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arXiv · 2509.26122

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

Emil Haugen·Alexei Stepanenko·Anders C. Hansen
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Abstract

We present new algorithms for a posteriori verification of neural networks (NNs) approximating solutions to PDEs. We use numerical quadrature to compute upper bounds for $L^2$ norms of NNs and their derivatives. When combined with energy estimates for specific PDEs, this yields verification algorithms which only output approximations with $\varepsilon$-accuracy (in a suitable norm) with respect to the true but unknown solution of the PDE -- for any given $\varepsilon > 0$. This framework enables trustworthy algorithms for NN-based PDE solvers, regardless of training method. Such a posteriori verification is essential because a priori error bounds generally cannot guarantee the accuracy of computed solutions due to the algorithmic undecidability of the optimisation problems used to train NNs

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Emil Haugen, Alexei Stepanenko, Anders C. Hansen. 2026-09-15. Trustworthy AI in numerics: On verification algorithms for neural network-based PDE solvers. https://arxiv.org/abs/2509.26122

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