Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
Abstract
We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative L2 and L∞ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
Related
- AKAPINN: Adaptive Kolmogorov-Arnold Physics-Informed Neural Networks for approximating solutions to quasilinear partial differential equations
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- Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
BibTeX
@misc{le2026kolmogorovarnold,
title = {{Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations}},
author = {Tan Phuong Dong Le},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02084},
eprint = {2610.02084},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02084},
}
RIS
TY - UNPB
TI - Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
AU - Tan Phuong Dong Le
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.02084
UR - https://arxiv.org/abs/2610.02084
AB - We study free-boundary problems within a physics-informed framework using Kolmogorov-Arnold network (KAN) approximations. The proposed approach incorporates obstacle constraints, partial differential equation (PDE) inequalities, complementarity conditions, and boundary conditions through residual-based loss functions. We consider a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. The proposed KAN solver is compared with physics-informed neural network (PINN) and residual-network baselines. Numerical experiments show that KANs achieve low relative L^2 and L^∞ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
ER -