Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
Abstract
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 S1 and on the flat tori 𝕋2 and 𝕋4. We train four neural parameterizations with ten seeds on each geometry, 120 runs in total. A standard network with periodic Fourier features converges to folded almost-everywhere solutions in all 30 runs: small residuals and the expected capture-layer gradient, yet values that are wrong over most of the domain. A network with periodicity imposed through the loss recovers the viscosity solution in one and two dimensions but becomes unreliable in four. 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. 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. A small sampled residual thus does not by itself validate a learned Hamilton-Jacobi solution; the parameterization and independent structural diagnostics can be essential for selecting the viscosity solution.
Related
- Pathological viscosity solutions of Hamilton--Jacobi equations
- Viscosity Solutions for Hamilton-Jacobi equations on Separable Banach Spaces
- Vanishing discount and viscosity selection problems for mechanical Hamilton--Jacobi equations
- Existence and Uniqueness of Viscosity Solutions for Stochastic Hamilton--Jacobi--Bellman Equations
- Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite Lévy measure
BibTeX
@misc{jimenez2026residuals,
title = {{Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs}},
author = {Raul Jimenez and Pablo Tejerina-Pérez and Leonid Sarieddine and Pedro Tarancón-Álvarez and Marisol Traforetti and Pau Solé-Vilaró and Oleksii Ignatenko and Nazar Tkach and Pavlos Protopapas},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27193},
eprint = {2610.27193},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27193},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
AU - Raul Jimenez
AU - Pablo Tejerina-Pérez
AU - Leonid Sarieddine
AU - Pedro Tarancón-Álvarez
AU - Marisol Traforetti
AU - Pau Solé-Vilaró
AU - Oleksii Ignatenko
AU - Nazar Tkach
AU - Pavlos Protopapas
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27193
UR - https://arxiv.org/abs/2610.27193
AB - 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 𝕋^2 and 𝕋^4. We train four neural parameterizations with ten seeds on each geometry, 120 runs in total. A standard network with periodic Fourier features converges to folded almost-everywhere solutions in all 30 runs: small residuals and the expected capture-layer gradient, yet values that are wrong over most of the domain. A network with periodicity imposed through the loss recovers the viscosity solution in one and two dimensions but becomes unreliable in four. 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. 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. A small sampled residual thus does not by itself validate a learned Hamilton-Jacobi solution; the parameterization and independent structural diagnostics can be essential for selecting the viscosity solution.
ER -