Title: Epigraph-Based Multigrid PINNs for Two-Player General-Sum Differential Games
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: Differential games with state constraints, epigraph technique, physics-informed neural networks, safe control

Abstract: In continuous-time, noncooperative, safety-critical settings, players with distinct goals and shared state constraints are naturally modeled by two-player general-sum differential games with state constraints. Solving such games requires numerically computing the Nash equilibrium of coupled Hamilton–Jacobi (HJ) PDEs, but these computations suffer from the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a scalable alternative, yet the resulting control is derived by maximizing the Hamiltonian with respect to gradients of the learned value function. Consequently, inaccurate value approximations can result in unsafe closed-loop control policies. Prior attempts to improve the accuracy of learned values have relied on supervised data, but these data are costly to obtain and might be unavailable in high-dimensional settings. To overcome these challenges, we propose Epigraph-based Multigrid PINNs (EMP), a fully self-supervised framework that eliminates dependence on supervised data. EMP introduces a learned value-driven rollout sampling strategy that leverages informative values and their gradients along closed-loop trajectories for PINN training, and applies multigrid refinement to improve the accuracy of value approximations. Together, these components yield safe and scalable control policies. Evaluations on 5D and 9D vehicle systems and a 13D drone system demonstrate that EMP achieves lower collision rates than all epigraph-based baselines under comparable budgets, highlighting its effectiveness for safety-critical multi-agent interactions without supervised data.

TL;DR: EMP is a fully self-supervised framework that combines epigraph-based PINNs, value gradients dynamics, and multigrid refinement to solve two-player general-sum differential games with state constraints.