Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees

TL;DR

This paper develops theoretical bounds linking neural network residual errors to the actual solution accuracy for PDEs, enabling rigorous error certification beyond traditional discretization methods.

Contribution

It introduces the first generalization bounds that connect residual minimization in neural PDE solvers to guarantees on the true solution error.

Findings

Residual control implies convergence to the true solution.

Certified bounds translate residual, boundary, and initial errors into solution error guarantees.

Probabilistic convergence results are established.

Abstract

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 subset of the solution space, vanishing residual error guarantees convergence to the true solution. We derive deterministic and probabilistic convergence results and provide certified generalization bounds translating residual, boundary, and initial errors into explicit solution error guarantees.