# Chaos into Order: Neural Framework for Expected Value Estimation of Stochastic Partial Differential Equations

arXiv: https://arxiv.org/abs/2502.03670 · HTML: https://mlsanity.com/paper/2502.03670/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 36 references that are in the map.
- First posted on arXiv 5 Feb 2025.

## Builds on (most cited first, 9 of 9)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Deep learning based numerical approximation algorithms for stochastic partial differential equations](https://mlsanity.com/paper/2012.01194/) (2020)
- [Pixel Normalization from Numeric Data as Input to Neural Networks](https://mlsanity.com/paper/1705.01809/) (2017)

## Built on by (most cited first, 1 of 1)

- [Latent-Variable Learning of SPDEs via Wiener Chaos](https://mlsanity.com/paper/2602.11794/) (2026)

## Abstract (arXiv)

Stochastic partial differential equations (SPDEs) describe the evolution of random processes over space and time, but their solutions are often analytically intractable and computationally expensive to estimate. In this paper, we propose the Learned Expectation Collapser (LEC), a physics-informed neural framework designed to approximate the expected value of linear SPDE solutions without requiring domain discretization. By leveraging randomized sampling of both space-time coordinates and noise realizations during training, LEC trains standard feedforward neural networks to minimize residual loss across multiple stochastic samples. We hypothesize and empirically confirm that this training regime drives the network to converge toward the expected value of the solution of the SPDE. Using the stochastic heat equation as a testbed, we evaluate performance across a diverse set of 144 experimental configurations that span multiple spatial dimensions, noise models, and forcing functions. The results show that the model consistently learns accurate approximations of the expected value of the solution in lower dimensions and a predictable decrease in accuracy with increased spatial dimensions, with improved stability and robustness under increased Monte Carlo sampling. Our findings offer new insight into how neural networks implicitly learn statistical structure from stochastic differential operators and suggest a pathway toward scalable, simulator-free SPDE solvers.