PaperlayerView exact revision on arXiv

Source-linked AI summary

Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations

Tan Phuong Dong Le

arXiv:2610.02084v1math.NAcs.LGcs.NA

TL;DR

Free-boundary PDE solvers must approximate both the solution and the associated contact or moving-interface geometry while satisfying the governing constraints. This paper uses KANs with residual-based physics-informed losses for obstacle and Stefan problems. In the tested examples, KANs resolve stationary contact regions and moving interfaces and outperform PINN and ResNet baselines on relative errors and parameter efficiency.

  • Problem

    Free-boundary PDEs require a numerical method to approximate both the solution and the location of contact regions or moving interfaces while satisfying governing constraints.

    Source: §1.1 · p. 1

  • Method

    The framework represents solutions with KANs and uses residual-based losses to enforce PDE, boundary, obstacle, complementarity, and applicable interface conditions.

    Source: §1 · p. 1, §1.3 · p. 2, §2.1 · p. 3

  • Results

    In the tested examples, KANs accurately approximate solutions and resolve stationary contact regions and moving interfaces, with lower relative errors and improved parameter efficiency than PINN and ResNet baselines.

    Source: §5 · p. 19

  • Takeaways & Limitations

    The tested results support KANs as an effective alternative for approximating free-boundary PDE solutions and their contact regions or moving interfaces.

    Source: §5 · p. 19

    The stated formulations cover the linear elliptic obstacle problem and a one-phase Stefan heat equation on a moving domain.

    Source: §2.2 · p. 4, §2.3 · p. 5

Abstract

from arXiv · show

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.

1 Introduction

Free-boundary PDEs require approximating both the solution and the contact or moving-boundary geometry while enforcing governing constraints. This work combines KAN representations with residual-based physics-informed losses and evaluates them on obstacle and Stefan problems against PINN and ResNet baselines.

1.1 Background and Motivation

  • Obstacle and Stefan problems require approximating both the solution and free-boundary geometry while satisfying governing constraints and interface conditions.The contact set is where an obstacle solution coincides with the obstacle function.Source: §1.1 · p. 1

1.2 Related Work

  • Classical free-boundary methods may require mesh refinement and explicit interface localization, while PINN losses must encode inequalities, complementarity, and interface geometry.KANs replace fixed node activations with learnable univariate edge functions, whose local flexibility may help resolve interfaces.Source: §1.2 · p. 2

1.3 Contributions

  • Residual-based losses enforce governing PDEs, boundary conditions, obstacle constraints, complementarity conditions, and moving-interface conditions.The framework combines these constraints with KAN representations.Source: §1.3 · p. 2
  • The framework is evaluated on elastic obstacle, p-Laplacian obstacle, and Stefan problems, with accuracy and parameter efficiency compared against PINN and ResNet baselines.The evaluation covers both stationary obstacle problems and a time-dependent Stefan problem.Source: §1.3 · p. 2

2 Problem Statement

The paper formulates KAN solution representations for stationary and time-dependent problems, then states the obstacle and one-phase Stefan constraints they must satisfy. The KAN uses learnable univariate edge functions, while the PDE formulations specify contact conditions and moving-interface behavior.

2.1 Kolmogorov-Arnold Network Architecture

  • KANs approximate solutions by composing layers of learnable univariate edge functions, parameterized by B-splines; refining the spline grid increases local resolution.For time-dependent problems, the input includes time as well as spatial coordinates.Source: §2.1 · p. 3, Figure 1 · p. 4

2.2 Elliptic Obstacle Problem

  • The elliptic obstacle problem requires the solution to lie above the obstacle, satisfy a PDE inequality, complementarity, and prescribed boundary data.In the contact set the solution equals the obstacle; outside it, the complementarity condition gives the linear PDE equality.Source: §2.2 · p. 4

2.3 Stefan Problem: Time-Dependent Free Boundary

  • The one-phase Stefan problem couples a temperature field on a moving domain to an interface represented by a free-surface function.The interface satisfies a zero-temperature condition and a Stefan condition coupling its normal velocity to heat flux.Source: §2.3 · pp. 4–5

3 Methodology

The methodology trains KAN approximations with residual-based losses to enforce the governing PDE and the problem’s obstacle, boundary, initial, and interface conditions. It applies this framework to stationary obstacle problems and a time-dependent Stefan problem.

3 Methodology

  • Adam training minimizes residual-based losses, with automatic differentiation supplying the spatial and temporal derivatives used in the residuals.The losses enforce the PDE, boundary or initial conditions, obstacle and complementarity constraints, and applicable interface conditions.Source: §3 · p. 5

3.1 Residual-Based Loss Formulation

  • A boundary mask makes the stationary trial solution satisfy the prescribed Dirichlet data exactly on the boundary.The representation uses the boundary data together with a KAN output multiplied by a mask that vanishes on the boundary.Source: §3.1 · p. 5
  • For the elastic obstacle problem, separate loss terms enforce the PDE inequality, obstacle constraint, and complementarity condition.Positive weights balance the PDE-inequality, obstacle, and complementarity losses.Source: §3.1 · pp. 5–6
  • For the nonlinear p-Laplacian obstacle problem, the formulation replaces the Laplacian residual with the p-Laplacian residual in the inequality and complementarity losses.The corresponding total loss is formed using those replaced residuals.Source: §3.1 · p. 6
  • For the Stefan problem, separate KANs represent temperature and the moving interface, which defines the predicted domain for the heat-equation residual.The total loss weights the heat-equation, initial-condition, interface-temperature, and Stefan free-boundary terms.Source: §3.1 · pp. 6–7

3.2 Algorithmic Frameworks

  • The algorithms train stationary obstacle solvers with either obstacle residual and jointly update the temperature and interface networks for the Stefan problem.For the Stefan problem, collocation points are evaluated in the current predicted domain and the networks are updated together.Source: §3.2 · pp. 7, 9

4 Numerical Experiments

The experiments assess KAN solution accuracy, residual convergence, free-boundary resolution, and parameter efficiency on two obstacle problems and a time-dependent Stefan problem. KANs closely approximate the tested solutions and interfaces, while the reported capacity comparisons favor KANs over the tested baselines.

4 Numerical Experiments

  • The experiments evaluate solution accuracy, residual-loss convergence, and contact-region or moving-boundary resolution, comparing KANs with PINN and ResNet baselines for accuracy and parameter efficiency.For time-dependent problems, norms are evaluated over the space-time domain or at selected snapshots.Source: §4 · p. 9

4.1 Radially Symmetric Obstacle Problem

  • 4.858 × 10^-5 relative L2 error and 6.517 × 10^-5 relative L∞ error accompany close agreement with the exact elastic obstacle solution; KANs also achieve lower relative L2 errors with fewer parameters than PINN and ResNet.The learned solution resolves the circular contact region.Source: Figure 3 · p. 11, §4.1 · p. 12, Figure 4 · p. 12

4.2 p-Laplacian Obstacle Problem; 4.3 Time-dependent Stefan Problem

  • Across the obstacle and Stefan spline-grid ablations, increasing grid resolution or network width does not guarantee lower error, and the Stefan results support selecting both jointly.The smallest reported errors are for width 8 at G = 15 in the p-Laplacian ablation and width 32 at G = 5 in the Stefan ablation.Source: §4.1 · p. 12, §4.2 · p. 13, §4.3 · p. 18

4.2 p-Laplacian Obstacle Problem

  • KAN's final relative L2 and L∞ errors on the p-Laplacian obstacle problem are 8.818 × 10^-4 and 2.836 × 10^-3, with remaining error concentrated mainly near the free boundary.The errors are largest where the solution transitions between contact and non-contact regions.Source: §4.2 · p. 13, Figure 5 · p. 14

4.3 Time-dependent Stefan Problem

  • 5.966 × 10^-4 final relative L2 error accompanies a Stefan temperature field whose predicted interface closely follows the exact interface; interface errors are approximately 10^-5 to 10^-2.Across snapshots, the largest temperature errors occur near the moving interface and reach approximately order 10^-2 at t = 0.8.Source: Figure 7 · p. 17, §4.3 · p. 18
  • KAN models reduce mean relative L2 error by about one order of magnitude versus the [23] baseline in the Stefan capacity comparison, while using fewer trainable parameters.The baseline widths 32, 64, and 100 yield mean errors of 3.7 × 10^-2, 3.2 × 10^-2, and 3.1 × 10^-2; KAN widths 16, 32, and 48 yield 3.6 × 10^-3, 3.7 × 10^-3, and 5.8 × 10^-3.Source: Figure 8 · p. 18, §4.3 · p. 18

5 Conclusion

The conclusion reports that KAN models accurately approximate solutions and resolve stationary contact regions and moving interfaces in the tested free-boundary PDEs. It also identifies future work in adaptive spline refinement, more complex geometries, higher dimensions, and theoretical error analysis.

5 Conclusion

  • In the tested examples, KANs achieve lower relative errors and improved parameter efficiency than the PINN and ResNet baselines.The conclusion also reports accurate approximation of solutions and resolution of stationary contact regions and moving interfaces.Source: §5 · p. 19
  • The framework represents solutions with learnable univariate edge functions and enforces governing equations and free-boundary constraints through residual-based losses.Source: §5 · p. 19
  • The evaluation covers the elastic obstacle problem, nonlinear p-Laplacian obstacle problem, and time-dependent one-phase Stefan problem.Source: §5 · p. 19
  • Increasing spline grid size or network width does not necessarily improve accuracy in the reported ablations.Source: §5 · p. 19
  • Future work will consider adaptive spline refinement, more complex geometries, higher-dimensional free-boundary problems, and theoretical error analysis.Source: §5 · p. 19
  • The study used no external datasets, and its source code and analysis scripts are publicly available.Source: §5 · p. 19

A Acronyms and General Notations

The appendix summarizes abbreviations and mathematical notation used in the paper.

A Acronyms and General Notations

  • The appendix summarizes the paper's abbreviations and mathematical notation, with Table 1 listing principal symbols used in formulations, losses, and experiments.Source: A Acronyms and General Notations · p. 22

List of General Notations

Table 1 lists notation used in the paper.

List of General Notations

B Hyperparameter Settings in the Numerical Experiments

The appendix records experimental settings and implementation details, including the default optimizer and derivative computation.

B Hyperparameter Settings in the Numerical Experiments

  • Unless otherwise stated, models use Adam and residual derivatives are computed with automatic differentiation.The appendix also summarizes problem settings, architectures, collocation-point sampling, and optimization hyperparameters.Source: B Hyperparameter Settings in the… · p. 22

B.1 Obstacle Problem Experiments

The obstacle experiments consider two stationary benchmarks, the EOP and nonlinear PLOP, with their problem and training settings listed in Table 2.

B.1 Obstacle Problem Experiments

  • The stationary obstacle benchmarks are the EOP and nonlinear PLOP, with problem and training settings listed in Table 2.Source: §B.1 · p. 22, Table 2 · p. 23

B.2 Stefan Problem Experiments

The Stefan experiments report problem and training settings for the two-dimensional one-phase Stefan problem.

B.2 Stefan Problem Experiments

  • The experimental settings cover the two-dimensional one-phase Stefan problem.Source: Table 3 · p. 24

C Comparative Study of Trainable Parameters and KAN Spline Basis Size

The study finds that KAN accuracy depends jointly on spline grid resolution and network width, rather than improving monotonically with grid size.

C Comparative Study of Trainable Parameters and KAN Spline Basis Size

  • Increasing spline grid size does not necessarily reduce relative L2 error monotonically; accuracy depends jointly on grid resolution and network width.Figure 9 covers elastic obstacle, nonlinear p-Laplacian obstacle, and Stefan problems across four network architectures.Source: Figure 9 · p. 25
Loading 2610.02084v1…