DeepSPoC: A Deep Learning-Based PDE Solver Governed by Sequential Propagation of Chaos

TL;DR

DeepSPoC introduces a novel deep learning framework that leverages sequential propagation of chaos to efficiently solve high-dimensional mean-field stochastic differential equations and nonlinear Fokker-Planck equations.

Contribution

The paper develops deepSPoC, combining SPoC theory with deep neural networks, including adaptive spatial methods, and provides convergence analysis and error estimation.

Findings

DeepSPoC effectively solves high-dimensional mean-field equations.

The method achieves improved accuracy with adaptive spatial techniques.

Experimental results demonstrate the method's versatility across various equations.

Abstract

Sequential propagation of chaos (SPoC) is a recently developed tool to solve mean-field stochastic differential equations and their related nonlinear Fokker-Planck equations. Based on the theory of SPoC, we present a new method (deepSPoC) that combines the interacting particle system of SPoC and deep learning. Under the framework of deepSPoC, two classes of frequently used deep models include fully connected neural networks and normalizing flows are considered. For high-dimensional problems, spatial adaptive method are designed to further improve the accuracy and efficiency of deepSPoC. We analysis the convergence of the framework of deepSPoC under some simplified conditions and also provide a posterior error estimation for the algorithm. Finally, we test our methods on a wide range of different types of mean-field equations.