Robust Deep FOSLS for Transmission Problems

TL;DR

This paper introduces a robust deep learning framework based on a weighted FOSLS formulation for solving transmission problems in heterogeneous media, maintaining accuracy even with high material contrast.

Contribution

It develops an energy-norm FOSLS approach with provable equivalence and variance reduction, improving neural network performance on discontinuous problems.

Findings

The method remains stable under high material contrast.

Gradient variance decreases as the loss diminishes.

Numerical experiments confirm effectiveness in 1D and 2D settings.

Abstract

This work presents a robust, energy-based deep learning framework for solving transmission problems in heterogeneous media, including cases with discontinuous material scenarios. We introduce a weighted First-Order System Least-Squares (FOSLS) formulation involving an energy-norm Poincar\'e constant and prove its equivalence to a natural energy norm of the underlying equations, with constants independent of material parameters. As a result, the optimization landscape remains aligned with a meaningful error approximation even under high material contrast, where standard neural network losses often deteriorate. We further prove that the FOSLS formulation, together with its integral-loss representation, exhibits a passive variance reduction property, whereby the gradient variance progressively decreases as the loss diminishes, in contrast to methods such as VPINNs and Deep Ritz. From a numerical standpoint, we adopt a reduced-order perspective by constructing a low-dimensional space described by a neural network. The optimal coefficients are computed via a least-squares solver, and the space is subsequently improved through gradient-based updates. By selecting the activation function ReQU, the method mitigates the spurious overshoots typically observed in smooth networks when approximating discontinuities. Numerical experiments in 1D and 2D interface settings corroborate these findings.