Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs

Key points

• 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.1
• 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 $\mathbb{T}^{4}$.1
• Two parameterizations that receive the geodesic distance do substantially better: a single network fed the wrapped displacement and its distance recovers the viscosity solution in all 30 runs, while a minimum over periodic image charts succeeds in 27 of 30 and introduces a wrong-image failure mode.1
• Finally, for the convex Hamiltonian considered here, we use the semiconcavity of the viscosity solution to build a mesh-free finite-difference test that needs only forward network evaluations; it detects the folded solutions and the harmful min-of-charts failures without access to the exact value function.1

Sources (1)

• [1]Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 19, 07:29 PM

  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.

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