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

TL;DR

This paper introduces the Learned Expectation Collapser, a neural framework that efficiently estimates the expected value of solutions to stochastic partial differential equations without discretization, demonstrating promising results across various configurations.

Contribution

The paper presents a novel neural network approach, LEC, that approximates SPDE expectations using randomized sampling, bypassing traditional domain discretization methods.

Findings

LEC accurately estimates expected values in lower dimensions.

Accuracy decreases with higher spatial dimensions.

Increased Monte Carlo sampling improves stability and robustness.

Abstract

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.