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arXiv · 2607.07506

Structure-Guided Gauss-Newton Method: Linear Advection-Reaction Equation

Zhiqiang Cai·César Herrera
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Abstract

The least-squares neural network (LSNN) method introduced in [5] for linear advection-reaction equations is capable of accurately approximating discontinuous solutions without a priori knowledge of the interface location. However, the resulting discretization is a non-convex optimization problem that is computationally intensive and complex. In this paper, we propose a structure-guided Gauss-Newton (SgGN) method that alternates between the linear (output) and the nonlinear (hidden layer) parameters. At each outer iteration, the linear parameters are computed by a linear solver, and the nonlinear parameters are updated by a modified Gauss-Newton (GN) method that explicitly removes the singularities of the GN matrix. Numerical experiments for all test problems presented in [5] show that the SgGN method is superior to the Adam optimizer [13], the commonly used first-order optimization algorithm, not only in computational cost but, more importantly, in accuracy

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Zhiqiang Cai, César Herrera. 2026-07-09. Structure-Guided Gauss-Newton Method: Linear Advection-Reaction Equation. https://arxiv.org/abs/2607.07506

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