Solving high dimensional optimal control problems and corresponding Hamilton-Jacobi PDEs are important but challenging problems in control engineering. In this paper, we propose two abstract neural network architectures which are respectively used to compute the value function and the optimal control for certain class of high dimensional optimal control problems. We provide the mathematical analysis for the two abstract architectures. We also show several numerical results computed using the deep neural network implementations of these abstract architectures. A preliminary implementation of our proposed neural network architecture on FPGAs shows promising speed up compared to CPUs. This work paves the way to leverage efficient dedicated hardware designed for neural networks to solve high dimensional optimal control problems and Hamilton-Jacobi PDEs.
Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
Universität Potsdam（波茨坦大学）；Freie Universität Berlin（柏林自由大学）；Brandenburgische Technische Universität Cottbus-Senftenberg（勃兰登堡工业大学科特布斯-森夫滕贝格）
Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs
Department of Mathematics, Imperial College London, South Kensington Campus（帝国理工学院伦敦数学系，南肯辛顿校区）；Department of Mathematics and Statistics, University of Konstanz（康斯坦茨大学数学与统计学系）；RICAM and Institute of Mathematics and Scientific Computing, University of Graz（格拉茨大学RICAM与数学与计算科学研究所）
Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks
提出了一种求解高维Hamilton-Jacobi-Bellman（HJB）型椭圆型偏微分方程（PDE）的新的数值方法.的HJB偏微分方程，重新制定为最优控制问题，处理的演员-评论家框架的启发，强化学习，基于神经网络参数化的价值和控制功能。在行动者-批评者框架内，我们采用了政策梯度的方法来改善控制，而对于价值函数，我们推导出一个方差减少的最小二乘时间差分法，使用随机分析。数值离散的随机控制问题，我们采用了自适应步长计划，以提高精度附近的区域边界。最后，通过对$20$空间维数的线性二次型调节器、随机Van der Pol振子、扩散Eikonal方程和完全非线性椭圆型偏微分方程的数值算例，验证了本文方法的有效性.
Policy iteration for Hamilton-Jacobi-Isaacs equations with control constraints and comparison with Hamilton-Jacobi-Bellman equations
Rajiv Gandhi Institute of Petroleum Technology（拉吉夫·甘地石油技术学院）；University of Graz（格拉茨大学）；Radon Institute for Computational and Applied Mathematics (RICAM)（雷登计算与应用数学研究所）
