A coupled Kolmogorov-Arnold Network and Level-Set framework for evolving interfaces

TL;DR

This paper introduces a novel shallow Kolmogorov-Arnold Network combined with a Level-set method to efficiently solve complex moving boundary PDEs, accurately modeling temperature and interface evolution without relying on measurement data.

Contribution

The work presents the first integration of KANs with level-set methods for moving boundary problems, demonstrating high accuracy and efficiency in physical simulations.

Findings

Accurately reconstructs temperature fields and interface dynamics in 1D and 2D.

Validates the model with semi-infinite analytical solutions.

Extends the framework to 2D interface propagation.

Abstract

Kolmogorov-Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)-based approaches, while retaining expressive power through spline-based activations. Moving boundary problems are ubiquitous in physical systems, whose numerical solutions are quite complex. We propose a shallow KAN framework combined with a Level-set formulation that directly approximates the temperature distribution $T(\mathbf{x},t)$ and the moving interface $\Gamma(t)$, enforcing the governing PDEs, phase equilibrium, and Stefan condition through physics-informed residuals. Numerical experiments in one and two dimensions show that the framework achieves accurate reconstructions of both temperature fields and interface dynamics, highlighting the potential of KANs as a compact and efficient alternative for moving boundary PDEs. First, we validate the model with semi-infinite analytical solutions. Subsequently, the model is extended to 2D using a level-set based formulation for interface propagation, which is solved within the KAN framework. This work demonstrates that KANs are capable of solving complex moving boundary problems without the need for measurement data.