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

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.

💡 Research Summary

The paper addresses a critical gap in the emerging field of physics‑informed neural networks (PINNs): the lack of mathematically rigorous error bounds for the solutions they produce. While classical numerical methods such as finite‑difference or finite‑element schemes come with well‑established convergence theory, PINNs are trained by stochastic optimization and therefore provide no a‑posteriori guarantees. To bridge this gap, the authors propose a two‑phase “Learn and Verify” framework that upgrades any PINN approximation into a provably correct enclosure of the true solution of an ordinary differential equation (ODE).

In the Learn phase, three neural networks are trained simultaneously. A primary network (\hat u_\theta(t)) approximates the solution in the usual PINN fashion. Two auxiliary networks with non‑negative outputs, (v_\theta(t)) and (w_\theta(t)), are then trained to construct a sub‑solution (u_\theta(t)=\hat u_\theta(t)-v_\theta(t)) and a super‑solution (\bar u_\theta(t)=\hat u_\theta(t)+w_\theta(t)). The key innovation is the Doubly Smoothed Maximum (DSM) loss, which provides a smooth approximation of the maximum violation of the differential inequalities \