Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations

Tan Phuong Dong Le

math.NAOct 1, 2026

Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations

Authors: Tan Phuong Dong Le

Organizations: Department of Applied Mathematics, University of Waterloo, Canada · Department of Statistics and Actuarial Science, University of Waterloo, Canada

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.

Figures & tables

Panel of Figure 1 — Figure 1 : Kolmogorov-Arnold network architecture and spline-based grid refinement. Left: the KAN represents the solution using trainable univariate edge functions $\phi_{l,j,i}$ , where $a_{l,i}$ denotes the scalar value of node $i$ in layer $l$ , and $\tilde{a}_{l,j,i}=\phi_{l,j,i}(a_{l,i})$ denotes the corresponding post-activation. Each node value $a_{l+1,j}$ is obtained by summing incoming post-activations. Right: each edge function is parameterized by a univariate B-spline expansion in the scalar variable $s$ . Refining the spline grid from $G_{1}=5$ to $G_{2}=10$ increases the local approximation resolution.

u_{\theta}(\vec{x})=g(\vec{x})+\rho(\vec{x})N_{\theta}(\vec{x}),\qquad\rho|_{\partial\Omega}=0.

Algorithm 1 KAN -based solver for stationary obstacle-type problems

\Omega_{\eta^{(k)}}(t)=\left\{(x_{1},x_{2})\in\Omega:0<x_{1}<\hat{s}_{\eta^{(k)}}(x_{2},t),\ 0<x_{2}<1\right\}.

Algorithm 2 KAN -based solver for the time-dependent SP

Panel of Figure 2 — Figure 2 : Obstacle function and collocation points. Panel (a) shows the radial obstacle function $\psi$ on $\Omega=[0,1]^{2}$ . Panel (b) shows the sampled collocation points, with interior points in green and boundary points on $\partial\Omega$ in red.

Panel of Figure 3 — Figure 3 : KAN approximation for the two-dimensional elastic obstacle problem. Comparison between the exact solution and the proposed KAN approximation on $\Omega=[0,1]^{2}$ . Panels (a) and (b) show the exact solution $u_{\mathrm{exact}}$ and the learned approximation $u_{\mathrm{KAN}}$ , respectively. Panel (c) shows the absolute pointwise error $|u_{\mathrm{KAN}}-u_{\mathrm{exact}}|$ , together with the circular free-boundary interface of radius $r^{*}=0.260$ . Panel (d) reports the evolution of the total loss and its obstacle, PDE, and complementarity components during training. Panel (e) shows the relative $L^{2}$ and $L^{\infty}$ errors over the training epochs. The final relative $L^{2}$ and $L^{\infty}$ errors are $4.858\times 10^{-5}$ and $6.517\times 10^{-5}$ , respectively.

Figure 4 : Model-capacity comparison for the EOP . Relative $L^{2}$ 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.

Panel of Figure 5 — Figure 5 : Comparison of PINN, ResNet, and KAN approximations for the two-dimensional $p$ -Laplacian obstacle problem. Panels (a)–(c) show the obstacle function, exact solution, and KAN approximation. Panels (d)–(f) compare the absolute pointwise errors of the PINN , ResNet , and KAN models. The largest KAN errors are concentrated near the free boundary. Panels (g) and (h) show the training loss components and relative errors over the training epochs. The final relative $L^{2}$ and $L^{\infty}$ errors of the KAN approximation are $8.818\times 10^{-4}$ and $2.836\times 10^{-3}$ , respectively.

Panel of Figure 6 — Figure 6 : KAN approximation for the two-dimensional one-phase Stefan problem at multiple time snapshots. The four rows correspond to $t=0.2,0.4,0.6,$ and $0.8$ . The columns show the exact solution $u_{\mathrm{exact}}(x_{1},x_{2},t)$ , the KAN approximation $u_{\mathrm{KAN}}(x_{1},x_{2},t)$ , and the absolute pointwise error $|u_{\mathrm{KAN}}-u_{\mathrm{exact}}|$ . The KAN model accurately approximates the temperature field on the moving domain, with the largest errors concentrated near the evolving interface.

Panel of Figure 7 — Figure 7 : Moving free-boundary approximation and training history for the two-dimensional one-phase Stefan problem. Panels (a) and (b) show the exact and predicted moving interfaces, respectively. Panel (c) shows the logarithmic absolute interface error, $\log_{10}|s_{\mathrm{KAN}}-s_{\mathrm{exact}}|$ . Panels (d) and (e) show the training loss components and relative solution errors over the training epochs. The final relative $L^{2}$ error of the temperature field is $5.966\times 10^{-4}$ .

Figure 8 : Model-capacity comparison for the two-dimensional Stefan problem. Relative $L^{2}$ error of the temperature field $u$ versus the total number of trainable parameters for the baseline method in [ 23 ] and the proposed KAN models. The KAN models achieve lower errors with fewer trainable parameters, indicating improved parameter efficiency for this experiment.

Notation	Description
$\Omega$	Spatial computational domain
$\partial\Omega$	Boundary of the spatial domain
$\vec{x}\in\Omega$	Spatial variable
$t\in[0,T]$	Time variable
$u_{\mathrm{exact}}$	Exact solution
$u_{\theta}$	KAN approximation of the solution

Table 1 : Notations in the paper.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Quantity	EOP	PLOP
Domain	$[0,1]^{2}$	$[0,2]^{2}$
Operator	$-\Delta u$	$-\nabla\cdot(\|\nabla u\|^{p-2}\nabla u)$
Source/forcing	$f=0$	$f=-1$
Order $p$	–	$p=4$
Obstacle	Eq. ( 36 )	Eq. ( 39 )
Exact solution	Eq. ( 37 )	Eq. ( 40 )

Appendix

Table 2 : Problem and training settings for the obstacle-type BVPs .

Quantity	Setting
Space-time sampling box	$[0,2.25]\times[0,1]\times[0,1]$
Governing equation	$\partial_{t}u-\Delta u=0$
Interface temperature condition	$u(s(x_{2},t),x_{2},t)=0$
Stefan free-boundary condition	Eq. ( 42 )
Exact temperature solution	Eq. ( 44 )
Exact moving free boundary	Eq. ( 43 )

Appendix

Table 3 : Problem and training settings for the two-dimensional one-phase SP .

Figure 9 : Effect of KAN grid size and model width on obstacle-type and Stefan-type problems. The top row reports the relative

L^{2}

error versus KAN grid size for four network architectures. Panel (a) corresponds to the elastic obstacle problem, while panel (b) corresponds to the nonlinear

p

-Laplacian obstacle problem. The bottom panel (c) reports the corresponding study for the Stefan problem. The results indicate that increasing the spline grid size does not necessarily lead to a monotonic reduction in error; rather, the approximation accuracy depends jointly on the spline grid resolution and network width.

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Quantum-classical physics-informed Kolmogorov-Arnold networks for PDEs

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Parametric Physics-Informed Neural Network Partial Differential Equations Kolmogorov-Arnold Networks Porous Media Integral

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We develop QCPIKAN, the first quantum-classical physics-informed Kolmogorov-Arnold network designed to solve partial differential equations (PDEs). Built upon Chebyshev-polynomial KAN layers and parameterized quantum circuits, this hybrid framework embeds physical constraints into the training loss to enforce physical consistency. Our theoretical investigations grounded in approximation theory prove that this design accelerates high-frequency error convergence to an exponential rate and effectively mitigates numerical dispersion. We validate the framework across three typical seepage scenarios in porous media, including single-phase flow, component transport and two-phase flow. Compared with existing quantum-classical physics-informed neural networks, QCPIKAN achieves superior performance in global prediction accuracy, local error control, dynamic evolution tracking and displacement front localization. This work provides a robust and efficient alternative for solving complex PDEs.

Abstract

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