mlsanityData to 14 Oct 2026

Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion

arXiv 2411.03384 ↗First posted 5 Nov 2024stat.ML cs.LG cs.NAcited by 6

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Abstract (arXiv)

In this paper, we solve stochastic partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. Moreover, we provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise. Finally, we apply our results in numerical examples to approximate the solution of three SPDEs: the stochastic heat equation, the Heath-Jarrow-Morton equation, and the Zakai equation.

Key facts. Cited by 6 papers in the map (data through 14 Oct 2026). Builds on 21 of its 89 references that are in the map. First posted on arXiv 5 Nov 2024.

Builds on

21 of 89 references.

References · 21

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  3. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
  4. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 67
  5. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 64
  6. 2015Neural Network with Unbounded Activation Functions is Universal Approximatorcited by 63
  7. 2019Deep splitting method for parabolic PDEscited by 53
  8. 2020Approximation Bounds for Random Neural Networks and Reservoir Systemscited by 52
  9. 2021Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamicscited by 47
  10. 2018Solving the Kolmogorov PDE by means of deep learningcited by 41
  11. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  12. 2021Optimal Stopping via Randomized Neural Networkscited by 28
9 more
  1. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  2. 2023Global universal approximation of functional input maps on weighted spacescited by 19
  3. 2023Universal approximation property of Banach space-valued random feature models including random neural networkscited by 13
  4. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  5. 2020Deep learning based numerical approximation algorithms for stochastic partial differential equationscited by 11
  6. 2022Chaotic Hedging with Iterated Integrals and Neural Networkscited by 10
  7. 2022An energy-based deep splitting method for the nonlinear filtering problemcited by 7
  8. 2024Universal approximation results for neural networks with non-polynomial activation function over non-compact domainscited by 7
  9. 2023Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricingcited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 5.

6 papers.

  1. 2026Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equationscited by 5
  2. 2025SPDEBench: An Extensive Benchmark for Learning Stochastic PDEscited by 1
  3. 2026Latent-Variable Learning of SPDEs via Wiener Chaoscited by 1
  4. 2026Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equationscited by 1
  5. 2026Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorizationcited by 0
  6. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0