Title: Beyond Minimax: Structure-Aware Learning for Differential Games
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: pursuit-evasion game, calculus of variations, pontryagin's maximum principle

Abstract: A central challenge in artificial intelligence is to design agents that solve structured engineering problems, such as zero-sum differential games, without handcrafted solutions or expert demonstrations. Differential games capture multi-agent interactions with opposing objectives, where optimal strategies are defined by equilibrium conditions. Classical theory based on Pontryagin’s Maximum Principle (PMP) and the Hamilton--Jacobi--Isaacs (HJI) equations provides principled foundations, but these conditions are rarely tractable in practice. Deep learning, by contrast, offers flexible function approximation but typically ignores such structure and depends on large datasets or extensive online interactions.

We introduce a framework that embeds equilibrium conditions and terminal constraints from the calculus of variations directly into the training objective. This enables neural networks to jointly learn state, control, and costate trajectories while handling variable terminal times and manifold-constrained terminal states, yielding approximate saddle-point equilibria. We illustrate our approach with the pursuit--evasion game \emph{Lady in the Lake}, showing that our method recovers structural properties of analytical solutions and generalizes to novel scenarios without supervision, pointing toward principled, structure-aware deep models for solving previously intractable differential games.

TL;DR: We propose SAL-DG, a structure-aware deep learning framework that solve pursuit–evasion games with variable time horizons and terminal constraints, without rewards or supervision