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.

翻译：序列混沌传播（SPoC）是近期发展的一种用于求解平均场随机微分方程及其相关非线性Fokker-Planck方程的工具。基于SPoC理论，我们提出了一种将SPoC的交互粒子系统与深度学习相结合的新方法（deepSPoC）。在deepSPoC框架下，我们考虑了两类常用的深度模型：全连接神经网络与归一化流。针对高维问题，我们设计了空间自适应方法以进一步提升deepSPoC的精度与效率。我们在简化条件下分析了deepSPoC框架的收敛性，并为算法提供了后验误差估计。最后，我们在多种不同类型的平均场方程上对所提方法进行了测试。