Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
This paper develops a physics-informed Kolmogorov-Arnold network (KAN) solver for free-boundary PDEs, including elliptic obstacle problems and the time-dependent Stefan problem. The core idea is to replace fixed MLP node activations with learnable univariate B-splines on network edges, aiming for localized resolution of contact sets and moving interfaces. Accurate, parameter-efficient approximation of these coupled unknowns matters because classical mesh-based methods require explicit interface tracking or costly refinement, while standard PINNs may blur sharp free boundaries.
The paper presents a competently executed proof-of-concept that KANs can approximate free-boundary PDEs within a residual-based physics-informed framework. On the tested benchmarks—a radially symmetric obstacle problem, a $p$-Laplacian obstacle problem, and a one-phase Stefan problem—the KAN solver reaches low relative errors (e.g., $4.858\times 10^{-5}$ in $L^2$ for the elastic obstacle). However, the evaluation is limited to smooth, manufactured, low-dimensional solutions, and the claim of parameter efficiency rests on baseline comparisons that lack full tuning details. The authors appropriately note that simply increasing spline grid size does not guarantee better accuracy, which tempers strong architectural conclusions.
The methodology is clearly structured: the authors construct hard-boundary trial solutions via $u_{\theta}(\vec{x}) = g(\vec{x}) + \rho(\vec{x}) N_{\theta}(\vec{x})$, employ separate KANs for the Stefan temperature and interface, and use automatic differentiation for all residual terms. The reported errors on exact benchmarks are genuinely low, with the radial obstacle reaching $4.858\times 10^{-5}$ and the Stefan temperature field $5.966\times 10^{-4}$. The open-source code and exhaustive hyperparameter tables in Appendix B support independent verification.
A fundamental limitation is that obstacle inequalities, complementarity, and Stefan conditions are enforced only through soft penalty losses such as $\mathcal{L}_{\mathrm{EOP}}(\theta)=\lambda_{\mathrm{pde}}\mathcal{L}_{\mathrm{pde}}+\lambda_{\mathrm{obs}}\mathcal{L}_{\mathrm{obs}}+\lambda_{\mathrm{comp}}\mathcal{L}_{\mathrm{comp}}$, so constraints are satisfied only approximately and sensitivity to the weights is not analyzed.\n\nThe experiments are further restricted to two-dimensional manufactured solutions with simple graph interfaces ($x_1=s(x_2,t)$), leaving open whether KANs can handle topology changes or noisy data. The ablation study also reveals non-monotonic behavior—errors at $G=20$ can be an order of magnitude larger than at $G=5$—which undercuts the intuition that finer splines uniformly improve free-boundary resolution and complicates model selection.
The numerical evidence supports the claim that the KAN solver reaches low error on the specific benchmarks tested, and the box-whisker plots in Figures 4 and 8 suggest favorable parameter efficiency relative to PINN and ResNet baselines for the obstacle problem. However, the Stefan comparison against Wang & Perdikaris reports much larger mean errors and huge variance for the baseline ($3.1\times 10^{-2}$ with standard deviation $1.4\times 10^{-1}$ for width 100), yet the paper does not state whether identical collocation budgets, loss weights, or training schedules were used, making the comparison suggestive rather than conclusive. The paper also omits comparisons with classical methods such as adaptive finite elements or level-set schemes, which remain the dominant practical approaches for these problems.
Reproducibility is moderately strong: the author provides a public GitHub repository and Appendix B lists exhaustive hyperparameter settings including network widths, spline grid sizes, learning rates, and point counts. Still, random seeds, hardware specifications, and wall-clock training times are absent, and the exact KAN implementation library (e.g., pykan or a custom build) is not stated. Because the ablation study shows that reproducing the best results requires careful joint selection of width and grid size—at $G=20$ errors can rise sharply—independent reproduction will require extensive tuning. Without reported timings, it is also unclear whether the parameter-efficiency gains offset any increased per-iteration cost of B-spline derivative evaluations.
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.
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