Trustworthy AI in numerics: On verification algorithms for neural network-based PDE solvers

• Main Authors: Haugen, Emil, Stepanenko, Alexei, Hansen, Anders C
• Format: Journal Article
• Language: English
• Published: 30.09.2025
• Subjects: Computer Science - Numerical Analysis; Mathematics - Numerical Analysis
• DOI: 10.48550/arxiv.2509.26122

We present new algorithms for a posteriori verification of neural networks (NNs) approximating solutions to PDEs. These verification algorithms compute accurate estimates of$L^p$norms of NNs and their derivatives. When combined with residual bounds for specific PDEs, the algorithms provide guarantees of$\eps$ -accuracy (in a suitable norm) with respect to the true, but unknown, solution of the PDE -- for arbitrary$\eps >0$ . In particular, if the NN fails to meet the desired accuracy, our algorithms will detect that and reject it, whereas any NN that passes the verification algorithms is certified to be$\eps$ -accurate. This framework enables trustworthy algorithms for NN-based PDE solvers, regardless of how the NN is initially computed. Such a posteriori verification is essential, since a priori error bounds in general cannot guarantee the accuracy of computed solutions, due to algorithmic undecidability of the optimization problems used to train NNs.