Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Siyuan Lang, Zhiyue Zhang

TL;DR

This work tackles quasi-linear elliptic and parabolic interface problems with discontinuous diffusion by combining Local Randomized Neural Networks (LRaNNs) with a Gauss–Newton initialization and a convex perturbation-correction step. The main idea is to first obtain a base, locally accurate but potentially stagnating solution, then solve a perturbation subproblem that yields a convex quadratic optimization in the perturbation coefficients, dramatically reducing the residual and improving accuracy. An a posteriori error analysis ties the overall generalization error to residual and quadrature errors while quantifying the truncation error from the perturbation expansion. Numerical experiments across moving interfaces, high-contrast media, and nonlinear diffusivities demonstrate a 4–6 order of magnitude improvement in $L^2$ accuracy, confirming robustness and efficiency of the method for challenging nonlinear interface problems.

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.

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

TL;DR

This work tackles quasi-linear elliptic and parabolic interface problems with discontinuous diffusion by combining Local Randomized Neural Networks (LRaNNs) with a Gauss–Newton initialization and a convex perturbation-correction step. The main idea is to first obtain a base, locally accurate but potentially stagnating solution, then solve a perturbation subproblem that yields a convex quadratic optimization in the perturbation coefficients, dramatically reducing the residual and improving accuracy. An a posteriori error analysis ties the overall generalization error to residual and quadrature errors while quantifying the truncation error from the perturbation expansion. Numerical experiments across moving interfaces, high-contrast media, and nonlinear diffusivities demonstrate a 4–6 order of magnitude improvement in accuracy, confirming robustness and efficiency of the method for challenging nonlinear interface problems.

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.
Paper Structure (18 sections, 89 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 18 sections, 89 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Methodology
  4. Local Randomized Neural Network (LRaNN) Architecture
  5. Discrete Least-Squares Functional
  6. Initialization Step
  7. The Perturbation–Correction Method
  8. Error Analysis
  9. Generalization error estimation
  10. Convexity Analysis of Perturbed Subproblems
  11. Numerical examples
  12. Example 1
  13. Example 2
  14. Example 3
  15. Example 4
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Two typical geometric configurations for interface problems. (a) A cut (open) interface that intersects the exterior boundary, partitioning the domain into disconnected subdomains. (b) An embedded (closed) interface located entirely within the domain, separating it into interior and exterior regions.
  • Figure 2: The structure of RaNN.
  • Figure 3: Example 1: Two-stage optimization results. Comparison of initialization-step and perturbation-step performance in terms of error reduction, residual decay, and interface accuracy.
  • Figure 4: Example 2 Results. The correction step effectively eliminates the structured residuals at the multi-material junction.
  • Figure 5: Example 3 Results. Performance on complex "plum-blossom" interfaces. Upper row: final pointwise error maps. Lower row: Error along the interface $\Gamma$ (blue) vs. the computed correction (red dashed).
  • ...and 3 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2