Recursive Regression with Neural Networks: Approximating the HJI PDE Solution

TL;DR

This paper introduces a neural network-based recursive regression method to approximate solutions of the HJI PDE, effectively reducing memory usage and addressing the curse of dimensionality in high-dimensional systems.

Contribution

It proposes a novel approximate dynamic programming algorithm that combines regression and minimax problems with neural networks for HJI PDE solutions.

Findings

Requires less memory than traditional grid-based methods

Successfully tested on systems with up to six dimensions

Addresses curse of dimensionality in HJI PDE approximation

Abstract

The majority of methods used to compute approximations to the Hamilton-Jacobi-Isaacs partial differential equation (HJI PDE) rely on the discretization of the state space to perform dynamic programming updates. This type of approach is known to suffer from the curse of dimensionality due to the exponential growth in grid points with the state dimension. In this work we present an approximate dynamic programming algorithm that computes an approximation of the solution of the HJI PDE by alternating between solving a regression problem and solving a minimax problem using a feedforward neural network as the function approximator. We find that this method requires less memory to run and to store the approximation than traditional gridding methods, and we test it on a few systems of two, three and six dimensions.