RUNNs: Ritz--Uzawa Neural Networks for Solving Variational Problems
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In this work, we propose a unified neural network framework for solving strong [1], weak [2], and ultraweak formulations based on the Ritz--Uzawa iterative scheme (RUNNs). This approach presents several advantages: a) In the case of singular solutions, one can rely on weak or ultraweak formulations to solve them, unlike PINNs and Deep Ritz Method, which are restricted to smooth problems. b) It allows one to easily initialize high-frequency residuals with Sinusoidal Fourier Feature Mapping [3] to increase convergence speed. Numerical experiments on the Poisson equation demonstrate that the proposed method significantly outperforms standard approaches in two challenging scenarios: regimes with limited computational budgets for quadrature (inexact integration) and low-regularity problems involving $H^{-2}$ distributional source terms.