Shallow-KAN Based Solution of Moving Boundary PDEs

TL;DR

The paper tackles moving-boundary PDEs in phase-change problems by introducing a shallow Kolmogorov–Arnold Network (KAN) framework that directly approximates the temperature fields $u_s$, $u_\ell$ and the level-set interface $\phi$ governing $\Gamma(t)$. By enforcing PDEs, interface conditions, and the Stefan balance through physics-informed residuals and employing an interface-focused collocation strategy within a level-set formulation, the approach achieves accurate interface tracking with dramatically fewer parameters than traditional MLP-based PINNs. Key results in 1D and 2D Stefan problems demonstrate high fidelity in both temperature distributions and interface dynamics, with $R^2$ values near 0.999 and low MAEs, using architectures with tens of parameters (e.g., around 640) and without relying on measurement data. This work underlines the potential of KANs as compact, efficient solvers for moving boundary problems, offering substantial gains in interpretability and training efficiency for PDE-constrained learning.

Abstract

Kolmogorov-Arnold Networks (KANs) require significantly smaller architectures compared to multilayer perceptron (MLP)-based approaches, while retaining expressive power through spline-based activations. We propose a shallow KAN framework that directly approximates the temperature distribution T(x,t) and the moving interface $Γ(t)$, enforcing the governing PDEs, phase equilibrium, and Stefan condition through physics-informed residuals. To enhance accuracy, we employ interface-focused collocation resampling. 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.

Figures (3)

• Figure 1: KAN vs. exact solution: (a) temperature, (b) interface.
• Figure 2: Evolution of the temperature field and error distribution for $n_{coll} = 8000$ at $t=1.5$.
• Figure 3: KAN vs. exact solution: (a) temperature, (b) interface.