DeepSPoC: A Deep Learning-Based PDE Solver Governed by Sequential Propagation of Chaos
This work addresses computational challenges in solving complex mean-field equations, which is important for applications in physics and finance, though it appears incremental as it builds on existing SPoC theory.
The authors tackled solving high-dimensional mean-field stochastic differential equations and nonlinear Fokker-Planck equations by proposing deepSPoC, a method combining sequential propagation of chaos with deep learning, which demonstrated improved accuracy and efficiency on various test cases.
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.