# Deep learning based numerical approximation algorithms for stochastic partial differential equations

arXiv: https://arxiv.org/abs/2012.01194 · HTML: https://mlsanity.com/paper/2012.01194/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2020.

## Built on by (most cited first, 11 of 11)

- [Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis](https://mlsanity.com/paper/2210.13300/) (2022)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
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- [An energy-based deep splitting method for the nonlinear filtering problem](https://mlsanity.com/paper/2203.17153/) (2022)
- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)
- [Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sense](https://mlsanity.com/paper/2409.20431/) (2024)
- [Computation of conditional expectations with guarantees](https://mlsanity.com/paper/2112.01804/) (2021)
- [A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting](https://mlsanity.com/paper/2409.14585/) (2024)
- [Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equations](https://mlsanity.com/paper/2502.03670/) (2025)
- [Nonlinear filtering based on density approximation and deep BSDE prediction](https://mlsanity.com/paper/2508.10630/) (2025)
- [High-dimensional Bayesian filtering through deep density approximation](https://mlsanity.com/paper/2511.07261/) (2025)

## Abstract (arXiv)

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.