Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
Physics-informed neural networks (PINNs) and other residual-trained solvers judge a candidate solution by the size of its sampled PDE residual. For first-order Hamilton-Jacobi equations this is not sufficient: the same equation and boundary data admit Lipschitz almost-everywhere solutions that differ from the viscosity solution by $O(1)$ while violating the equation only on sets of measure zero. We study whether residual minimization selects such solutions in a fixed-horizon pursuit-evasion game whose Hamilton-Jacobi-Isaacs value function is known in closed form on the circle $S^1$ and on the flat tori $\mathbb{T}^{2}$ and $\mathbb{T}^{4}$. We train four neural parameterizations with ten ...
| Authors | Raul Jimenez, Pablo Tejerina-Pérez, Leonid Sarieddine et al. |
|---|---|
| Primary category | math.NA |
| Published | 19 October 2026 |
| Last updated at source | 19 October 2026 |
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