aiqbench analysis
This research demonstrates the efficacy of Kolmogorov-Arnold Networks (KANs) as a robust alternative to Physics-Informed Neural Networks (PINNs) for solving complex free-boundary partial differential equations. By incorporating obstacle constraints and moving interfaces through residual-based loss functions, KANs achieve superior accuracy in resolving contact regions compared to standard baselines. The study validates KANs as a promising architecture for advancing physics-informed machine learning in mathematical modeling.
Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
arXiv:2610.02084v1 Announce Type: cross 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 $L^2$ and $L^\infty$ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
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 $L^2$ and $L^\infty$ errors while accurately resolving contact regions and moving interfaces. The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2610.02084 [math.NA]
(or arXiv:2610.02084v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2610.02084 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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