Least squares extreme learning machine collocation method for solving elliptic interface problems with sharp corners

doi:10.1016/j.enganabound.2026.107063

Least squares extreme learning machine collocation method for solving elliptic interface problems with sharp corners

2026-09-25

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Jilong He

DOI：10.1016/j.enganabound.2026.107063

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Abstract

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Elliptic interface problems with discontinuous coefficients, nontrivial transmission conditions, and complicated geometries arise in many scientific and engineering applications. Such problems may involve curved interfaces, irregular external boundaries, and polygonal interfaces with sharp corners. Although sharp interface corners can, in general, induce local solution singularities whose behavior depends on the corner angles, interface configuration, and coefficient contrast, the present work focuses primarily on the numerical treatment of nonsmooth interface geometries and discontinuous material coefficients. In particular, the numerical examples are mainly based on piecewise smooth manufactured solutions and are intended to assess the enforcement of governing equations, boundary conditions, and interface transmission conditions on geometrically complicated domains. We propose a least-squares extreme learning machine (LS-ELM) collocation method for elliptic interface problems. The physical domain is decomposed into subdomains separated by the interface, and an independent single-hidden-layer extreme learning machine is constructed in each subdomain. The hidden-layer weights and biases are randomly generated and fixed, while only the output-layer coefficients are computed. This piecewise representation allows the numerical approximation to accommodate discontinuous coefficients and distinct solution behaviors in different subdomains. The governing differential equation, Dirichlet boundary conditions, and interface conditions are imposed at collocation points and assembled into a global linear algebraic system. The neural-network basis functions and their first- and second-order derivatives are evaluated analytically using the chain rule. Coordinate normalization is incorporated to ensure the consistent transformation of derivatives from normalized coordinates to physical coordinates. The resulting least-squares formulation includes PDE residuals within each subdomain, boundary residuals on the exterior boundary, and residuals associated with solution and flux transmission conditions across the interface. The unknown output coefficients are obtained from a regularized linear least-squares problem, thereby avoiding the iterative nonlinear optimization and repeated backpropagation required by conventional physics-informed neural networks. Numerical experiments are conducted for a variety of smooth and nonsmooth interface geometries, including curved interfaces and polygonal interfaces with sharp corners, as well as problems with high-contrast diffusion coefficients. The proposed method is compared with radial-basis-function meshless methods, physics-informed neural networks, improved physics-informed neural networks, and extreme-theory-of-functional-connections approaches. The results show that LS-ELM can provide accurate approximations while requiring substantially less computational time than gradient-based neural-network methods. In many tested cases, the method achieves accuracy comparable to, and occasionally better than, the considered reference methods. A sensitivity study is further performed to investigate the effects of activation functions, random initialization ranges, collocation-point distributions, constraint weights, and Tikhonov regularization parameters. The results indicate that the activation function and initialization scale strongly affect approximation accuracy and matrix conditioning, while appropriate weighting of boundary and interface constraints improves the enforcement of physical conditions. Regularization can alleviate numerical difficulties caused by ill-conditioned collocation matrices, although excessive regularization may introduce approximation bias and reduce accuracy. Repeated computations with different random seeds and condition-number analyses demonstrate the overall robustness of the proposed method. The results suggest that LS-ELM is an efficient and flexible meshless approach for elliptic interface problems with discontinuous coefficients and geometrically nonsmooth interfaces.

Keywords:

Elliptic interface problems

Least squares

Extreme learning machine

Sharp corners

Tikhonov regularization

Least squares extreme learning machine collocation method for solving elliptic interface problems with sharp corners

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