Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
Amartya Mukherjee, Maxwell Fitzsimmons, David C. Del Rey Fern\'andez, Jun Liu

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…
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Taxonomy
TopicsModel Reduction and Neural Networks · Advanced Graph Neural Networks · Neural Networks and Reservoir Computing
