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

TL;DR

A new method is presented that combines the interacting particle system of SPoC and deep learning that combines the interacting particle system of SPoC and deep learning and also provides a posterior error estimation for the algorithm.

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.

Figures (13)

• Figure 1: The flow chart of the deepSPoC
• Figure 2: Loss v.s. relative $L^2$ error in the 1D porous medium equation.
• Figure 3: Result of 1D porous medium equation computed by deepSPoC with fully connected neural networks and loss function $L_{sq}$.
• Figure 4: Result of 3D porous medium equation by deepSPoC with fully connected neural network and loss function $L_{sq}$.
• Figure 5: Result of 5D porous medium equation computed by deepSPoC with fully connected neural networks.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• proof
• proof
• proof : Proof of Theorem \ref{['theorem']}