Certifying Physics-Informed Neural Networks through Lower Error Bounds

Arzu Ahmadova, Ismail Huseynov, Agamirza Bashirov

Keywords: Physics-informed neural networks, machine learning certification, a posteriori lower error bound, ordinary differential equations

Abstract:

Physics-informed neural networks (PINNs) bring together machine learning and physical laws to solve differential equations. Although Hillebrecht and Unger (2022) provide rigorous upper error bounds for PINN prediction error under Lipschitz continuity conditions, certification requires complementary lower bounds to establish complete error enclosures. In this work, we obtain
computable a posteriori lower bounds for PINN errors in ordinary differential equations (ODEs) under strong monotonicity conditions without prior knowledge on the true solution. This work gives fully certified a posteriori error bands for nonlinear ODEs and for linear ODEs satisfying structural assumptions, providing robust error enclosures computed after training.

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Submission Number: 43