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

2026-03-19

AI Summary

Existing neural PDE solvers lack rigorous theoretical guarantees linking residual errors to solution-space errors, making their generalization performance difficult to quantify. This work establishes a unified theoretical framework that, under the assumption of a compact solution subset, leverages functional space analysis, generalization theory, and probabilistic inequalities to derive, for the first time, explicit and certifiable deterministic and probabilistic generalization bounds relating pointwise collocation residuals, initial condition errors, and boundary condition errors to the overall solution error. By bridging this gap, the study fills a critical theoretical void in physics-informed neural networks regarding solution error control and provides rigorous convergence and reliability guarantees for residual-based training methodologies.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

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.

Problem

• generalization error
• physics-informed neural networks
• partial differential equations
• residual error
• uncertainty quantification

Innovation

• error certification
• physics-informed neural networks
• generalization bounds
• residual control
• PDE solvers