AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-21
Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
Physics-informed neural networks judge candidate solutions by the size of their sampled PDE residual, but for first-order Hamilton-Jacobi equations the same equation and boundary data admit solutions that differ from the viscosity solution by O(1) while violating the equation only on sets of measure zero. In a pursuit-evasion game with a closed-form value function on the circle and flat tori, a standard network with periodic Fourier features converged to folded almost-everywhere solutions in all 30 of its runs — small residuals but wrong values over most of the domain — while parameterizations fed the geodesic distance recovered the viscosity solution, one of them in all 30 of its runs. Using the semiconcavity of the viscosity solution, the authors also build a mesh-free finite-difference test requiring only forward network evaluations, which detects the folded solutions and other failures without access to the exact value function.