深度学习与进化算法耦合下的最优多维随机控制问题  

Solving Multi-Dimensional Optimal Stochastic Control Problems with Deep Learning and Evolution Algorithm

作　　者：张智豪 徐勉 

机构地区：[1]上海理工大学,理学院,上海 [2]河南中烟工业有限责任公司洛阳卷烟厂,河南 洛阳

出　　处：《应用数学进展》2022年第3期1222-1241,共20页Advances in Applied Mathematics

摘　　要：受困于维数诅咒,能够求解高维偏微分方程(PDEs)的算法一直以来都极其有限。鄂维南和韩劼群在2017年提出的算法通过将未知解的梯度看作策略函数,利用深度学习可以较为有效的解决高维偏微分方程,但却无法解决带有真正策略函数的问题。本文提出了一种新算法,通过多层神经网络表示策略函数映射,将方程的解映射为适应度函数,把网络中的参数看作自变量,通过进化算法优化整个策略函数;同时配合鄂维南和韩劼群的算法求解问题。通过在Riccati方程和投资消费问题等的实际算例模拟下,表明了算法的准确性和实际意义。Because of the curse of dimensionality, developing efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an extremely difficult task. The algorithm which Weinan E and Jiequn Han proposed in 2017 views the gradient of the unknown solution as policy function, and through deep learning can effectively solve high-dimensional partial differential equations, but this method cannot deal with stochastic control problems with real policy function. We propose a new algorithm for solving this problem, which use multilayer neural network to represent the map of policy function and view the parameters in the neural network as independent variables. Then, we use the evolution algorithm to optimal the policy function. At the same time, we cooperate with Weinan E and Jiequn Han’s algorithm to solve this problem. Numerical results on 5-dimensional Riccati equation and 12-dimensional Investment and Consumption Problem suggest the accuracy and practical significance of the algorithm.

关 键 词：偏微分方程 倒向随机微分方程 高维 深度学习 随机控制 进化算法 

分 类 号：O175[理学—数学]