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

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.