Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks

2026-01-27

AI Summary

This work addresses the lack of rigorous error bounds in physics-informed neural networks (PINNs), which hinders their reliability for trustworthy scientific computing. The authors propose a novel “learn-and-verify” framework that, for the first time, endows PINNs with mathematically provable a posteriori error bounds. By integrating a newly designed doubly smoothed maximum (DSM) loss function with interval arithmetic, the method generates machine-verifiable certificates that rigorously bound the neural network approximation of solutions to differential equations. The approach successfully constructs sharp, guaranteed enclosures around the true solutions of nonlinear ordinary differential equations featuring time-varying coefficients and finite-time blow-up, thereby demonstrating its effectiveness and robustness in delivering certified, reliable results.

Technology Category

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

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems. However, unlike classical numerical methods that offer established convergence guarantees, neural network-based approximations typically lack rigorous error bounds. Furthermore, the non-deterministic nature of their optimization makes it difficult to mathematically certify their accuracy. To address these challenges, we propose a"Learn and Verify"framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations. By combining a novel Doubly Smoothed Maximum (DSM) loss for training with interval arithmetic for verification, we compute rigorous a posteriori error bounds as machine-verifiable proofs. Numerical experiments on nonlinear Ordinary Differential Equations (ODEs), including problems with time-varying coefficients and finite-time blow-up, demonstrate that the proposed framework successfully constructs rigorous enclosures of the true solutions, establishing a foundation for trustworthy scientific machine learning.

Problem

• Physics-Informed Neural Networks
• rigorous verification
• error bounds
• differential equations
• scientific machine learning

Innovation

• Physics-Informed Neural Networks
• rigorous error bounds
• interval arithmetic
• Doubly Smoothed Maximum loss
• verified scientific machine learning