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

TL;DR

This paper introduces a 'Learn and Verify' framework that combines novel training and verification techniques to provide rigorous, mathematically certified error bounds for physics-informed neural network solutions of differential equations.

Contribution

It proposes a new framework integrating a novel loss function and interval arithmetic to produce verifiable error bounds for PINNs, enhancing their reliability.

Findings

Successfully computes rigorous solution enclosures for nonlinear ODEs

Demonstrates effectiveness on problems with time-varying coefficients

Establishes a foundation for trustworthy scientific machine learning

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.