Neural Network Dual Norms for Minimal Residual Finite Element Methods

Abstract: Minimal-residual methods for PDEs with a residual in a dual space are non-trivial to guarantee stability. We present a minimal-residual finite element method in which the solution space is a standard finite element space, but neural networks are used as test functions for the evaluation of residual dual norms. The use of a neural network improves the approximation of the residual representer, and thereby improves the stability of the method. Our hybrid approach is implemented through a deep residual Uzawa algorithm that alternates finite element updates with neural network training. We prove consistency and convergence results for the Uzawa methodology. We also prove an a priori error estimate that relies on a suitable Fortin compatibility condition. Numerical experiments on advection-reaction problems with singular or discontinuous data show that the proposed framework delivers robust and accurate approximations.