KANs solve free-boundary PDEs: low relative L2 and L-infinity errors with accurate contact regions and moving interfaces on a linear elliptic obstacle problem, a p-Laplacian obstacle problem, and a one-phase Stefan problem
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Synopsis
The work embeds Kolmogorov-Arnold network (KAN) approximations in a physics-informed framework whose residual-based loss functions encode obstacle constraints, PDE inequalities, complementarity conditions, and boundary conditions, and on a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem it reports that KANs achieve low relative L2 and L-infinity errors and accurately resolve contact regions and moving interfaces compared with PINN and residual-network baselines.
Figure 4 : Model-capacity comparison for the EOP . Relative L 2 L^{2} errors are plotted against the total number of trainable parameters for PINN , ResNet , and KAN models. Across the different architectures, KAN achieves lower errors with fewer parameters, demonstrating improved parameter efficiency for the obstacle problem.
arXivInterpretation
A physics-informed free-boundary solver based on KAN approximations is proposed, with obstacle constraints, PDE inequalities, complementarity conditions, and boundary conditions all incorporated through residual-based loss functions. Physics-informed frameworks have mostly targeted forward problems on fixed domains; here the inequality and complementarity structure specific to free boundaries is written explicitly into the loss, so a KAN representation can be applied to problems with unknown contact sets or phase interfaces. The abstract describes a residual-based physics-informed framework instantiated on three problems: a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem.
On the three free-boundary test problems, the KAN solver attains low relative L2 and L-infinity errors and accurately resolves contact regions and moving interfaces. Compared with PINN and residual-network baselines, KANs are reported as an effective alternative in free-boundary settings rather than only as solvers for fixed-boundary PDEs. The evidence comes from numerical experiments using relative L2 and L-infinity errors as metrics, together with observations of contact-region and moving-interface resolution; the abstract does not give specific error values, mesh sizes, or problem scales.
The results indicate that KAN representations provide an effective alternative for solving free-boundary PDEs. The applicability of KANs is extended from ordinary function fitting and fixed-domain PDEs to free-boundary problems involving inequality constraints and complementarity conditions. This conclusion is supported by the numerical comparison across the three problems described in the abstract; it is numerical feasibility evidence rather than a proof of convergence or an error bound.
Perspective
The result is aimed at researchers and practitioners solving free-boundary PDEs within a physics-informed framework, and it applies to settings where obstacle constraints, PDE inequalities, complementarity conditions, and boundary conditions can be expressed as residual losses; the abstract covers a linear elliptic obstacle problem, a nonlinear p-Laplacian obstacle problem, and a time-dependent one-phase Stefan problem. Its value lies in demonstrating KAN resolution of contact regions and moving interfaces and in providing a starting point for transferring similar constraint-based losses to other free-boundary models.
The abstract does not report specific relative L2 and L-infinity error values, problem scales, training configurations, or ablation results, so the size of the KAN advantage over PINN and residual-network baselines and its sensitivity to hyperparameters remain hard to judge. How contact-region and moving-interface resolution accuracy is quantified, and whether it holds in higher dimensions or under stronger nonlinearity, are open questions. The abstract also does not address theoretical convergence or error bounds, so stability and scalability await follow-up work.