Neural Network First-Order Least-Squares for Elliptic Transmission Problems
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In recent years, Deep Neural Networks (DNNs) have emerged as a new framework for solving Partial Differential Equations (PDEs). The most widely used approach, Physics-Informed Neural Networks (PINNs), minimizes pointwise residuals and therefore rely on strong regularity assumptions on the solution. These assumptions are violated in the case of transmission problems governed by the Poisson Equation. To address these issues, other formulations have been proposed. The Deep Ritz Method (DRM) minimizes an energy functional and Variational PINNs (VPINNs) minimize a weak residual projected onto a finite-dimensional test space. However, these functionals suffer from training instabilities due to stochastic integration errors. In this work, we consider a Neural Network solver based on a robust First-Order System Least Squares (FOSLS) formulation of the Poisson Problem with respect to a novel energy-consistent norm for transmission problems. Numerical experiments and theoretical results show that the proposed method is less susceptible to quadrature errors compared to DRM. Problems with discontinuous coefficients illustrate the advantages of the robust FOSLS functional with respect to existing methods such as PINNs, VPINNs, and DRM.