mlsanityData to 7 Oct 2026

Solving Poisson Equations using Neural Walk-on-Spheres

arXiv 2406.03494 ↗First posted 5 Jun 2024cs.LG cs.NA math.NAcited by 11

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Abstract (arXiv)

We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. Leveraging stochastic representations and Walk-on-Spheres methods, we develop novel losses for neural networks based on the recursive solution of Poisson equations on spheres inside the domain. The resulting method is highly parallelizable and does not require spatial gradients for the loss. We provide a comprehensive comparison against competing methods based on PINNs, the Deep Ritz method, and (backward) stochastic differential equations. In several challenging, high-dimensional numerical examples, we demonstrate the superiority of NWoS in accuracy, speed, and computational costs. Compared to commonly used PINNs, our approach can reduce memory usage and errors by orders of magnitude. Furthermore, we apply NWoS to problems in PDE-constrained optimization and molecular dynamics to show its efficiency in practical applications.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 25 of its 60 references that are in the map. First posted on arXiv 5 Jun 2024.

Builds on

25 of 60 references.

References · 25

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  2. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  3. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  5. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  6. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  7. 2016Deep Convolutional Neural Network for Inverse Problems in Imagingcited by 431
  8. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  9. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  10. 2023Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systemscited by 118
  11. 2020Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path spacecited by 80
  12. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
13 more
  1. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 62
  2. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  3. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  4. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 44
  5. 2021Solving PDE-constrained Control Problems Using Operator Learningcited by 41
  6. 2020Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learningcited by 22
  7. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  8. 2020A Derivative-Free Method for Solving Elliptic Partial Differential Equations with Deep Neural Networkscited by 13
  9. 2022Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learningcited by 13
  10. 2022Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficientscited by 12
  11. 2023Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditionscited by 8
  12. 2023Boundary Value Caching for Walk on Spherescited by 6
  13. 2022From Monte Carlo to neural networks approximations of boundary value problemscited by 3

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 4; 2026: 5.

11 papers.

  1. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  2. 2025Neural Green's Functionscited by 6
  3. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  4. 2025DeepRWCap: Neural-Guided Random-Walk Capacitance Solver for IC Designcited by 2
  5. 2026Mollified Value Learningcited by 1
  6. 2026Operator Learning Using Weak Supervision from Walk-on-Spherescited by 1
  7. 2025Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Conditioncited by 0
  8. 2025WoSNN: Stochastic Solver for PDEs with Machine Learningcited by 0
  9. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  10. 2026Layerwise Decoupling for Stable Structured Sparsification of Fully Connected Layerscited by 0
  11. 2026Neural Harmonic Measure Operatorcited by 0