Pontryagin Neural Network Solutions for Hamilton–Jacobi–Bellman Optimality of High-Order Systems

Author(s):

Bradley T. Burchett

Abstract:

Recently, Physics Informed Neural Networks (PINNs) have come to the forefront as a novel means to solve partial differential equations (PDEs). Nonlinear optimal control problems often require the solution of a nonlinear PDE to satisfy necessary conditions for optimality. PINNs can be trained to solve such a PDE; however, for high-dimensional systems, discretizing the operating space leads to very large training datasets unless current space-filling methods are used. The use of the rank one lattice method is shown to reduce the training set size. In this report, analytic gradients are derived and used to streamline the gradient-based neural network training required to solve the Hamilton–Jacobi–Bellman (HJB) PDE. These two methods provide a means of offline training to render a PINN-based solution to HJB optimal control. The method is demonstrated on a four-state nonlinear chaotic system.