Kolmogorov-arnold networks improve solving free-boundary pdes accurately
Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
Machine Learning
Summary
Solving certain math problems called free-boundary partial differential equations (PDEs) is challenging because the boundaries of the solution region can change or move. This paper studies a new method using Kolmogorov-Arnold networks (KANs) that incorporate the problem’s physical constraints directly into the solving process. The authors compare this method to other neural network-based solvers and show that KANs achieve accurate results with low errors and can handle complex boundary conditions effectively. This offers a promising alternative way to solve these tricky PDEs.
What this means in practice
- •For engineering simulation teams: Simulate physical systems with moving boundaries more accurately using KAN-based solvers that handle interface changes and constraints effectively.
- •For computational physics developers: Develop software tools for nonlinear PDEs involving free boundaries by integrating KAN frameworks for improved numerical behavior.
Authors
Tan Phuong Dong Le
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.