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

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.

💡 Research Summary

The paper addresses the challenging class of quasi‑linear elliptic interface problems in which the diffusion coefficient is discontinuous across an internal surface. Traditional mesh‑based methods (finite differences, finite volumes, immersed interface, cut‑FEM, etc.) achieve high accuracy but require conforming or sophisticated unfitted meshes, which become cumbersome for moving or highly irregular interfaces, high‑contrast media, and high‑dimensional problems. Recent mesh‑free approaches such as physics‑informed neural networks (PINNs), deep Ritz, or extreme learning machines (ELMs) avoid meshing but suffer from the non‑convexity of the residual‑minimization objective, often stagnating at a moderate error level.

The authors propose a three‑stage perturbation‑correction framework built on Local Randomized Neural Networks (LRaNNs). The computational domain Ω is split by the interface Γ into two subdomains Ω⁺ and Ω⁻. A single‑hidden‑layer randomized neural network (RaNN) is assigned to each subdomain; hidden‑layer weights and biases are drawn randomly once and then frozen, leaving only the output‑layer coefficients (α⁺, α⁻) as trainable variables. This locality allows each network to capture the possibly discontinuous solution behavior without the need for a globally expressive network.

Stage 1 – Initialization (non‑convex least‑squares).
All governing equations (the PDE in each subdomain, jump conditions on Γ, and Dirichlet boundary conditions) are assembled into a residual vector F(α). The residuals are evaluated at Monte‑Carlo collocation points in Ω⁺, Ω⁻, Γ, and ∂Ω, yielding a discrete least‑squares functional
( G(α)=\frac12|F(α)|_2^2).
Because only the output coefficients appear, the Jacobian J(α)=∂F/∂α has a block‑structured form. The authors solve the resulting non‑linear least‑squares problem with a regularized Gauss‑Newton iteration: at each iteration they form (J^T J) and solve (J^T J,δα = -J^T F) using a truncated singular‑value decomposition (SVD) to mitigate ill‑conditioning. This yields a base approximation (u_0) that already respects the interface geometry but typically stalls at an L² error of order 10⁻²–10⁻³ due to the inherent non‑convexity.

Stage 2 – Perturbation‑Correction.
To break the stagnation, the authors expand the nonlinear operator around the base solution (u_0). Keeping only the first‑order term in the perturbation leads to a linearized correction equation for a new output vector β that multiplies the same hidden‑layer basis functions. Crucially, the correction residual becomes convex in β, turning the subproblem into a standard linear least‑squares problem that can be solved directly (or with a few Gauss‑Newton steps). Because the correction equation is derived from a Taylor expansion, the truncation error can be controlled by the order of expansion; the authors work with first order, which already yields dramatic error reduction.

Stage 3 – A‑posteriori Error Estimate.
The paper provides a rigorous a‑posteriori bound for the total generalization error in the H¹‑seminorm:
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