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

For quasi-linear elliptic interface problems with discontinuous diffusion coefficients, randomized neural network approximations may exhibit stagnation because the associated objective functional is generally nonconvex.

This paper proposes a Local Randomized Neural Network (LRaNN) perturbation-correction method, denoted by LRaNN-PC, to alleviate this stagnation.

The method represents the solution using an LRaNN on each subdomain, coupled through the interface conditions in a domain-decomposed framework.

It consists of a primary stage and a perturbation-correction (PC) stage.

The primary stage computes the primary approximation by minimizing the original nonconvex objective functional.

The PC stage constructs a residual-driven correction by performing a local expansion of the residual around the primary approximation and representing the correction in an independently generated randomized trial space.

The correction coefficients are obtained by solving a least-squares residual-correction subproblem in this trial space.

For the solution-dependent quasi-linear elliptic model under the stated sufficient assumptions, we derive a residual-controlled upper bound for the broken $H^1$ seminorm error.

The bound involves the discrete residual, the quadrature error, the residual of the perturbation-correction subproblem, and the truncation remainder of the perturbation expansion.

Numerical experiments further test irregular interfaces and high-contrast coefficients within this setting, and also examine gradient-dependent diffusivities and moving-interface extensions.

In the tested benchmarks, LRaNN-PC reduces the relative $L^2$ error by up to $4$--$7$ orders of magnitude compared with the primary LRaNN stage.