arXiv subjects

Siyuan Lang

Publications and source records attributed to Siyuan Lang.

3 recordsLinked to original sources

arXiv2026

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.

math.NA↗

arXiv2026

PCELM: Perturbation-Correction Extreme Learning Machine for the Stefan problem

For Stefan problems, characterized by moving boundaries and discontinuous coefficients due to phase changes, the inherent nonconvexity of the objective functional frequently causes optimization difficulty in randomized neural network approximations; to address this, we propose a Perturbation-Correction Extreme Learning Machine (PCELM) framework, built upon the extreme learning machine framework. This method first establishes a basic approximation during an initialization step by minimizing the original nonconvex residual, typically achieving only moderate accuracy, and then, in a subsequent correction step, determines a correction term by solving a subproblem based on a perturbation expansion around this basic approximation, thereby transforming it into a convex optimization problem for the output coefficients that ensures rapid convergence. We further provide a rigorous a convexity analysis, demonstrating that PCELM method solves a convex sub-problem. Numerical experiments on various Stefan problems, including multi-phase and multi-dimensional Stefan problems, confirm that the proposed PCELM method successfully overcomes optimization plateaus, with the correction step consistently delivering a significant improvement of 2-6 orders of magnitude in the relative L2 accuracy.

math.NA↗

arXiv2026

A Predictor Corrector Convex Splitting Method for Stefan Problems Based on Extreme Learning Machines

Solving Stefan problems via neural networks is inherently challenged by the nonlinear coupling between the solutions and the free boundary, which results in a non-convex optimization problem. To address this, this work proposes an Operator Splitting Method (OSM) based on Extreme Learning Machines (ELM) to decouple the geometric interface evolution from the physical field reconstruction. Within a predictor-corrector framework, the method splits the coupled system into an alternating sequence of two linear and convex subproblems: solving the diffusion equation on fixed subdomains and updating the interface geometry based on the Stefan condition. A key contribution is the formulation of both steps as linear least-squares problems; this transforms the computational strategy from a non-convex gradient-based optimization into a stable fixed-point iteration composed of alternating convex solvers. From a theoretical perspective, the relaxed iterative operator is shown to be locally contractive, and its fixed points are consistent with stationary points of the coupled residual functional. Benchmarks across 1D to 3D domains demonstrate the stability and high accuracy of the method, confirming that the proposed framework provides a highly accurate and efficient numerical solution for free boundary problems.

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