Dimension-Free Differential Games via Latent Representation

We develop a spectral Latent method for Hamilton-Jacobi-Isaacs (HJI) equations arising from two-player and multi-player differential games in high dimensions. The value function \(V(t,x)\) is represented in the Latent basis with \(N^\) modes per dimension, reducing the PDE to an ODE system. For \(\rho\)-analytic value functions, we prove convergence: \(\|V - V^{N^}\| \leq C\rho^{-N^}\), independent of the state dimension \(d\). We address the key technical challenge — singular surfaces where the value function loses smoothness — via Latent Number monitoring: \(\rho(t,x)\) is computed locally and drops toward 1 near barriers, dispersal surfaces, and focal lines, enabling adaptive resolution. For multi-player differential games (\(N \geq 3\)), we combine the HJI Latent solver with the grade decomposition (Nagy, 2026b): the \(N\)-player value function is approximated by its grade-\(R^\) truncation, reducing the \(Nd\)-dimensional problem to a collection of coupled \(R^* d\)-dimensional problems. We demonstrate the method on three benchmark problems: a 3D pursuit-evasion game (exact solution available), a 12-dimensional multi-vehicle reach-avoid game, and a 50-dimensional stylized climate negotiation game. The Latent method solves the 12D problem in minutes on a single GPU; grid-based methods cannot solve it at all.