A Causality-Free Neural Network Method for High-Dimensional Hamilton-Jacobi-Bellman Equations

Computing optimal feedback controls for nonlinear systems generally requires solving Hamilton-Jacobi-Bellman (HJB) equations, which, in high dimensions, are notoriously difficult. Existing strategies often rely on specific, restrictive problem structures, or are valid only locally around some nominal trajectory. In this paper, we propose a data-driven method to approximate semi-global solutions to HJB equations for general high-dimensional nonlinear systems and compute optimal feedback controls in real-time. To accomplish this, we model solutions to HJB equations with neural networks (NNs) trained on data generated without discretizing the state space. Training is made more effective and data-efficient by leveraging the known problem structure and using the partially-trained NN to aid in further data generation. We demonstrate the effectiveness of our method by learning solutions to HJB equations for nonlinear systems of dimension up to 30 arising from the stabilization of a Burgers'-type partial differential equation. The trained NNs are then used for real-time optimal feedback control of these systems.