aiqbench analysis
This research establishes rigorous theoretical bounds connecting residual minimization in physics-informed neural networks to explicit solution error guarantees, addressing a critical gap in uncertainty quantification for PDEs. By proving that vanishing residual error implies convergence within compact subsets of the solution space, the work provides deterministic and probabilistic certification methods essential for deploying AI solvers in safety-critical scientific applications.
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
arXiv:2603.19165v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Analysis of PDEs (math.AP); Functional Analysis (math.FA)
arXiv:2603.19165 [cs.LG]
(or arXiv:2603.19165v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2603.19165 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 224 words
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