Fermi operator. Hence multiple evaluations are needed to be sequentially performed to compute the chemical potential to ensure the correct number of electrons within a given tolerance. This hinders the performance of FOE methods in practice. In this paper we develop an efficient and robust strategy to determine the chemical potential in the context of the PEXSI method. The main idea of the new method is not to find the exact chemical potential at each self-consistent-field (SCF) iteration iteration, but to dynamically and rigorously update the upper and lower bounds for the true chemical potential, so that the chemical potential reaches its convergence along the SCF iteration. Instead of evaluating the Fermi operator for multiple times sequentially, our method uses a two-level strategy that evaluates the Fermi operators in parallel. In the regime of full parallelization, the wall clock time of each SCF iteration is always close to the time for one single evaluation of the Fermi operator, even when the initial guess is far away from the converged solution. We demonstrate the effectiveness of the new method using examples with metallic and insulating characters, as well as results from ab initio molecular dynamics.

Diffusion models may be formulated as a time-indexed sequence of energy-based models, where the score corresponds to the negative gradient of an energy function. As opposed to learning the score directly, an energy parameterization is attractive as the energy itself can be used to control generation via Monte Carlo samplers. Architectural constraints and training instability in energy parameterized models have so far yielded inferior performance compared to directly approximating the score or denoiser. We address these deficiencies by introducing a novel training regime for the energy function through distillation of pre-trained diffusion models, resembling a Helmholtz decomposition of the score vector field. We further showcase the synergies between energy and score by casting the diffusion sampling procedure as a Feynman Kac model where sampling is controlled using potentials from the learnt energy functions. The Feynman Kac model formalism enables composition and low temperature sampling through sequential Monte Carlo.

We consider Proca field perturbations in a five-dimensional Schwarzschild-anti-de Sitter (Schwarzschild-AdS_{5}) black hole geometry. Using the vector spherical harmonic (VSH) method, we show that the Proca field decomposes into scalar-type and vector-type components according to their tensorial behavior on the three-sphere. Two degrees of freedom of the field are described by scalar-type components, which are coupled due to the mass term, while the remaining two degrees of freedom are described by a vector-type component, which decouples completely. Motivated by the Frolov-Krtouš-Kubizňák-Santos (FKKS) ansatz in the limit of zero spin, we use a field transformation to decouple the scalar-type components at the expense of introducing a complex separation parameter β. This parameter can be determined analytically, and its values correspond to two distinct polarizations of the scalar-type sector: "electromagnetic" and "non-electromagnetic", denoted by β_{+} and β_{-}, respectively. In the scalar-type sector, the radial differential equation for each polarization is a Fuchsian differential equation with five singularities, whereas in the vector-type sector, the radial equation has four singularities. By means of the isomonodromy method, we reformulate the boundary value problem in terms of the initial conditions of the Painlevé VI τ function and, using a series expansion of the τ function, we compute the scalar-type and vector-type quasinormal modes (QNMs) in the small horizon limit. Our results are in overall very good agreement with those obtained via the numerical integration method. This shows that the isomonodromy method is a reliable method to compute quasinormal modes in the small horizon limit with high accuracy.

Geomagnetic map interpolation aims to infer unobserved geomagnetic data at spatial points, yielding critical applications in navigation and resource exploration. However, existing methods for scattered data interpolation are not specifically designed for geomagnetic maps, which inevitably leads to suboptimal performance due to detection noise and the laws of physics. Therefore, we propose a Physics-informed Diffusion Generation framework~(PDG) to interpolate incomplete geomagnetic maps. First, we design a physics-informed mask strategy to guide the diffusion generation process based on a local receptive field, effectively eliminating noise interference. Second, we impose a physics-informed constraint on the diffusion generation results following the kriging principle of geomagnetic maps, ensuring strict adherence to the laws of physics. Extensive experiments and in-depth analyses on four real-world datasets demonstrate the superiority and effectiveness of each component of PDG.

Singly-TASE operators for the numerical solution of stiff differential equations were proposed by Calvo et al. in J.Sci. Comput. 2023 to reduce the computational cost of Runge-Kutta-TASE (RKTASE) methods when the involved linear systems are solved by some LU factorization. In this paper we propose a modification of these methods to improve the efficiency by considering different TASE operators for each stage of the Runge-Kutta. We prove that the resulting RKTASE methods are equivalent to W-methods (Steihaug and Wolfbrandt, Mathematics of Computation,1979) and this allows us to obtain the order conditions of the proposed Modified Singly-RKTASE methods (MSRKTASE) through the theory developed for the W-methods. We construct new MSRKTASE methods of order two and three and demonstrate their effectiveness through numerical experiments on both linear and nonlinear stiff systems. The results show that the MSRKTASE schemes significantly enhance efficiency and accuracy compared to previous Singly-RKTASE schemes.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN). We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP in terms of accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

Machine learning based solvers have garnered much attention in physical simulation and scientific computing, with a prominent example, physics-informed neural networks (PINNs). However, PINNs often struggle to solve high-frequency and multi-scale PDEs, which can be due to spectral bias during neural network training. To address this problem, we resort to the Gaussian process (GP) framework. To flexibly capture the dominant frequencies, we model the power spectrum of the PDE solution with a student t mixture or Gaussian mixture. We apply the inverse Fourier transform to obtain the covariance function (by Wiener-Khinchin theorem). The covariance derived from the Gaussian mixture spectrum corresponds to the known spectral mixture kernel. Next, we estimate the mixture weights in the log domain, which we show is equivalent to placing a Jeffreys prior. It automatically induces sparsity, prunes excessive frequencies, and adjusts the remaining toward the ground truth. Third, to enable efficient and scalable computation on massive collocation points, which are critical to capture high frequencies, we place the collocation points on a grid, and multiply our covariance function at each input dimension. We use the GP conditional mean to predict the solution and its derivatives so as to fit the boundary condition and the equation itself. As a result, we can derive a Kronecker product structure in the covariance matrix. We use Kronecker product properties and multilinear algebra to promote computational efficiency and scalability, without low-rank approximations. We show the advantage of our method in systematic experiments. The code is released at https://github.com/xuangu-fang/Gaussian-Process-Slover-for-High-Freq-PDE.

Maxwell's equations, a system of linear partial differential equations (PDEs), describe the behavior of electric and magnetic fields in time and space and are essential for many important electromagnetic applications. Although numerical methods have been applied successfully in the past, the primary challenge in solving these equations arises from the frequency of electromagnetic fields, which depends on the shape and size of the objects to be resolved. Since the domain of influence for these equations is compactly supported, even a small perturbation in frequency necessitates a new discretization of Maxwell's equations, resulting in substantial computational costs. In this work, we investigate the potential of neural operators, particularly the Deep Operator Network (DeepONet) and its variants, as a surrogate model for Maxwell's equations. Existing DeepONet implementations are restricted to real-valued data in R^n, but since the time-harmonic Maxwell's equations yield solutions in the complex domain C^n, a specialized architecture is required to handle complex algebra. We propose a formulation of DeepONet for complex data, define the forward pass in the complex domain, and adopt a reparametrized version of DeepONet for more efficient training. We also propose a unified framework to combine a plurality of DeepONets, trained for multiple electromagnetic field components, to incorporate the boundary condition. We conduct computational experiments on a 3D metallic sphere without singularities and on a metallic almond-shaped target to demonstrate the effectiveness of the proposed method for problems involving singularity-prone solutions. As shown by computational experiments, our method significantly enhances the efficiency of predicting scattered fields from a spherical object at arbitrary high frequencies.

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

We present the wakefield conformal mapping technique that can be readily applied to the analysis of the radiation generated by an ultra-relativistic particle in the step transition and a collimator. We derive simple analytical expressions for the lower and upper bounds of both longitudinal and transverse wake potentials. We test the derived expressions against well-known formulas in several representative examples. The proposed method can greatly simplify the optimization of collimating sections, as well as become a useful tool in the shape optimization problems.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

The extended kernel ridge regression (EKRR) method with odd-even effects was adopted to improve the description of the nuclear charge radius using five commonly used nuclear models. These are: (i) the isospin dependent A^{1/3} formula, (ii) relativistic continuum Hartree-Bogoliubov (RCHB) theory, (iii) Hartree-Fock-Bogoliubov (HFB) model HFB25, (iv) the Weizs\"acker-Skyrme (WS) model WS^ast, and (v) HFB25^ast model. In the last two models, the charge radii were calculated using a five-parameter formula with the nuclear shell corrections and deformations obtained from the WS and HFB25 models, respectively. For each model, the resultant root-mean-square deviation for the 1014 nuclei with proton number Z geq 8 can be significantly reduced to 0.009-0.013~fm after considering the modification with the EKRR method. The best among them was the RCHB model, with a root-mean-square deviation of 0.0092~fm. The extrapolation abilities of the KRR and EKRR methods for the neutron-rich region were examined and it was found that after considering the odd-even effects, the extrapolation power was improved compared with that of the original KRR method. The strong odd-even staggering of nuclear charge radii of Ca and Cu isotopes and the abrupt kinks across the neutron N=126 and 82 shell closures were also calculated and could be reproduced quite well by calculations using the EKRR method.

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

We introduced a novel method to solve Bayesian inverse problems governed by PDE equations with a hybrid two-level MCMC where we took advantage of the AI surrogate model speed and the accuracy of numerical models. We show theoretically the potential to solve Bayesian inverse problems accurately with only a small number of numerical samples when the AI surrogate model error is small. Several numerical experiment results are included which demonstrates the advantage of the hybrid method.

Orbit recovery problems are a class of problems that often arise in practice and various forms. In these problems, we aim to estimate an unknown function after being distorted by a group action and observed via a known operator. Typically, the observations are contaminated with a non-trivial level of noise. Two particular orbit recovery problems of interest in this paper are multireference alignment and single-particle cryo-EM modelling. In order to suppress the noise, we suggest using the method of moments approach for both problems while introducing deep neural network priors. In particular, our neural networks should output the signals and the distribution of group elements, with moments being the input. In the multireference alignment case, we demonstrate the advantage of using the NN to accelerate the convergence for the reconstruction of signals from the moments. Finally, we use our method to reconstruct simulated and biological volumes in the cryo-EM setting.

Nesterov's accelerated gradient (AG) is a popular technique to optimize objective functions comprising two components: a convex loss and a penalty function. While AG methods perform well for convex penalties, such as the LASSO, convergence issues may arise when it is applied to nonconvex penalties, such as SCAD. A recent proposal generalizes Nesterov's AG method to the nonconvex setting. The proposed algorithm requires specification of several hyperparameters for its practical application. Aside from some general conditions, there is no explicit rule for selecting the hyperparameters, and how different selection can affect convergence of the algorithm. In this article, we propose a hyperparameter setting based on the complexity upper bound to accelerate convergence, and consider the application of this nonconvex AG algorithm to high-dimensional linear and logistic sparse learning problems. We further establish the rate of convergence and present a simple and useful bound to characterize our proposed optimal damping sequence. Simulation studies show that convergence can be made, on average, considerably faster than that of the conventional proximal gradient algorithm. Our experiments also show that the proposed method generally outperforms the current state-of-the-art methods in terms of signal recovery.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the L^2 difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

This paper presents a comprehensive study on the unified module for accelerating stable-diffusion processes, specifically focusing on the lcm-lora module. Stable-diffusion processes play a crucial role in various scientific and engineering domains, and their acceleration is of paramount importance for efficient computational performance. The standard iterative procedures for solving fixed-source discrete ordinates problems often exhibit slow convergence, particularly in optically thick scenarios. To address this challenge, unconditionally stable diffusion-acceleration methods have been developed, aiming to enhance the computational efficiency of transport equations and discrete ordinates problems. This study delves into the theoretical foundations and numerical results of unconditionally stable diffusion synthetic acceleration methods, providing insights into their stability and performance for model discrete ordinates problems. Furthermore, the paper explores recent advancements in diffusion model acceleration, including on device acceleration of large diffusion models via gpu aware optimizations, highlighting the potential for significantly improved inference latency. The results and analyses in this study provide important insights into stable diffusion processes and have important ramifications for the creation and application of acceleration methods specifically, the lcm-lora module in a variety of computing environments.

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given precision level. One challenge in PCG solvers is the selection of preconditioners, as different problem-dependent systems can benefit from different preconditioners. We present a new method to introduce inductive bias in preconditioning conjugate gradient algorithm. Given a system matrix and a set of solution vectors arise from an underlying distribution, we train a graph neural network to obtain an approximate decomposition to the system matrix to be used as a preconditioner in the context of PCG solvers. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach in solving various 2D and 3D linear second-order PDEs.

A faster than real time forecast system for solar wind and interplanetary magnetic field transients that is driven by hourly updated solar magnetograms is proposed to provide a continuous nowcast of the solar corona (<0.1AU) and 24-hours forecast of the solar wind at 1 AU by solving a full 3-D MHD model. This new model has been inspired by the concept of relativity of simultaneity used in the theory of special relativity. It is based on time transformation between two coordinate systems: the solar rest frame and a boosted system in which the current observations of the solar magnetic field and tomorrow's measurement of the solar wind at 1 AU are simultaneous. In this paper we derive the modified governing equations for both hydrodynamics (HD) and magnetohydrodynamics (MHD) and present a new numerical algorithm that only modifies the conserved quantities but preserves the original HD/MHD numerical flux. The proposed method enables an efficient numerical implementation, and thus a significantly longer forecast time than the traditional method.

Ever since the original algorithm by Nesterov (1983), the true nature of the acceleration phenomenon has remained elusive, with various interpretations of why the method is actually faster. The diagnosis of the algorithm through the lens of Ordinary Differential Equations (ODEs) and the corresponding dynamical system formulation to explain the underlying dynamics has a rich history. In the literature, the ODEs that explain algorithms are typically derived by considering the limiting case of the algorithm maps themselves, that is, an ODE formulation follows the development of an algorithm. This obfuscates the underlying higher order principles and thus provides little evidence of the working of the algorithm. Such has been the case with Nesterov algorithm and the various analogies used to describe the acceleration phenomena, viz, momentum associated with the rolling of a Heavy-Ball down a slope, Hessian damping etc. The main focus of our work is to ideate the genesis of the Nesterov algorithm from the viewpoint of dynamical systems leading to demystifying the mathematical rigour behind the algorithm. Instead of reverse engineering ODEs from discrete algorithms, this work explores tools from the recently developed control paradigm titled Passivity and Immersion approach and the Geometric Singular Perturbation theory which are applied to arrive at the formulation of a dynamical system that explains and models the acceleration phenomena. This perspective helps to gain insights into the various terms present and the sequence of steps used in Nesterovs accelerated algorithm for the smooth strongly convex and the convex case. The framework can also be extended to derive the acceleration achieved using the triple momentum method and provides justifications for the non-convergence to the optimal solution in the Heavy-Ball method.

We consider in this work quantities that can be obtained as limits of powers of parametrized matrices, for instance the inverse matrix or the logarithm of the determinant. Under the assumption of affine dependence in the parameters, we use the Empirical Interpolation Method (EIM) to derive an approximation for powers of these matrices, from which we derive a nonintrusive approximation for the aforementioned limits. We derive upper bounds of the error made by the obtained formula. Finally, numerical comparisons with classical intrusive and nonintrusive approximation techniques are provided: in the considered test-cases, our algorithm performs well compared to the nonintrusive ones.

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

The inspiral, merger, and ringdown of Massive Black Hole Binaries (MBHBs) is one the main sources of Gravitational Waves (GWs) for the future Laser Interferometer Space Antenna (LISA), an ESA-led mission in the implementation phase. It is expected that LISA will detect these systems throughout the entire observable universe. Robust and efficient data analysis algorithms are necessary to detect and estimate physical parameters for these systems. In this work, we explore the application of Sequential Neural Likelihood, a simulation-based inference algorithm, to detect and characterize MBHB GW signals in synthetic LISA data. We describe in detail the different elements of the method, their performance and possible alternatives that can be used to enhance the performance. Instead of sampling from the conventional likelihood function, which requires a forward simulation for each evaluation, this method constructs a surrogate likelihood that is ultimately described by a neural network trained from a dataset of simulations of the MBHB signals and noise. One important advantage of this method is that, given that the likelihood is independent of the priors, we can iteratively train models that target specific observations in a fraction of the time and computational cost that other traditional and machine learning-based strategies would require. Because of the iterative nature of the method, we are able to train models to obtain qualitatively similar posteriors with less than 2\% of the simulator calls that Markov Chain Monte Carlo methods would require. We compare these posteriors with those obtained from Markov Chain Monte Carlo techniques and discuss the differences that appear, in particular in relation with the important role that data compression has in the modular implementation of the method that we present. We also discuss different strategies to improve the performance of the algorithms.

Two benchmark problems for linear radiation transport and derived from the literature are presented in detail and several quantities of interest are defined. High-resolution simulations are computed using standard, robust numerical methods and implemented using HPC resources. The goal of these simulations is to provide reference solutions for new discretization approaches, to aid in the development of multi-fidelity uncertainty quantification and optimization algorithms, to provide data for training surrogates models. The code used to perform the simulations and post-process the data is publicly available, as is the raw data that used to generate the results presented in the paper.

Accurate aerodynamic modeling of satellites in very low Earth orbit (VLEO) requires gas-surface interaction (GSI) models that capture the full velocity spectrum from thermal to orbital speeds. Atmospheric particles initially strike spacecraft surfaces at hypersonic velocities of 6 000 - 10 000 m/s. Due to surface roughness and complex geometries, especially within air-breathing electric propulsion (ABEP) intake systems, multiple collisions occur, progressively reducing the particle velocities. A recent machine learning framework for deriving scattering kernels from molecular dynamics (MD) simulations has shown promise, but remains limited to high-velocity single impacts and possibly violates fundamental equilibrium principles such as detailed balance. This work extends this machine learning based scattering kernel to cover the complete velocity range using conditional normalizing flows trained with physics-informed constraints, enabling accurate modeling of multi-bounce scenarios in realistic VLEO applications. We train a conditional Real-valued Non-Volume Preserving (cRealNVP) model on expanded molecular dynamics simulations covering velocities from thermal to hypersonic speeds, incorporating a detailed balance loss term. The resulting model demonstrates improved accuracy compared to previous approaches even in the original high-velocity regime, while successfully capturing thermal-velocity scattering. Quantitative assessment shows that thermalization is approximated within acceptable tolerances. This framework provides essential capabilities for accurate ABEP intake optimization and VLEO mission planning while offering a general methodology applicable to broader rarefied gas dynamics problems requiring thermodynamic consistency.

The field of plasma physics heavily relies on simulations to model various phenomena, such as instabilities, turbulence, and nonlinear behaviors that would otherwise be difficult to study from a purely theoretical approach. Simulations are fundamental in accurately setting up experiments, which can be extremely costly and complex. As high-fidelity tools, gyrokinetic simulations play a crucial role in discovering new physics, interpreting experimental results, and improving the design of next-generation devices. However, their high computational costs necessitate the use of acceleration platforms to reduce execution time. This work revolves around the TRIangular MEsh based Gyrokinetic (TRIMEG) code, which performs high-accuracy particle-in-cell plasma simulations in tokamak geometries, leveraging a novel finite element approach. The rise of graphical processing units (GPUs) constitutes an occasion to satisfy such computational needs, by offloading the most expensive portion of the code to the accelerators. The chosen approach features GPU offloading with the OpenMP API, which grants portability of the code to different architectures, namely AMD and NVIDIA. The particle pushing as well as the grid-to-particle operations have been ported to GPU platforms. Compiler limitations had to be overcome, and portions of the code were restructured to be suitable for GPU acceleration. Kernel performance was evaluated by carrying out GPU grid size exploration, as well as scalability studies. In addition, the efficiency of hybrid MPI-OpenMP offloading parallelization was assessed. The speedup of the GPU implementation was calculated by comparing it with the pure CPU version using different rationales. The Ion Temperature Gradient (ITG) mode was simulated using the GPU-accelerated version, and its correctness was verified in terms of the energy growth rate and the two-dimensional mode structures.

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

Physics-Informed Neural Networks (PINNs) provide a mesh-free approach for solving differential equations by embedding physical constraints into neural network training. However, PINNs tend to overfit within the training domain, leading to poor generalization when extrapolating beyond trained spatiotemporal regions. This work presents SPIKE (Sparse Physics-Informed Koopman-Enhanced), a framework that regularizes PINNs with continuous-time Koopman operators to learn parsimonious dynamics representations. By enforcing linear dynamics dz/dt = Az in a learned observable space, both PIKE (without explicit sparsity) and SPIKE (with L1 regularization on A) learn sparse generator matrices, embodying the parsimony principle that complex dynamics admit low-dimensional structure. Experiments across parabolic, hyperbolic, dispersive, and stiff PDEs, including fluid dynamics (Navier-Stokes) and chaotic ODEs (Lorenz), demonstrate consistent improvements in temporal extrapolation, spatial generalization, and long-term prediction accuracy. The continuous-time formulation with matrix exponential integration provides unconditional stability for stiff systems while avoiding diagonal dominance issues inherent in discrete-time Koopman operators.

We introduce a new family of particle evolution samplers suitable for constrained domains and non-Euclidean geometries. Stein Variational Mirror Descent and Mirrored Stein Variational Gradient Descent minimize the Kullback-Leibler (KL) divergence to constrained target distributions by evolving particles in a dual space defined by a mirror map. Stein Variational Natural Gradient exploits non-Euclidean geometry to more efficiently minimize the KL divergence to unconstrained targets. We derive these samplers from a new class of mirrored Stein operators and adaptive kernels developed in this work. We demonstrate that these new samplers yield accurate approximations to distributions on the simplex, deliver valid confidence intervals in post-selection inference, and converge more rapidly than prior methods in large-scale unconstrained posterior inference. Finally, we establish the convergence of our new procedures under verifiable conditions on the target distribution.

A Milstein-type method is proposed for some highly non-linear non-autonomous time-changed stochastic differential equations (SDEs). The spatial variables in the coefficients of the time-changed SDEs satisfy the super-linear growth condition and the temporal variables obey some H\"older's continuity condition. The strong convergence in the finite time is studied and the convergence order is obtained.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schr\"odinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the kappa=pm1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10^{-8} Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schr\"odinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines.

Nonlinear model order reduction has opened the door to parameter optimization and uncertainty quantification in complex physics problems governed by nonlinear equations. In particular, the computational cost of solving these equations can be reduced by means of local reduced-order bases. This article examines the benefits of a physics-informed cluster analysis for the construction of cluster-specific reduced-order bases. We illustrate that the choice of the dissimilarity measure for clustering is fundamental and highly affects the performances of the local reduced-order bases. It is shown that clustering with an angle-based dissimilarity on simulation data efficiently decreases the intra-cluster Kolmogorov N-width. Additionally, an a priori efficiency criterion is introduced to assess the relevance of a ROM-net, a methodology for the reduction of nonlinear physics problems introduced in our previous work in [T. Daniel, F. Casenave, N. Akkari, D. Ryckelynck, Model order reduction assisted by deep neural networks (ROM-net), Advanced Modeling and Simulation in Engineering Sciences 7 (16), 2020]. This criterion also provides engineers with a very practical method for ROM-nets' hyperparameters calibration under constrained computational costs for the training phase. On five different physics problems, our physics-informed clustering strategy significantly outperforms classic strategies for the construction of local reduced-order bases in terms of projection errors.

This thesis addresses a fundamental problem in deformation quantization: the difficulty of calculating the star-exponential, the symbol of the evolution operator, due to convergence issues. Inspired by the formalism that connects the star-exponential with the quantum propagator for bosonic systems, this work develops the analogous extension for the fermionic case. A rigorous method, based on Grassmann variables and coherent states, is constructed to obtain a closed-form expression for the fermionic star-exponential from its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, allowing for the calculation of the ground state energy directly in phase space. Finally, the method is validated by successfully applying it to the simple and driven harmonic oscillators, where it is demonstrated that a simplified ("naive") approach (with an ad-hoc "remediation") is a valid weak-coupling limit of the rigorous ("meticulous") formalism, thereby providing a new and powerful computational tool for the study of fermionic systems.

The Reduced Basis Method can be exploited in an efficient way only if the so-called affine dependence assumption on the operator and right-hand side of the considered problem with respect to the parameters is satisfied. When it is not, the Empirical Interpolation Method is usually used to recover this assumption approximately. In both cases, the Reduced Basis Method requires to access and modify the assembly routines of the corresponding computational code, leading to an intrusive procedure. In this work, we derive variants of the EIM algorithm and explain how they can be used to turn the Reduced Basis Method into a nonintrusive procedure. We present examples of aeroacoustic problems solved by integral equations and show how our algorithms can benefit from the linear algebra tools available in the considered code.

We present a vector-based method to balance chemical reactions. The algorithm builds candidates in a deterministic way, removes duplicates, and always prints coefficients in the lowest whole-number form. For redox cases, electrons and protons/hydroxide are treated explicitly, so both mass and charge are balanced. We also outline the basic principles of the vector formulation of stoichiometry, interpreting reactions as integer vectors in composition space, this geometric view supports compact visualizations of reagent-product interactions and helps surface distinct reaction families. The method enumerates valid balances for arbitrary user-specified species lists without special-case balancing rules or symbolic tricks, and it provides a clean foundation for developing new algorithmic variants (e.g., alternative objectives or constraints). On representative examples (neutralization, double displacement, decomposition, classical redox, small multicomponent sets) and a negative control, the method produced correct integer balances. When multiple balances exist, we report a canonical one - minimizing the total coefficient sum with a simple tie-breaker - without claiming global optimality beyond the solutions the search enumerates. The procedure applies per reaction and extends to reaction networks via consistent per-reaction application. We do not report runtimes, broader benchmarking and code/data release are planned.

In this paper, we propose a novel neural network framework, the Legendre-Kolmogorov-Arnold Network (Legendre-KAN) method, designed to solve fully nonlinear Monge-Ampère equations with Dirichlet boundary conditions. The architecture leverages the orthogonality of Legendre polynomials as basis functions, significantly enhancing both convergence speed and solution accuracy compared to traditional methods. Furthermore, the Kolmogorov-Arnold representation theorem provides a strong theoretical foundation for the interpretability and optimization of the network. We demonstrate the effectiveness of the proposed method through numerical examples, involving both smooth and singular solutions in various dimensions. This work not only addresses the challenges of solving high-dimensional and singular Monge-Ampère equations but also highlights the potential of neural network-based approaches for complex partial differential equations. Additionally, the method is applied to the optimal transport problem in image mapping, showcasing its practical utility in geometric image transformation. This approach is expected to pave the way for further enhancement of KAN-based applications and numerical solutions of PDEs across a wide range of scientific and engineering fields.

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

Implicit layer deep learning techniques, like Neural Differential Equations, have become an important modeling framework due to their ability to adapt to new problems automatically. Training a neural differential equation is effectively a search over a space of plausible dynamical systems. However, controlling the computational cost for these models is difficult since it relies on the number of steps the adaptive solver takes. Most prior works have used higher-order methods to reduce prediction timings while greatly increasing training time or reducing both training and prediction timings by relying on specific training algorithms, which are harder to use as a drop-in replacement due to strict requirements on automatic differentiation. In this manuscript, we use internal cost heuristics of adaptive differential equation solvers at stochastic time points to guide the training toward learning a dynamical system that is easier to integrate. We "close the black-box" and allow the use of our method with any adjoint technique for gradient calculations of the differential equation solution. We perform experimental studies to compare our method to global regularization to show that we attain similar performance numbers without compromising the flexibility of implementation on ordinary differential equations (ODEs) and stochastic differential equations (SDEs). We develop two sampling strategies to trade off between performance and training time. Our method reduces the number of function evaluations to 0.556-0.733x and accelerates predictions by 1.3-2x.

Training neural PDE solvers is often bottlenecked by expensive data generation or unstable physics-informed neural network (PINN) involving challenging optimization landscapes due to higher-order derivatives. To tackle this issue, we propose an alternative approach using Monte Carlo approaches to estimate the solution to the PDE as a stochastic process for weak supervision during training. Leveraging the Walk-on-Spheres method, we introduce a learning scheme called Walk-on-Spheres Neural Operator (WoS-NO) which uses weak supervision from WoS to train any given neural operator. We propose to amortize the cost of Monte Carlo walks across the distribution of PDE instances using stochastic representations from the WoS algorithm to generate cheap, noisy, estimates of the PDE solution during training. This is formulated into a data-free physics-informed objective where a neural operator is trained to regress against these weak supervisions, allowing the operator to learn a generalized solution map for an entire family of PDEs. This strategy does not require expensive pre-computed datasets, avoids computing higher-order derivatives for loss functions that are memory-intensive and unstable, and demonstrates zero-shot generalization to novel PDE parameters and domains. Experiments show that for the same number of training steps, our method exhibits up to 8.75times improvement in L_2-error compared to standard physics-informed training schemes, up to 6.31times improvement in training speed, and reductions of up to 2.97times in GPU memory consumption. We present the code at https://github.com/neuraloperator/WoS-NO

A large-scale database of two-dimensional UEDGE simulations has been developed to study detachment physics in KSTAR and to support surrogate models for control applications. Nearly 70,000 steady-state solutions were generated, systematically scanning upstream density, input power, plasma current, impurity fraction, and anomalous transport coefficients, with magnetic and electric drifts across the magnetic field included. The database identifies robust detachment indicators, with strike-point electron temperature at detachment onset consistently Te around 3-4 eV, largely insensitive to upstream conditions. Scaling relations reveal weaker impurity sensitivity than one-dimensional models and show that heat flux widths follow Eich's scaling only for uniform, low D and Chi. Distinctive in-out divertor asymmetries are observed in KSTAR, differing qualitatively from DIII-D. Complementary time-dependent simulations quantify plasma response to gas puffing, with delays of 5-15 ms at the outer strike point and approximately 40 ms for the low-magnetic-field-side (LFS) radiation front. These dynamics are well captured by first-order-plus-dead-time (FOPDT) models and are consistent with experimentally observed detachment-control behavior in KSTAR [Gupta et al., submitted to Plasma Phys. Control. Fusion (2025)]

Solving complex Partial Differential Equations (PDEs) accurately and efficiently is an essential and challenging problem in all scientific and engineering disciplines. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without increasing the overall mesh degree of freedom count. Conventional sophisticated mesh movement methods are extremely expensive and struggle to handle scenarios with complex boundary geometries. However, existing learning-based methods require re-training from scratch given a different PDE type or boundary geometry, which limits their applicability, and also often suffer from robustness issues in the form of inverted elements. In this paper, we introduce the Universal Mesh Movement Network (UM2N), which -- once trained -- can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries. UM2N consists of a Graph Transformer (GT) encoder for extracting features and a Graph Attention Network (GAT) based decoder for moving the mesh. We evaluate our method on advection and Navier-Stokes based examples, as well as a real-world tsunami simulation case. Our method outperforms existing learning-based mesh movement methods in terms of the benchmarks described above. In comparison to the conventional sophisticated Monge-Amp\`ere PDE-solver based method, our approach not only significantly accelerates mesh movement, but also proves effective in scenarios where the conventional method fails. Our project page is at https://erizmr.github.io/UM2N/.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerically partial differential equations (PDEs) in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in 100,000 effective dimensions in 12 hours on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

In this paper, a mathematical model is developed to describe the evolution of the concentration of compounds through a gas chromatography column. The model couples mass balances and kinetic equations for all components. Both single and multiple-component cases are considered with constant or variable velocity. Non-dimensionalisation indicates the small effect of diffusion. The system where diffusion is neglected is analysed using Laplace transforms. In the multiple-component case, it is demonstrated that the competition between the compounds is negligible and the equations may be decoupled. This reduces the problem to solving a single integral equation to determine the concentration profile for all components (since they are scaled versions of each other). For a given analyte, we then only two parameters need to be fitted to the data. To verify this approach, the full governing equations are also solved numerically using the finite difference method and a global adaptive quadrature method to integrate the Laplace transformation. Comparison with the Laplace solution verifies the high degree of accuracy of the simpler Laplace form. The Laplace solution is then verified against experimental data from BTEX chromatography. This novel method, which involves solving a single equation and fitting parameters in pairs for individual components, is highly efficient. It is significantly faster and simpler than the full numerical solution and avoids the computationally expensive methods that would normally be used to fit all curves at the same time.

The kernel thinning (KT) algorithm of Dwivedi and Mackey (2021) compresses a probability distribution more effectively than independent sampling by targeting a reproducing kernel Hilbert space (RKHS) and leveraging a less smooth square-root kernel. Here we provide four improvements. First, we show that KT applied directly to the target RKHS yields tighter, dimension-free guarantees for any kernel, any distribution, and any fixed function in the RKHS. Second, we show that, for analytic kernels like Gaussian, inverse multiquadric, and sinc, target KT admits maximum mean discrepancy (MMD) guarantees comparable to or better than those of square-root KT without making explicit use of a square-root kernel. Third, we prove that KT with a fractional power kernel yields better-than-Monte-Carlo MMD guarantees for non-smooth kernels, like Laplace and Mat\'ern, that do not have square-roots. Fourth, we establish that KT applied to a sum of the target and power kernels (a procedure we call KT+) simultaneously inherits the improved MMD guarantees of power KT and the tighter individual function guarantees of target KT. In our experiments with target KT and KT+, we witness significant improvements in integration error even in 100 dimensions and when compressing challenging differential equation posteriors.

Existing model reduction techniques for high-dimensional models of conservative partial differential equations (PDEs) encounter computational bottlenecks when dealing with systems featuring non-polynomial nonlinearities. This work presents a nonlinear model reduction method that employs lifting variable transformations to derive structure-preserving quadratic reduced-order models for conservative PDEs with general nonlinearities. We present an energy-quadratization strategy that defines the auxiliary variable in terms of the nonlinear term in the energy expression to derive an equivalent quadratic lifted system with quadratic system energy. The proposed strategy combined with proper orthogonal decomposition model reduction yields quadratic reduced-order models that conserve the quadratized lifted energy exactly in high dimensions. We demonstrate the proposed model reduction approach on four nonlinear conservative PDEs: the one-dimensional wave equation with exponential nonlinearity, the two-dimensional sine-Gordon equation, the two-dimensional Klein-Gordon equation with parametric dependence, and the two-dimensional Klein-Gordon-Zakharov equations. The numerical results show that the proposed lifting approach is competitive with the state-of-the-art structure-preserving hyper-reduction method in terms of both accuracy and computational efficiency in the online stage while providing significant computational gains in the offline stage.

The interdiffusion coefficients are estimated either following the Wagner's method expressed with respect to the composition (mol or atomic fraction) normalized variable after considering the molar volume variation or the den Broeder's method expressed with respect to the concentration (composition divided by the molar volume) normalized variable. On the other hand, the relations for estimation of the intrinsic diffusion coefficients of components as established by van Loo and integrated diffusion coefficients in a phase with narrow homogeneity range as established by Wagner are currently available with respect to the composition normalized variable only. In this study, we have first derived the relation proposed by den Broeder following the line of treatment proposed by Wagner. Further, the relations for estimation of the intrinsic diffusion coefficients of the components and integrated interdiffusion coefficient are established with respect to the concentration normalized variable, which were not available earlier. The veracity of these methods is examined based on the estimation of data in Ni-Pd, Ni-Al and Cu-Sn systems. Our analysis indicates that both the approaches are logically correct and there is small difference in the estimated data in these systems although a higher difference could be found in other systems. The integrated interdiffusion coefficients with respect to the concentration (or concentration normalized variable) can only be estimated considering the ideal molar volume variation. This might be drawback in certain practical systems.

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

We introduce ensembles within score-based sampling methods to develop gradient-free approximate sampling techniques that leverage the collective dynamics of particle ensembles to compute approximate reverse diffusion drifts. We introduce the underlying methodology, emphasizing its relationship with generative diffusion models and the previously introduced F\"ollmer sampler. We demonstrate the efficacy of ensemble strategies through various examples, ranging from low- to medium-dimensionality sampling problems, including multi-modal and highly non-Gaussian probability distributions, and provide comparisons to traditional methods like NUTS. Our findings highlight the potential of ensemble strategies for modeling complex probability distributions in situations where gradients are unavailable. Finally, we showcase its application in the context of Bayesian inversion problems within the geophysical sciences.

We design a physics-aware auto-encoder to specifically reduce the dimensionality of solutions arising from convection-dominated nonlinear physical systems. Although existing nonlinear manifold learning methods seem to be compelling tools to reduce the dimensionality of data characterized by a large Kolmogorov n-width, they typically lack a straightforward mapping from the latent space to the high-dimensional physical space. Moreover, the realized latent variables are often hard to interpret. Therefore, many of these methods are often dismissed in the reduced order modeling of dynamical systems governed by the partial differential equations (PDEs). Accordingly, we propose an auto-encoder type nonlinear dimensionality reduction algorithm. The unsupervised learning problem trains a diffeomorphic spatio-temporal grid, that registers the output sequence of the PDEs on a non-uniform parameter/time-varying grid, such that the Kolmogorov n-width of the mapped data on the learned grid is minimized. We demonstrate the efficacy and interpretability of our approach to separate convection/advection from diffusion/scaling on various manufactured and physical systems.

Nuclear fusion plays a pivotal role in the quest for reliable and sustainable energy production. A major roadblock to viable fusion power is understanding plasma turbulence, which significantly impairs plasma confinement, and is vital for next-generation reactor design. Plasma turbulence is governed by the nonlinear gyrokinetic equation, which evolves a 5D distribution function over time. Due to its high computational cost, reduced-order models are often employed in practice to approximate turbulent transport of energy. However, they omit nonlinear effects unique to the full 5D dynamics. To tackle this, we introduce GyroSwin, the first scalable 5D neural surrogate that can model 5D nonlinear gyrokinetic simulations, thereby capturing the physical phenomena neglected by reduced models, while providing accurate estimates of turbulent heat transport.GyroSwin (i) extends hierarchical Vision Transformers to 5D, (ii) introduces cross-attention and integration modules for latent 3Dleftrightarrow5D interactions between electrostatic potential fields and the distribution function, and (iii) performs channelwise mode separation inspired by nonlinear physics. We demonstrate that GyroSwin outperforms widely used reduced numerics on heat flux prediction, captures the turbulent energy cascade, and reduces the cost of fully resolved nonlinear gyrokinetics by three orders of magnitude while remaining physically verifiable. GyroSwin shows promising scaling laws, tested up to one billion parameters, paving the way for scalable neural surrogates for gyrokinetic simulations of plasma turbulence.

One of the major limits of kernel ridge regression (KRR) is that storing and manipulating the kernel matrix K_n for n samples requires O(n^2) space, which rapidly becomes unfeasible for large n. Nystrom approximations reduce the space complexity to O(nm) by sampling m columns from K_n. Uniform sampling preserves KRR accuracy (up to epsilon) only when m is proportional to the maximum degree of freedom of K_n, which may require O(n) columns for datasets with high coherence. Sampling columns according to their ridge leverage scores (RLS) gives accurate Nystrom approximations with m proportional to the effective dimension, but computing exact RLS also requires O(n^2) space. (Calandriello et al. 2016) propose INK-Estimate, an algorithm that processes the dataset incrementally and updates RLS, effective dimension, and Nystrom approximations on-the-fly. Its space complexity scales with the effective dimension but introduces a dependency on the largest eigenvalue of K_n, which in the worst case is O(n). In this paper we introduce SQUEAK, a new algorithm that builds on INK-Estimate but uses unnormalized RLS. As a consequence, the algorithm is simpler, does not need to estimate the effective dimension for normalization, and achieves a space complexity that is only a constant factor worse than exact RLS sampling.

Diffusion model-based inverse problem solvers have demonstrated state-of-the-art performance in cases where the forward operator is known (i.e. non-blind). However, the applicability of the method to blind inverse problems has yet to be explored. In this work, we show that we can indeed solve a family of blind inverse problems by constructing another diffusion prior for the forward operator. Specifically, parallel reverse diffusion guided by gradients from the intermediate stages enables joint optimization of both the forward operator parameters as well as the image, such that both are jointly estimated at the end of the parallel reverse diffusion procedure. We show the efficacy of our method on two representative tasks -- blind deblurring, and imaging through turbulence -- and show that our method yields state-of-the-art performance, while also being flexible to be applicable to general blind inverse problems when we know the functional forms.

We propose an efficient method for approximating natural gradient descent in neural networks which we call Kronecker-Factored Approximate Curvature (K-FAC). K-FAC is based on an efficiently invertible approximation of a neural network's Fisher information matrix which is neither diagonal nor low-rank, and in some cases is completely non-sparse. It is derived by approximating various large blocks of the Fisher (corresponding to entire layers) as being the Kronecker product of two much smaller matrices. While only several times more expensive to compute than the plain stochastic gradient, the updates produced by K-FAC make much more progress optimizing the objective, which results in an algorithm that can be much faster than stochastic gradient descent with momentum in practice. And unlike some previously proposed approximate natural-gradient/Newton methods which use high-quality non-diagonal curvature matrices (such as Hessian-free optimization), K-FAC works very well in highly stochastic optimization regimes. This is because the cost of storing and inverting K-FAC's approximation to the curvature matrix does not depend on the amount of data used to estimate it, which is a feature typically associated only with diagonal or low-rank approximations to the curvature matrix.

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

The study conducted by Shumailov et al. (2024) demonstrates that repeatedly training a generative model on synthetic data leads to model collapse. This finding has generated considerable interest and debate, particularly given that current models have nearly exhausted the available data. In this work, we investigate the effects of fitting a distribution (through Kernel Density Estimation, or KDE) or a model to the data, followed by repeated sampling from it. Our objective is to develop a theoretical understanding of the phenomenon observed by Shumailov et al. (2024). Our results indicate that the outcomes reported are a statistical phenomenon and may be unavoidable.

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

We derive a Fast Multipole Method (FMM) where a low-rank approximation of the kernel is obtained using the Empirical Interpolation Method (EIM). Contrary to classical interpolation-based FMM, where the interpolation points and basis are fixed beforehand, the EIM is a nonlinear approximation method which constructs interpolation points and basis which are adapted to the kernel under consideration. The basis functions are obtained using evaluations of the kernel itself. We restrict ourselves to translation-invariant kernels, for which a modified version of the EIM approximation can be used in a multilevel FMM context; we call the obtained algorithm Empirical Interpolation Fast Multipole Method (EIFMM). An important feature of the EIFMM is a built-in error estimation of the interpolation error made by the low-rank approximation of the far-field behavior of the kernel: the algorithm selects the optimal number of interpolation points required to ensure a given accuracy for the result, leading to important gains for inhomogeneous kernels.

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

We introduce a data-driven diagnostic that combines the singular value decomposition (SVD) with an information-theoretic entropy to quantify the phase-space complexity of perturbed distribution functions in gyrokinetic turbulence. Applying this framework to nonlinear flux-tube simulations that solve the time evolution of the ion distribution function represented by Fourier modes with the wavenumber for real space, we define the von Neumann entropy (vNE) to analyze the velocity-space structure. A global survey in the wavenumber space reveals a wavenumber-dependent variation of the vNE in velocity-space structure: the vNE remains low at low wavenumber but increases across k_perpρ_{ti}sim 1. Hermite/Laguerre decompositions revealed that the finite Larmor radius (FLR) phase mixing in the perpendicular (magnetic-moment) direction is active. Simultaneously, the systematic increase in vNE for k_perpρ_{ti} correlates with the broadening of the Hermite spectrum, suggesting enhanced parallel phase mixing (Landau resonance) as the primary mechanism for the observed wave number dependence. These results demonstrate that the SVD-based vNE provides a compact measure of kinetic complexity without assuming a predefined basis and enables a global mapping of its wavenumber dependence of phase-mixing processes in gyrokinetic turbulence.

Score-based (denoising diffusion) generative models have recently gained a lot of success in generating realistic and diverse data. These approaches define a forward diffusion process for transforming data to noise and generate data by reversing it (thereby going from noise to data). Unfortunately, current score-based models generate data very slowly due to the sheer number of score network evaluations required by numerical SDE solvers. In this work, we aim to accelerate this process by devising a more efficient SDE solver. Existing approaches rely on the Euler-Maruyama (EM) solver, which uses a fixed step size. We found that naively replacing it with other SDE solvers fares poorly - they either result in low-quality samples or become slower than EM. To get around this issue, we carefully devise an SDE solver with adaptive step sizes tailored to score-based generative models piece by piece. Our solver requires only two score function evaluations, rarely rejects samples, and leads to high-quality samples. Our approach generates data 2 to 10 times faster than EM while achieving better or equal sample quality. For high-resolution images, our method leads to significantly higher quality samples than all other methods tested. Our SDE solver has the benefit of requiring no step size tuning.

Physics-informed neural networks (PINNs) are emerging as popular mesh-free solvers for partial differential equations (PDEs). Recent extensions decompose the domain, applying different PINNs to solve the equation in each subdomain and aligning the solution at the interface of the subdomains. Hence, they can further alleviate the problem complexity, reduce the computational cost, and allow parallelization. However, the performance of the multi-domain PINNs is sensitive to the choice of the interface conditions for solution alignment. While quite a few conditions have been proposed, there is no suggestion about how to select the conditions according to specific problems. To address this gap, we propose META Learning of Interface Conditions (METALIC), a simple, efficient yet powerful approach to dynamically determine the optimal interface conditions for solving a family of parametric PDEs. Specifically, we develop two contextual multi-arm bandit models. The first one applies to the entire training procedure, and online updates a Gaussian process (GP) reward surrogate that given the PDE parameters and interface conditions predicts the solution error. The second one partitions the training into two stages, one is the stochastic phase and the other deterministic phase; we update a GP surrogate for each phase to enable different condition selections at the two stages so as to further bolster the flexibility and performance. We have shown the advantage of METALIC on four bench-mark PDE families.

The task of locating first order saddle points on high-dimensional surfaces describing the variation of energy as a function of atomic coordinates is an essential step for identifying the mechanism and estimating the rate of thermally activated events within the harmonic approximation of transition state theory. When combined directly with electronic structure calculations, the number of energy and atomic force evaluations needed for convergence is a primary issue. Here, we describe an efficient implementation of Gaussian process regression (GPR) acceleration of the minimum mode following method where a dimer is used to estimate the lowest eigenmode of the Hessian. A surrogate energy surface is constructed and updated after each electronic structure calculation. The method is applied to a test set of 500 molecular reactions previously generated by Hermez and coworkers [J. Chem. Theory Comput. 18, 6974 (2022)]. An order of magnitude reduction in the number of electronic structure calculations needed to reach the saddle point configurations is obtained by using the GPR compared to the dimer method. Despite the wide range in stiffness of the molecular degrees of freedom, the calculations are carried out using Cartesian coordinates and are found to require similar number of electronic structure calculations as an elaborate internal coordinate method implemented in the Sella software package. The present implementation of the GPR surrogate model in C++ is efficient enough for the wall time of the saddle point searches to be reduced in 3 out of 4 cases even though the calculations are carried out at a low Hartree-Fock level.

Despite the remarkable success of diffusion models in image generation, slow sampling remains a persistent issue. To accelerate the sampling process, prior studies have reformulated diffusion sampling as an ODE/SDE and introduced higher-order numerical methods. However, these methods often produce divergence artifacts, especially with a low number of sampling steps, which limits the achievable acceleration. In this paper, we investigate the potential causes of these artifacts and suggest that the small stability regions of these methods could be the principal cause. To address this issue, we propose two novel techniques. The first technique involves the incorporation of Heavy Ball (HB) momentum, a well-known technique for improving optimization, into existing diffusion numerical methods to expand their stability regions. We also prove that the resulting methods have first-order convergence. The second technique, called Generalized Heavy Ball (GHVB), constructs a new high-order method that offers a variable trade-off between accuracy and artifact suppression. Experimental results show that our techniques are highly effective in reducing artifacts and improving image quality, surpassing state-of-the-art diffusion solvers on both pixel-based and latent-based diffusion models for low-step sampling. Our research provides novel insights into the design of numerical methods for future diffusion work.

Partial differential equations (PDEs) are important tools to model physical systems, and including them into machine learning models is an important way of incorporating physical knowledge. Given any system of linear PDEs with constant coefficients, we propose a family of Gaussian process (GP) priors, which we call EPGP, such that all realizations are exact solutions of this system. We apply the Ehrenpreis-Palamodov fundamental principle, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs. Our approach can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise defined initial and boundary conditions. Constructing EPGP-priors is algorithmic, generally applicable, and comes with a sparse version (S-EPGP) that learns the relevant spectral frequencies and works better for big data sets. We demonstrate our approach on three families of systems of PDE, the heat equation, wave equation, and Maxwell's equations, where we improve upon the state of the art in computation time and precision, in some experiments by several orders of magnitude.

We pursue the development and application of the recently-introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely the Jastrow, configuration state function and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C_2 molecule up to the dissociation limit, and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations of the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.

Brain tumor growth is unique to each glioma patient and extends beyond what is visible in imaging scans, infiltrating surrounding brain tissue. Understanding these hidden patient-specific progressions is essential for effective therapies. Current treatment plans for brain tumors, such as radiotherapy, typically involve delineating a uniform margin around the visible tumor on pre-treatment scans to target this invisible tumor growth. This "one size fits all" approach is derived from population studies and often fails to account for the nuances of individual patient conditions. We present the GliODIL framework, which infers the full spatial distribution of tumor cell concentration from available multi-modal imaging, leveraging a Fisher-Kolmogorov type physics model to describe tumor growth. This is achieved through the newly introduced method of Optimizing the Discrete Loss (ODIL), where both data and physics-based constraints are softly assimilated into the solution. Our test dataset comprises 152 glioblastoma patients with pre-treatment imaging and post-treatment follow-ups for tumor recurrence monitoring. By blending data-driven techniques with physics-based constraints, GliODIL enhances recurrence prediction in radiotherapy planning, challenging traditional uniform margins and strict adherence to the Fisher-Kolmogorov partial differential equation (PDE) model, which is adapted for complex cases.

In machine learning applications, it is well known that carefully designed learning rate (step size) schedules can significantly improve the convergence of commonly used first-order optimization algorithms. Therefore how to set step size adaptively becomes an important research question. A popular and effective method is the Polyak step size, which sets step size adaptively for gradient descent or stochastic gradient descent without the need to estimate the smoothness parameter of the objective function. However, there has not been a principled way to generalize the Polyak step size for algorithms with momentum accelerations. This paper presents a general framework to set the learning rate adaptively for first-order optimization methods with momentum, motivated by the derivation of Polyak step size. It is shown that the resulting methods are much less sensitive to the choice of momentum parameter and may avoid the oscillation of the heavy-ball method on ill-conditioned problems. These adaptive step sizes are further extended to the stochastic settings, which are attractive choices for stochastic gradient descent with momentum. Our methods are demonstrated to be more effective for stochastic gradient methods than prior adaptive step size algorithms in large-scale machine learning tasks.

Diffusion models (DMs) have proven to be effective in modeling high-dimensional distributions, leading to their widespread adoption for representing complex priors in Bayesian inverse problems (BIPs). However, current DM-based posterior sampling methods proposed for solving common BIPs rely on heuristic approximations to the generative process. To exploit the generative capability of DMs and avoid the usage of such approximations, we propose an ensemble-based algorithm that performs posterior sampling without the use of heuristic approximations. Our algorithm is motivated by existing works that combine DM-based methods with the sequential Monte Carlo (SMC) method. By examining how the prior evolves through the diffusion process encoded by the pre-trained score function, we derive a modified partial differential equation (PDE) governing the evolution of the corresponding posterior distribution. This PDE includes a modified diffusion term and a reweighting term, which can be simulated via stochastic weighted particle methods. Theoretically, we prove that the error between the true posterior distribution can be bounded in terms of the training error of the pre-trained score function and the number of particles in the ensemble. Empirically, we validate our algorithm on several inverse problems in imaging to show that our method gives more accurate reconstructions compared to existing DM-based methods.

We develop a fast, tractable technique called Net-Trim for simplifying a trained neural network. The method is a convex post-processing module, which prunes (sparsifies) a trained network layer by layer, while preserving the internal responses. We present a comprehensive analysis of Net-Trim from both the algorithmic and sample complexity standpoints, centered on a fast, scalable convex optimization program. Our analysis includes consistency results between the initial and retrained models before and after Net-Trim application and guarantees on the number of training samples needed to discover a network that can be expressed using a certain number of nonzero terms. Specifically, if there is a set of weights that uses at most s terms that can re-create the layer outputs from the layer inputs, we can find these weights from O(slog N/s) samples, where N is the input size. These theoretical results are similar to those for sparse regression using the Lasso, and our analysis uses some of the same recently-developed tools (namely recent results on the concentration of measure and convex analysis). Finally, we propose an algorithmic framework based on the alternating direction method of multipliers (ADMM), which allows a fast and simple implementation of Net-Trim for network pruning and compression.

Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible in closed form, so prior methods resort to imprecise approximations of the score matching training objective that degrade performance and preclude applications in high dimensions. In this work, we reexamine these approximations and propose several practical improvements. Our key observation is that most relevant manifolds are symmetric spaces, which are much more amenable to computation. By leveraging and combining various ans\"{a}tze, we can quickly compute relevant quantities to high precision. On low dimensional datasets, our correction produces a noticeable improvement, allowing diffusion to compete with other methods. Additionally, we show that our method enables us to scale to high dimensional tasks on nontrivial manifolds. In particular, we model QCD densities on SU(n) lattices and contrastively learned embeddings on high dimensional hyperspheres.

MATSim (Multi-Agent Transport Simulation Toolkit) is an open source large-scale agent-based transportation planning project applied to various areas like road transport, public transport, freight transport, regional evacuation, etc. BEAM (Behavior, Energy, Autonomy, and Mobility) framework extends MATSim to enable powerful and scalable analysis of urban transportation systems. The agents from the BEAM simulation exhibit 'mode choice' behavior based on multinomial logit model. In our study, we consider eight mode choices viz. bike, car, walk, ride hail, driving to transit, walking to transit, ride hail to transit, and ride hail pooling. The 'alternative specific constants' for each mode choice are critical hyperparameters in a configuration file related to a particular scenario under experimentation. We use the 'Urbansim-10k' BEAM scenario (with 10,000 population size) for all our experiments. Since these hyperparameters affect the simulation in complex ways, manual calibration methods are time consuming. We present a parallel Bayesian optimization method with early stopping rule to achieve fast convergence for the given multi-in-multi-out problem to its optimal configurations. Our model is based on an open source HpBandSter package. This approach combines hierarchy of several 1D Kernel Density Estimators (KDE) with a cheap evaluator (Hyperband, a single multidimensional KDE). Our model has also incorporated extrapolation based early stopping rule. With our model, we could achieve a 25% L1 norm for a large-scale BEAM simulation in fully autonomous manner. To the best of our knowledge, our work is the first of its kind applied to large-scale multi-agent transportation simulations. This work can be useful for surrogate modeling of scenarios with very large populations.

We introduce Power Bundle Adjustment as an expansion type algorithm for solving large-scale bundle adjustment problems. It is based on the power series expansion of the inverse Schur complement and constitutes a new family of solvers that we call inverse expansion methods. We theoretically justify the use of power series and we prove the convergence of our approach. Using the real-world BAL dataset we show that the proposed solver challenges the state-of-the-art iterative methods and significantly accelerates the solution of the normal equation, even for reaching a very high accuracy. This easy-to-implement solver can also complement a recently presented distributed bundle adjustment framework. We demonstrate that employing the proposed Power Bundle Adjustment as a sub-problem solver significantly improves speed and accuracy of the distributed optimization.

We describe and analyse Levenberg-Marquardt methods for solving systems of nonlinear equations. More specifically, we propose an adaptive formula for the Levenberg-Marquardt parameter and analyse the local convergence of the method under Hölder metric subregularity of the function defining the equation and Hölder continuity of its gradient mapping. Further, we analyse the local convergence of the method under the additional assumption that the Łojasiewicz gradient inequality holds. We finally report encouraging numerical results confirming the theoretical findings for the problem of computing moiety conserved steady states in biochemical reaction networks. This problem can be cast as finding a solution of a system of nonlinear equations, where the associated mapping satisfies the Łojasiewicz gradient inequality assumption.

Estimating composition dependent diffusion coefficients in multicomponent alloys was a longstanding challenge due to limitations in experimental methods. In this study, we have first demonstrated a strategic design of producing only three diffusion couples to estimate all types, that is tracer, intrinsic, and interdiffusion coefficients at the Kirkendall marker planes. This establishes a systematic variation of diffusion coefficients with composition in a very wide composition range of the NiCoFeCrMn system in comparison to the data available on impurity diffusion coefficients in pure elements and tracer diffusion coefficients at the equiatomic composition estimated by the radiotracer method. Following, a physics-informed Neural Network based numerical inverse method is developed to extract composition-dependent diffusivities over the whole composition range of the diffusion couples.

Various tasks in data science are modeled utilizing the variational regularization approach, where manually selecting regularization parameters presents a challenge. The difficulty gets exacerbated when employing regularizers involving a large number of hyperparameters. To overcome this challenge, bilevel learning can be employed to learn such parameters from data. However, neither exact function values nor exact gradients with respect to the hyperparameters are attainable, necessitating methods that only rely on inexact evaluation of such quantities. State-of-the-art inexact gradient-based methods a priori select a sequence of the required accuracies and cannot identify an appropriate step size since the Lipschitz constant of the hypergradient is unknown. In this work, we propose an algorithm with backtracking line search that only relies on inexact function evaluations and hypergradients and show convergence to a stationary point. Furthermore, the proposed algorithm determines the required accuracy dynamically rather than manually selected before running it. Our numerical experiments demonstrate the efficiency and feasibility of our approach for hyperparameter estimation on a range of relevant problems in imaging and data science such as total variation and field of experts denoising and multinomial logistic regression. Particularly, the results show that the algorithm is robust to its own hyperparameters such as the initial accuracies and step size.

Data-driven surrogate models improve the efficiency of simulating continuous dynamical systems, yet their autoregressive rollouts are often limited by instability and spectral blow-up. While global regularization techniques can enforce contractive dynamics, they uniformly damp high-frequency features, introducing a contraction-dissipation dilemma. Furthermore, long-horizon trajectory optimization methods that explicitly correct drift are bottlenecked by memory constraints. In this work, we propose Jacobian-Adaptive Weighting for Stability (JAWS), a probabilistic regularization strategy designed to mitigate these limitations. By framing operator learning as Maximum A Posteriori (MAP) estimation with spatially heteroscedastic uncertainty, JAWS dynamically modulates the regularization strength based on local physical complexity. This allows the model to enforce contraction in smooth regions to suppress noise, while relaxing constraints near singular features to preserve gradients, effectively realizing a behavior similar to numerical shock-capturing schemes. Experiments demonstrate that this spatially-adaptive prior serves as an effective spectral pre-conditioner, which reduces the base operator's burden of handling high-frequency instabilities. This reduction enables memory-efficient, short-horizon trajectory optimization to match or exceed the long-term accuracy of long-horizon baselines. Evaluated on the 1D viscous Burgers' equation, our hybrid approach improves long-term stability, shock fidelity, and out-of-distribution generalization while reducing training computational costs.

Preconditioning is at the heart of iterative solutions of large, sparse linear systems of equations in scientific disciplines. Several algebraic approaches, which access no information beyond the matrix itself, are widely studied and used, but ill-conditioned matrices remain very challenging. We take a machine learning approach and propose using graph neural networks as a general-purpose preconditioner. They show attractive performance for many problems and can be used when the mainstream preconditioners perform poorly. Empirical evaluation on over 800 matrices suggests that the construction time of these graph neural preconditioners (GNPs) is more predictable and can be much shorter than that of other widely used ones, such as ILU and AMG, while the execution time is faster than using a Krylov method as the preconditioner, such as in inner-outer GMRES. GNPs have a strong potential for solving large-scale, challenging algebraic problems arising from not only partial differential equations, but also economics, statistics, graph, and optimization, to name a few.

The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a Runge-Kutta collocation method at Radau points) and Dassl, based on backward differentiation formulas, among the others. When solving fractional ordinary differential equations (ODEs), the derivative operator is replaced by a non-local one and the fractional ODE is reformulated as a Volterra integral equation, to which these codes cannot be directly applied. This article is a follow-up of the article by the authors (Guglielmi and Hairer, SISC, 2025) for differential equations with distributed delays. The main idea is to approximate the fractional kernel t^{α-1}/ Γ(α) (α>0) by a sum of exponential functions or by a sum of exponential functions multiplied by a monomial, and then to transform the fractional integral (of convolution type) into a set of ordinary differential equations. The augmented system is typically stiff and thus requires the use of an implicit method. It can have a very large dimension and requires a special treatment of the arising linear systems. The present work presents an algorithm for the construction of an approximation of the fractional kernel by a sum of exponential functions, and it shows how the arising linear systems in a stiff time integrator can be solved efficiently. It is explained how the code Radau5 can be used for solving fractional differential equations. Numerical experiments illustrate the accuracy and the efficiency of the proposed method. Driver examples are publicly available from the homepages of the authors.

We consider a family of linear systems A_mu alpha=C with system matrix A_mu depending on a parameter mu and for simplicity parameter-independent right-hand side C. These linear systems typically result from the finite-dimensional approximation of a parameter-dependent boundary-value problem. We derive a procedure based on the Empirical Interpolation Method to obtain a separated representation of the system matrix in the form A_muapproxsum_{m}beta_m(mu)A_{mu_m} for some selected values of the parameter. Such a separated representation is in particular useful in the Reduced Basis Method. The procedure is called nonintrusive since it only requires to access the matrices A_{mu_m}. As such, it offers a crucial advantage over existing approaches that instead derive separated representations requiring to enter the code at the level of assembly. Numerical examples illustrate the performance of our new procedure on a simple one-dimensional boundary-value problem and on three-dimensional acoustic scattering problems solved by a boundary element method.

We propose an adaptive step size with an energy approach for a suitable class of preconditioned gradient descent methods. We focus on settings where the preconditioning is applied to address the constraints in optimization problems, such as the Hessian-Riemannian and natural gradient descent methods. More specifically, we incorporate these preconditioned gradient descent algorithms in the recently introduced Adaptive Energy Gradient Descent (AEGD) framework. In particular, we discuss theoretical results on the unconditional energy-stability and convergence rates across three classes of objective functions. Furthermore, our numerical results demonstrate excellent performance of the proposed method on several test bed optimization problems.

In astrophysical and cosmological analyses, the increasing quality and volume of astronomical data demand efficient and precise computational tools. This work introduces a novel adaptive algorithm for automatic knots (AutoKnots) allocation in spline interpolation, designed to meet user-defined precision requirements. Unlike traditional methods that rely on manually configured knot distributions with numerous parameters, the proposed technique automatically determines the optimal number and placement of knots based on interpolation error criteria. This simplifies configuration, often requiring only a single parameter. The algorithm progressively improves the interpolation by adaptively sampling the function-to-be-approximated, f(x), in regions where the interpolation error exceeds the desired threshold. All function evaluations contribute directly to the final approximation, ensuring efficiency. While each resampling step involves recomputing the interpolation table, this process is highly optimized and usually computationally negligible compared to the cost of evaluating f(x). We show the algorithm's efficacy through a series of precision tests on different functions. However, the study underscores the necessity for caution when dealing with certain function types, notably those featuring plateaus. To address this challenge, a heuristic enhancement is incorporated, improving accuracy in flat regions. This algorithm has been extensively used and tested over the years. NumCosmo includes a comprehensive set of unit tests that rigorously evaluate the algorithm both directly and indirectly, underscoring its robustness and reliability. As a practical application, we compute the surface mass density Sigma(R) and the average surface mass density Sigma(<R) for Navarro-Frenk-White and Hernquist halo density profiles, which provide analytical benchmarks. (abridged)

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the d-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

We address the computational challenges of solving parametric PDEs with non parametrized geometric variations and non-reducible problems, such as those involving shocks and discontinuities of variable positions. Traditional dimensionality reduction methods like POD struggle with these scenarios due to slowly decaying Kolmogorov widths. To overcome this, we propose a novel non-linear dimensionality reduction technique to reduce the required modes for representation. The non-linear reduction is obtained through a POD after applying a transformation on the fields, which we call optimal mappings, and is a solution to an optimization problem in infinite dimension. The proposed learning framework combines morphing techniques, non-linear dimensionality reduction, and Gaussian Process Regression (GPR). The problem is reformulated on a reference geometry before applying the dimensionality reduction. Our method learns both the optimal mapping, and the solution fields, using a series of GPR models, enabling efficient and accurate modeling of complex parametric PDEs with geometrical variability. The results obtained concur with current state-of-the-art models. We mainly compare our method with the winning solution of the ML4CFD NeurIPS 2024 competition.

Offline goal-conditioned reinforcement learning (GCRL) learns goal-conditioned policies from static pre-collected datasets. However, accurate value estimation remains a challenge due to the limited coverage of the state-action space. Recent physics-informed approaches have sought to address this by imposing physical and geometric constraints on the value function through regularization defined over first-order partial differential equations (PDEs), such as the Eikonal equation. However, these formulations can often be ill-posed in complex, high-dimensional environments. In this work, we propose a physics-informed regularization derived from the viscosity solution of the Hamilton-Jacobi-Bellman (HJB) equation. By providing a physics-based inductive bias, our approach grounds the learning process in optimal control theory, explicitly regularizing and bounding updates during value iterations. Furthermore, we leverage the Feynman-Kac theorem to recast the PDE solution as an expectation, enabling a tractable Monte Carlo estimation of the objective that avoids numerical instability in higher-order gradients. Experiments demonstrate that our method improves geometric consistency, making it broadly applicable to navigation and high-dimensional, complex manipulation tasks. Open-source codes are available at https://github.com/HrishikeshVish/phys-fk-value-GCRL.

Large-scale kernel ridge regression (KRR) is limited by the need to store a large kernel matrix K_t. To avoid storing the entire matrix K_t, Nystrom methods subsample a subset of columns of the kernel matrix, and efficiently find an approximate KRR solution on the reconstructed matrix. The chosen subsampling distribution in turn affects the statistical and computational tradeoffs. For KRR problems, recent works show that a sampling distribution proportional to the ridge leverage scores (RLSs) provides strong reconstruction guarantees for the approximation. While exact RLSs are as difficult to compute as a KRR solution, we may be able to approximate them well enough. In this paper, we study KRR problems in a sequential setting and introduce the INK-ESTIMATE algorithm, that incrementally computes the RLSs estimates. INK-ESTIMATE maintains a small sketch of K_t, that at each step is used to compute an intermediate estimate of the RLSs. First, our sketch update does not require access to previously seen columns, and therefore a single pass over the kernel matrix is sufficient. Second, the algorithm requires a fixed, small space budget to run dependent only on the effective dimension of the kernel matrix. Finally, our sketch provides strong approximation guarantees on the distance between the true kernel matrix and its approximation, and on the statistical risk of the approximate KRR solution at any time, because all our guarantees hold at any intermediate step.