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

• URL: http://arxiv.org/abs/2601.19818v1
• Date: Tue, 27 Jan 2026 17:21:33 GMT
• Title: Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks
• Authors: Kazuaki Tanaka, Kohei Yatabe,
• Abstract summary: We propose a framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations.<n>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.
• Score: 12.111053304637808
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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