Scalable Safety Analysis and Control Synthesis for Nonlinear Differential Games

Abstract: Hamilton-Jacobi (HJ) Reachability analysis is a powerful tool for solving differential games with bounded inputs; it can provide safety and liveness guarantees for each player and the corresponding optimal control law. However, control-theoretic approaches to solving nonlinear differential games struggle with the ​“curse of dimensionality.” We will explore two methods for overcoming this issue, one with conservative guarantees and one with probabilistic guarantees:

1. Recently, the applied math community has been exploring the use of the Hopf formula for efficiently solving linear differential games with bounded inputs via Alternating Direction Method of Multipliers. We will show how we can lift a nonlinear game to a linear space wherein we can bound the linearization error. We can then treat this error as an adversary in a linear game solvable by the Hopf formula, with results that can map back to the original space for conservative guarantees on the true nonlinear game.

2. More recently, solving HJ reachability using model-based supervised learning via physics-informed neural networks (PINNs) has become increasingly popular because the computation time scales with complexity rather than dimensionality. However, these PINNs suffer from learning errors and catastrophic forgetting. We will show how linear supervision (e.g. from the same Hopf formula above) of the nonlinear game can significantly improve the learning results, resulting in tighter probabilistic guarantees. This work was recently nominated for the Best Paper Award at the Learning for Dynamics and Control conference.

Bio: Sylvia Herbert is an assistant professor of Mechanical and Aerospace Engineering at the University of California, San Diego.

See all upcoming talks at: https://​www​.anl​.gov/​m​c​s​/​l​a​n​s​-​s​e​m​inars.