Discontinuous extreme learning machine for interface and free boundary problems

Abstract

We present a machine-learning framework for interface and free-boundary problems, focusing on physics-informed neural networks (PINNs). Two major challenges are addressed: (i) interface-induced discontinuities and (ii) moving boundaries inherent to free-boundary problems. To meet these challenges, we introduce the discontinuous extreme learning machine (DELM), a mesh-free method that leverages an “artificial discontinuity” mechanism, and the local extreme learning machine (locELM) architecture. Our first innovation augments the input of a single-layer neural network with two additional variables: a piecewise-constant indicator that enforces discontinuities in the solution itself, and the absolute value of a signed-distance level-set function that produces the correct gradient jump across the interface. This design captures discontinuities without splitting the network into multiple pieces or inflating the parameter count. For problems with evolving interfaces (e.g., the Stefan problem), we devise a decoupled discrete-DELM strategy that integrates seamlessly with the classical front-tracking and time-discretization technique. At each time step, the front-tracking module updates the interface geometry, and DELM subsequently solves the governing PDE in the updated domain. To further reduce complexity while maintaining accuracy, the computational domain is partitioned, and an independent single-layer ELM is trained within each subdomain. Various numerical experiments validate the proposed framework, demonstrating high accuracy and fast computational speed across a wide range of benchmark problems.