Pointwise Error Estimates for Numerical Physics-Informed Neural Networks

What the math gives to ML

The paper turns a sampled PINN residual into a query-specific, deterministic pointwise error certificate by reconstructing the network prediction as a compatible piecewise-linear finite-element field. For a square linear discrete PDE system, an adjoint Green vector converts the residual into the exact signed error at each query, rather than merely bounding a global norm. This is transferable as a post-training certification layer for neural PDE solvers: it can identify spatially localized failure, drive adaptive collocation, and produce intervals whose width reflects operator sensitivity at each query. The most practical first implementation is an adjoint-weighted residual estimator on a small mesh, augmented by a computable discretization-error term when comparison with the continuous PDE solution is required.