AI RESEARCH
Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees
arXiv CS.LG
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ArXi:2603.19165v1 Announce Type: new 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, Our main theoretical contribution establishes generalization bounds that connect residual control to solution-space error.