A posteriori certification for neural network approximations to PDEs

TL;DR

This paper introduces a method to rigorously estimate errors in neural network approximations to PDEs using efficient computation of Riesz representations, applicable to complex geometries and various PDE types.

Contribution

It presents a novel framework for a posteriori error certification of neural network solutions to PDEs, combining variational analysis with efficient numerical techniques.

Findings

Derived tight upper and lower error bounds for NN PDE approximations.

Applicable to complex geometries with minimal regularity assumptions.

Numerical experiments confirm the effectiveness of the error bounds.

Abstract

We propose rigorous lower and upper error bounds for neural network (NN) approximations to PDEs by efficiently computing the Riesz representations of suitable extension and restrictions of the NN residual towards geometrically simpler domains, which are either embedded or enveloping the original domain, enabling the use of fast numerical solvers. The resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behaviour of the derived upper and lower error bounds.