Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs

TL;DR

This paper introduces a data-driven deep learning method to efficiently approximate solutions to Hamilton-Jacobi-Bellman PDEs for control problems, combining supervised learning with residual minimization.

Contribution

It proposes a novel hybrid approach that uses a Riccati-based control law to generate training data and then refines the solution through residual minimization, improving accuracy and efficiency.

Findings

The method effectively avoids spurious solutions.

It reduces data inefficiency compared to supervised learning alone.

Numerical tests demonstrate improved approximation accuracy.

Abstract

A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a warm start for the minimization of a loss function based on the residual of the HJB PDE. The combination of supervised learning and residual minimization avoids spurious solutions and mitigate the data inefficiency of a supervised learning-only approach. Numerical tests validate the different advantages of the proposed methodology.