Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
Uncertainty quantification for partial differential equations is traditionally grounded in discretization theory, where solution error is controlled via mesh/grid refinement. Physics-informed neural networks fundamentally depart from this paradigm: they approximate solutions by minimizing residual losses at collocation points, introducing new sources of error arising from optimization, sampling, representation, and overfitting. As a result, the generalization error in the solution space remains an open problem. Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error. We prove that when neural approximations lie in a compact...
| Authors | Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fernández et al. |
|---|---|
| Primary category | cs.LG |
| Published | 19 March 2026 |
| Last updated at source | 1 September 2026 |
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