A Perturbation-Correction Method Based on Local Randomized Neural Networks for Quasi-Linear Interface Problems

TL;DR

This paper introduces a perturbation-correction method using Local Randomized Neural Networks to effectively solve quasi-linear interface problems with discontinuous coefficients, overcoming optimization stagnation and significantly improving accuracy.

Contribution

The paper presents a novel perturbation-correction framework based on LRaNNs that transforms a nonconvex optimization into a convex problem, ensuring rapid convergence and improved accuracy.

Findings

Achieves 4-6 orders of magnitude improvement in L^2 accuracy.

Effectively overcomes optimization stagnation in complex interface problems.

Provides a rigorous a posteriori error estimate for the method.

Abstract

For quasi-linear interface problems with discontinuous diffusion coefficients, the nonconvex objective functional often leads to optimization stagnation in randomized neural network approximations. This paper Proposes a perturbation-correction framework based on Loacal Randomized Neural Networks(LRaNNs) to overcome this limitation. In the initialization step, a satisisfactory based approximation is obtained by minimizing the original nonconvex residual, typically stagnating at a moderate accuracy level. Subsequently, in the correction step, a correction term is determined by solving a subproblem governed by a perturbation expansion around the base approximation. This reformulation yields a convex optimization problem for the output coefficients, which guarantees rapic convergence. We rigorously derive an a posteriori error estitmate, demonstrating that the total generalization error is governed by the discrete residual norm, quadrature error, and a controllable truncation error. Numerical experiments on nonlinear diffusion problems with irregular moving interfaces, gradient-dependent diffusivities, and high-contrast media demonstrate that the proposed method effectively overcomes the optimization plateau. The correction step yields a significant improvement of 4-6 order of magnitude in L^2 accuracy.