PDE Solving

PDE: Partial Differential Equation

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19 papers in the last four weeks, up 111% on the four weeks before. 0.2% of all new papers.

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Latest papers 170

Apr 17, 2026physics.comp-ph

A Structure-Preserving Graph Neural Solver for Parametric Hyperbolic Conservation Laws

Hyperbolic conservation laws govern a wide range of transport-driven dynamics featuring shocks, contact discontinuities, and complex wave interactions, posing distinct challenges for deep-learning-based surrogate modeling. While classical numerical methods provide robust and physically admissible solutions, their computational cost restricts applicability in many-query tasks such as parametric studies and design optimization. Conversely, existing neural surrogates offer rapid inference but often fail to respect intrinsic PDE structures, leading to non-physical artifacts, rollout instability, and poor generalization. We present an interpretable, structure-preserving graph neural solver that bridges classical numerical principles with graph neural networks (GNNs). The network is designed as a learned reconstruction-and-flux operator rather than a black-box state updater, thereby inherently preserving key properties such as local conservation and upwinding. Inspired by Arbitrary high-order DERivatives schemes, we further recast message-passing GNNs as high-order space-time predictors, enabling conservative and stable neural updates with large time steps. Evaluation is performed on challenging supersonic flow benchmarks spanning broad parametric variations in geometry, initial/boundary conditions, and flow regimes. The neural solver achieves superior long-horizon rollout stability and accuracy compared with strong surrogate baselines, outperforms low-order discretizations, and delivers orders-of-magnitude runtime speedups over high-resolution simulations.

Jiamin Jiang, Shanglin Lv, Jingrun Chen

School of Artificial Intelligence and Data Science and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China · School of Mathematical Sciences, University of Science and Technology of China, China · School of Mathematical Sciences and Suzhou Institute for Advanced Research, University of Science and Technology of China, and Suzhou Big Data & AI Research and Engineering Center, China

Graph Neural Networks Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Apr 16, 2026cs.LG

Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)

We explore the possibility of solving Partial Differential Equations (PDEs) using discrete weak formulations. We propose a programming environment for defining a discrete computational domain, introducing discrete functions defined over a set of points, constructing discrete inner products, and introducing discrete weak formulations employing Kronecker delta test functions. Building on this setup, we propose a discrete neural network representation, training the solution function defined over a discrete set of points and employing discrete finite difference derivatives in the automatic differentiation procedures. As a challenging computational model example, we focus on Stokes equations in two-dimensions, defined over a discrete set of points. We train the solution using the discrete weak residual and the Adamax algorithm with discrete automatic differentiation of the discrete gradients. Despite introducing the python environment, we also provide a rigorous mathematical formulation based on discrete weak formulations, proving the well-posedness and robustness of the loss function. The solution of the discrete weak formulations is based on neural network training employing a robust loss function that is related to the true error. In this way, we have a robust control of the numerical error during the training of the neural networks. Besides the Stokes formulation, we also explain the functionality of the proposed library using the Laplace problem formulation.

Tomasz Służalec, Marcin Łoś, Askold Vilkha +1

AGH University of Krakow, Poland

PDE Solving Physics-Informed ML

Apr 10, 2026cs.LG

Stochastic-Dimension Frozen Sampled Neural Network for High-Dimensional Gross-Pitaevskii Equations on Unbounded Domains

This paper introduces the Stochastic-Dimension Frozen Sampled Neural Network (SD-FSNN), a novel computational framework for solving high-dimensional Gross-Pitaevskii equation (GPE) on unbounded domain. The proposed method circumvents the curse-of-dimensionality that plagues traditional discretizations and the computational bottlenecks of gradient-based neural network solvers through a synergistic combination of techniques. First, a prescribed Gaussian envelope encodes the far-field decay of the wavefunction, enabling a space-time separation where the spatial approximation is handled by a frozen, single-hidden-layer neural network with data-driven sampled features. This yields a gradient-free formalism where spatial derivatives are analytically precomputed and time-dependence is evolved via reduced ODEs. Second, a stochastic-dimension sampler provides a conditionally unbiased estimate of the spatial operator by evaluating only a small subset of spatial dimensions at each time step, essentially reducing computational and memory costs. Discrete conservation laws are also enforced, ensuring long-term stability. Extensive numerical experiments on GPE in up to 1000 dimensions demonstrate that SD-FSNN achieves significantly higher accuracy and efficiency compared to state-of-the-art methods, including PINNs, randomized feature methods, and tensor-network approaches. The results confirm that SD-FSNN effectively mitigates the Kolmogorov

n

-width barrier for frozen-basis models on structured solution manifolds.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China. · School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130025, China.

Neural PDE Solvers PDE Solving

Apr 8, 2026physics.comp-ph

Modelling Gas-Phase Reaction Kinetics with Guided Particle Diffusion Sampling

Physics-guided sampling with diffusion priors has recently shown strong performance in solving complex systems of partial differential equations (PDEs) from sparse observations. However, these methods are typically evaluated on benchmark problems that do not fully demonstrate their ability to generate temporally consistent solutions of time-dependent PDEs, often focusing instead on reconstructing a single snapshot. In this work, we apply these methods to gas-phase reaction kinetics problems governed by the advection-reaction-diffusion (ARD) equation, providing a setting that more closely reflects realistic laboratory experiments. We demonstrate that guided sampling can be used to reconstruct full spatiotemporal trajectories, rather than isolated states. Furthermore, we show that these methods generalise to previously unseen parameter regimes, highlighting their potential for real-world applications.

Andrew Millard, Zheng Zhao, Henrik Pedersen

Division of Statistics and Machine Learning (STIMA), Linköping University, Sweden · STIMA, Linköping University, Sweden · Division of Chemistry, Linköping University, Sweden

PDE Solving Diffusion Posterior Sampling

Mar 19, 2026cs.LG

Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.

Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández +1

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada

Neural Network Generalization Neural PDE Solvers PDE Solving Physics-Informed ML Generalization Bounds

Mar 19, 2026cs.LG

Neural Galerkin Normalizing Flow for Transition Probability Density Functions of Diffusion Models

We propose a new Neural Galerkin Normalizing Flow framework to approximate the transition probability density function of a diffusion process by solving the corresponding Fokker-Planck equation with an atomic initial distribution, parametrically with respect to the location of the initial mass. By using Normalizing Flows, we look for the solution as a transformation of the transition probability density function of a reference stochastic process, ensuring that our approximation is structure-preserving and automatically satisfies positivity and mass conservation constraints. By extending Neural Galerkin schemes to the context of Normalizing Flows, we derive a system of ODEs for the time evolution of the Normalizing Flow's parameters. Adaptive sampling routines are used to evaluate the Fokker-Planck residual in meaningful locations, which is of vital importance to address high-dimensional PDEs. Numerical results show that this strategy captures key features of the true solution and enforces the causal relationship between the initial datum and the density function at subsequent times. After completing an offline training phase, online evaluation becomes significantly more cost-effective than solving the PDE from scratch. The proposed method serves as a promising surrogate model, which could be deployed in many-query problems associated with stochastic differential equations, like Bayesian inference, simulation, and diffusion bridge generation.

Riccardo Saporiti, Fabio Nobile

CSQI, École Polytechnique Féderale de Lausanne, Switzerland

Normalizing Flows Neural PDE Solvers PDE Solving PDE Surrogate Modeling

Mar 13, 2026cs.LG

Disentangled Latent Dynamics Manifold Fusion for Solving Parameterized PDEs

Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable. The problem becomes even more severe when the model must also predict beyond the training time range. Existing methods usually cannot handle parameter generalization and temporal extrapolation at the same time. Standard parameterized models treat time as just another input and therefore fail to capture intrinsic dynamics, while recent continuous-time latent methods often rely on expensive test-time auto-decoding for each instance, which is inefficient and can disrupt continuity across the parameterized solution space. To address this, we propose Disentangled Latent Dynamics Manifold Fusion (DLDMF), a physics-informed framework that explicitly separates space, time, and parameters. Instead of unstable auto-decoding, DLDMF maps PDE parameters directly to a continuous latent embedding through a feed-forward network. This embedding initializes and conditions a latent state whose evolution is governed by a parameter-conditioned Neural ODE. We further introduce a dynamic manifold fusion mechanism that uses a shared decoder to combine spatial coordinates, parameter embeddings, and time-evolving latent states to reconstruct the corresponding spatiotemporal solution. By modeling prediction as latent dynamic evolution rather than static coordinate fitting, DLDMF reduces interference between parameter variation and temporal evolution while preserving a smooth and coherent solution manifold. As a result, it performs well on unseen parameter settings and in long-term temporal extrapolation. Experiments on several benchmark problems show that DLDMF consistently outperforms state-of-the-art baselines in accuracy, parameter generalization, and extrapolation robustness.

Zhangyong Liang, Huanhuan Gao

National Center for Applied Mathematics, Tianjin University, Tianjin, 300072, PR China · School of Mechanical and Aerospace Engineering, Jilin University, Changchun, 130025, China

Latent Dynamics Modeling PDE Solving PDE Surrogate Modeling Physics-Informed ML

Mar 9, 2026cs.LG

MUSA-PINN: Multi-scale Weak-form Physics-Informed Neural Networks for Fluid Flow in Complex Geometries

While Physics-Informed Neural Networks (PINNs) offer a mesh-free approach to solving fluid-flow PDEs, standard point-wise residual minimization suffers from convergence pathologies in topologically complex domains like Triply Periodic Minimal Surfaces (TPMS). The locality bias of point-wise constraints fails to propagate global information through tortuous channels, causing unstable gradients and conservation violations. To address this, we propose the Multi-scale Weak-form PINN (MUSA-PINN), which reformulates Navier-Stokes equation constraints as integral conservation laws over hierarchical spherical control volumes. We enforce continuity and momentum conservation via flux-balance residuals on control surfaces. Our method utilizes a three-scale subdomain strategy-comprising large volumes for long-range coupling, skeleton-aware meso-scale volumes aligned with transport pathways, and small volumes for local refinement-alongside a two-stage training schedule prioritizing continuity. Experiments on steady incompressible flow in TPMS geometries show MUSA-PINN outperforms state-of-the-art baselines, reducing relative errors by up to 93% and preserving mass conservation.

Weizheng Zhang, Xunjie Xie, Hao Pan +4

Shandong University, Qingdao, China · Tsinghua University, Beijing, China · Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing, China

Neural PDE Solvers PDE Solving Physics-Informed ML Computational Fluid Dynamics

Feb 16, 2026cs.LG

Pseudo-differential-enhanced physics-informed neural networks

We present pseudo-differential enhanced physics-informed neural networks (PINNs), an extension of gradient enhancement but in Fourier space. Gradient enhancement of PINNs dictates that the PDE residual is taken to a higher differential order than prescribed by the PDE, added to the objective as an augmented term in order to improve training and overall learning fidelity. We propose the same procedure after application via Fourier transforms, since differentiating in Fourier space is multiplication with the Fourier wavenumber under suitable decay. Our methods are fast and efficient. Our methods oftentimes achieve superior PINN versus numerical error in fewer training iterations, potentially pair well with few/fixed samples in collocation, and can achieve breakthrough rather than gradual effects on the error. Moreover, our methods are suitable for fractional derivatives. We establish that our methods, due to the dynamical effects, improve spectral eigenvalue decay of the neural tangent kernel (NTK), and so our methods contribute towards the learning of high frequencies in early training, mitigating the effects of frequency bias up to the polynomial order and possibly greater with smooth activations. In particular, our primary contribution can be characterized as gradient enhancement affects spectral bias, and we specialize to Fourier space, although our theory does not prove the ambient case exactly due to a result with the Plancherel theorem on the descent paths. Our methods accommodate advanced techniques in PINNs, such as Fourier feature embeddings. A pitfall of discrete Fourier transforms via the Fast Fourier Transform (FFT) is mesh subjugation, and so we demonstrate compatibility of our methods for greater mesh flexibility and invariance on alternative Euclidean and non-Euclidean domains via Monte Carlo methods and otherwise, although possessing trade-offs of their own.

Andrew Gracyk

Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

PDE Solving Spectral Bias Physics-Informed ML

Jan 19, 2026cs.LG

Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems

The present study investigates the numerical solution of Black-Scholes partial differential equation (PDE) for option valuation with multiple underlying assets. We develop a physics-informed (PI) machine learning algorithm based on a radial basis function neural network (RBFNN) that concurrently optimizes the network architecture and predicts the target option price. The physics-informed radial basis function neural network (PIRBFNN) combines the strengths of the traditional radial basis function collocation method and the physics-informed neural network machine learning approach to effectively solve PDE problems in the financial context. By employing a PDE residual-based technique to adaptively refine the distribution of hidden neurons during the training process, the PIRBFNN facilitates accurate and efficient handling of multidimensional option pricing models featuring non-smooth payoff conditions. The validity of the proposed method is demonstrated through a set of experiments encompassing a single-asset European put option, a double-asset exchange option, and a four-asset basket call option.

Yan Ma, Yumeng Ren, Elisabeth Larsson

aCollege of Mathematics and Statistics, Northwest Normal University, Lanzhou, China · bDepartment of Mathematics, City University of Hong Kong, Hong Kong SAR, China · cDepartment of Information Technology, Uppsala University, Sweden

Quantitative Finance PDE Solving Physics-Informed ML

Dec 22, 2025physics.comp-ph

Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations

Solving high-dimensional Fokker-Planck (FP) equations remains a challenging problem in computational physics and stochastic dynamics, due to the curse of dimensionality, unbounded domains, and complex probability landscapes. In this work, we propose an adaptive probability flow residual minimization (A-PFRM) method for this problem. The second-order FP equation is reformulated as an equivalent first-order continuity equation associated with the probability flow ordinary differential equation, based on which a loss function is constructed to train neural network approximations without Hessian computation. To further improve the computational efficiency, the Hutchinson trace estimator is applied to compute the divergence in the corresponding score function, such that the training time can be dimension-independent on GPUs. Adaptive sampling strategies are employed to generate the collocation points, and our analysis shows that the Kullback-Leibler divergence between our A-PFRM approximation and the exact solution is bounded by the residual loss weighted by the estimated density function. Numerical experiments are presented to demonstrate the performance of A-PFRM, which include Ornstein-Uhlenbeck (OU) processes problems, Brownian motions with time-varying diffusion, and Geometric OU processes featuring non-Gaussian solutions up to one hundred dimensions.

Xiaolong Wu, Qifeng Liao

School of Information Science and Technology, ShanghaiTech University, Shanghai, 201210, China

Neural PDE Solvers PDE Solving Adaptive Sampling

Nov 20, 2025math.NA

Hybrid coupling with operator inference and the overlapping Schwarz alternating method

This paper presents a novel hybrid approach for coupling subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) with each other and with subdomain-local high-fidelity full order models (FOMs) with using the overlapping Schwarz alternating method (O-SAM). The proposed methodology addresses significant challenges in multiscale modeling and simulation, particularly the long runtime and complex mesh generation requirements associated with traditional high-fidelity simulations. By leveraging the flexibility of O-SAM, we enable the seamless integration of disparate models, meshes, and time integration schemes, enhancing computational efficiency while maintaining high accuracy. Our approach is demonstrated through a series of numerical experiments on complex three-dimensional (3D) solid dynamics problems, showcasing speedups of up to 106x compared to conventional FOM-FOM couplings. This work paves the way for more efficient simulation workflows in engineering applications, with potential extensions to a wide range of partial differential equations.

Irina Tezaur, Eric Parish, Anthony Gruber +3

Sandia National Laboratories, Livermore, CA and Albuquerque, NM, USA · Rice University, Houston, TX, USA

PDE Solving Reduced-Order Modeling

Nov 19, 2025math.NA

Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation

Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and finite element approaches, are widely used to solve governing partial differential equations (PDEs) such as the Helmholtz equation. However, these methods face significant computational challenges when applied to high-frequency wave problems in complex two-dimensional domains. This work investigates Finite Basis Physics-Informed Neural Networks (FBPINNs) and their multilevel extensions as a promising alternative. These methods leverage domain decomposition, partitioning the computational domain into overlapping sub-domains, each governed by a local neural network. We assess their accuracy and computational efficiency in solving the Helmholtz equation for the homogeneous case, demonstrating their potential to mitigate the limitations of traditional approaches.

Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri +1

Department of Mathematics and Computer Science, Eindhoven University of Technology, The Netherlands · Atlantis project-team, Inria Research Center at Université Côte d’Azur, France · MACARON project-team, Université de Strasbourg, CNRS, Inria, IRMA, France

Neural PDE Solvers PDE Solving Physics-Informed ML

Oct 29, 2025cs.LG

LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, our model embeds the underlying physical laws and learns the solution solely from initial and boundary data. Consequently, the boundary loss directly quantifies domain-wide error, enabling rigorous error estimation for well-posed IBVPs. We implement LieSolver and demonstrate its application to linear homogeneous PDEs, showing that it outperforms physics-informed neural networks (PINNs) in both speed and accuracy while yielding compact models. Overall, our approach significantly enhances the efficiency and reliability of predictions for PDE-constrained problems.

René P. Klausen, Ivan Timofeev, Jonas Naujoks +4

Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute, Berlin, Germany · Department of Electrical Engineering and Computer Science, Technische Universität Berlin, Germany · BIFOLD Berlin Institute for the Foundations of Learning and Data, Berlin, Germany +1

Neural PDE Solvers PDE Solving Physics-Informed ML

Sep 19, 2025math.NA

A Flow-rate-conserving CNN-based Domain Decomposition Method for Blood Flow Simulations

This work aims to predict blood flow with non-Newtonian viscosity in stenosed arteries using convolutional neural network (CNN) surrogate models. An alternating Schwarz domain decomposition method is proposed which uses CNN-based subdomain solvers. A universal subdomain solver (USDS) is trained on a single, fixed geometry and then applied for each subdomain solve in the Schwarz method. Results for two-dimensional stenotic arteries of varying shape and length for different inflow conditions are presented and statistically evaluated. One key finding, when using a limited amount of training data, is that incorporating a physics-aware constraint, as, in our case, flow rate conservation, into the USDS improves the prediction accuracy and convergence behavior of the Schwarz method compared to a purely data-driven USDS. As the USDS is a data-driven, inexact subdomain solver, admissible parameter ranges for the geometry and inflow configurations must be defined and tested.

Simon Klaes, Axel Klawonn, Natalie Kubicki +4

Department of Mathematics and Computer Science, University of Cologne, Germany · Center for Data and Simulation Science, University of Cologne, Germany · Institute for Advanced Simulation (IAS-2), Forschungszentrum Jülich, Germany +1

PDE Solving PDE Surrogate Modeling Physics-Informed ML Convolutional Neural Networks Computational Fluid Dynamics

Aug 29, 2025cs.LG

Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equation

Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential equations. In practice, one often employs stochastic gradient descent type algorithms to train the neural network. Therefore, the convergence guarantee of stochastic gradient descent is of fundamental importance. In this work, we establish the linear convergence of stochastic gradient descent / flow in training over-parameterized two layer PINNs with a general class of activation functions for solving one model second-order elliptic problem, i.e., the Poisson equation, in the sense of high probability. These results extend the existing result [20] in which gradient descent was analyzed. The challenge of the analysis lies in handling the dynamic randomness introduced by stochastic optimization methods. The key of the analysis lies in ensuring the positive definiteness of suitable Gram matrices during the training. The analysis sheds insight into the dynamics of the optimization process, and provides guarantees on physics informed neural networks trained by stochastic algorithms.

Bangti Jin, Longjun Wu

Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China

Neural PDE Solvers PDE Solving Stochastic Optimization Convergence Physics-Informed ML Stochastic Gradient Descent

Jun 23, 2025math.NA

DPG loss functions for learning parameter-to-solution maps by neural networks

We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance the prediction capability of the resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishing the variational correctness of a loss function from an ultraweak Discontinuous Petrov Galerkin (DPG) discretization are worked out. Despite the focus on the example, the proposed concepts apply to a much wider scope of problems, namely problems for which stable DPG formulations are available. The issue of high-contrast diffusion fields and ensuing difficulties with degrading ellipticity are discussed. Both numerical results and theoretical arguments illustrate that for high-contrast diffusion parameters the proposed DPG loss functions deliver much more robust performance than simpler least-squares losses.

Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan

PDE Solving PDE Surrogate Modeling Robust Loss Functions PDE Operator Learning Physics-Informed ML

Oct 18, 2024math.OC

Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address \emph{complexity}: for broad operator classes described only through regularity (e.g.\ uniform continuity or

C^r

-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale \emph{exponentially} in the reciprocal accuracy

1/\varepsilon

. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in

1/\varepsilon

. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of \emph{non-Markovian} BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows \emph{polynomially} in

1/\varepsilon

. We unlock this polynomial scaling regime by \emph{informing the NO's inductive bias} by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Doléans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.

Takashi Furuya, Anastasis Kratsios

Doshisha University, Japan · Department of Mathematics McMaster University and Vector Institute, Canada

Stochastic Differential Equations PDE Solving Neural Network Approximation Theory Neural Operators

Dec 15, 2021math.NA

Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations

We construct and analyze approximation rates of deep operator networks (ONets) between infinite-dimensional spaces that emulate with an exponential rate of convergence the coefficient-to-solution map of elliptic second-order partial differential equations. In particular, we consider problems set in

d

-dimensional periodic domains,

d=1, 2, \dots

, and with analytic right-hand sides and coefficients. Our analysis covers linear, elliptic second order divergence-form PDEs as, e.g., diffusion-reaction problems, parametric diffusion equations, and elliptic systems such as linear isotropic elastostatics in heterogeneous materials. We leverage the exponential convergence of spectral collocation methods for boundary value problems whose solutions are analytic. In the present periodic and analytic setting, this follows from classical elliptic regularity. Within the ONet branch and trunk construction of [Chen and Chen, 1993] and of [Lu et al., 2021], we show the existence of deep ONets which emulate the coefficient-to-solution map to a desired accuracy in the

H^1

norm, uniformly over the coefficient set. We prove that the neural networks in the ONet have size

\mathcal{O}(\left|\log(\varepsilon)\right|^κ)

, where

\varepsilon>0

is the approximation accuracy, for some

κ>0

depending on the physical space dimension.

Carlo Marcati, Christoph Schwab

Seminar for Applied Mathematics, ETH Zurich, Switzerland

PDE Solving PDE Operator Learning Neural Network Approximation Theory Deep Operator Networks

Dec 2, 2020math.NA

Deep learning based numerical approximation algorithms for stochastic partial differential equations

In this article, we introduce a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Christian Beck, Sebastian Becker, Patrick Cheridito +2

Department of Mathematics, ETH Zurich · School of Data Science and School of Artificial Intelligence,2025 The Chinese University of Hong Kong, Shenzhen (CUHK-Shenhen) · Applied Mathematics: Institute for Analysis and Numerics, University of Münster +1

Stochastic Differential Equations Neural PDE Solvers PDE Solving