Scaled-cPIKANs: Domain Scaling in Chebyshev-based Physics-informed Kolmogorov-Arnold Networks

TL;DR

Scaled-cPIKAN integrates Chebyshev-based Kolmogorov-Arnold networks with a domain-scaling transformation to $[-1,1]^d$, enabling efficient, accurate representation of oscillatory PDEs over large spatial domains. By reformulating governing equations on the scaled domain and coupling physics-informed losses with Chebyshev expansions, the method achieves superior convergence and accuracy across diffusion, Helmholtz, Allen-Cahn, and reaction-diffusion problems in both forward and inverse settings. Across extensive experiments, Scaled-cPIKAN consistently outperforms Scaled-PINN, cPIKAN, and PINN by orders of magnitude in relative $L^2$ error and exhibits robust performance under noise and domain enlargement. This approach advances scalable scientific machine learning for high-frequency and multi-scale PDEs, with prospects for domain decomposition and adaptive scaling in future work.

Abstract

Partial Differential Equations (PDEs) are integral to modeling many scientific and engineering problems. Physics-informed Neural Networks (PINNs) have emerged as promising tools for solving PDEs by embedding governing equations into the neural network loss function. However, when dealing with PDEs characterized by strong oscillatory dynamics over large computational domains, PINNs based on Multilayer Perceptrons (MLPs) often exhibit poor convergence and reduced accuracy. To address these challenges, this paper introduces Scaled-cPIKAN, a physics-informed architecture rooted in Kolmogorov-Arnold Networks (KANs). Scaled-cPIKAN integrates Chebyshev polynomial representations with a domain scaling approach that transforms spatial variables in PDEs into the standardized domain \([-1,1]^d\), as intrinsically required by Chebyshev polynomials. By combining the flexibility of Chebyshev-based KANs (cKANs) with the physics-driven principles of PINNs, and the spatial domain transformation, Scaled-cPIKAN enables efficient representation of oscillatory dynamics across extended spatial domains while improving computational performance. We demonstrate Scaled-cPIKAN efficacy using four benchmark problems: the diffusion equation, the Helmholtz equation, the Allen-Cahn equation, as well as both forward and inverse formulations of the reaction-diffusion equation (with and without noisy data). Our results show that Scaled-cPIKAN significantly outperforms existing methods in all test cases. In particular, it achieves several orders of magnitude higher accuracy and faster convergence rate, making it a highly efficient tool for approximating PDE solutions that feature oscillatory behavior over large spatial domains.

Figures (24)

• Figure 1: Comparison of the solutions predicted for the diffusion equation (Example \ref{['Exam.DiffEqu']}) in $[-2, 2] \times [0, 1]$. Each figure displays the outcomes of Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN (from top to bottom). From left to right in each row: the loss function, the predicted solution $u$, and the absolute error of the prediction are shown. The first plot in the top row shows the ground truth solution.
• Figure 2: Comparison of the solutions predicted for the diffusion equation (Example \ref{['Exam.DiffEqu']}) in $[-4, 4] \times [0, 1]$. Each figure displays the outcomes of Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN (from top to bottom). From left to right in each row: the loss function, the predicted solution $u$, and the absolute error of the prediction are shown. The first plot in the top row shows the ground truth solution.
• Figure 3: Comparison of the solutions predicted for the diffusion equation (Example \ref{['Exam.DiffEqu']}) in $[-6, 6] \times [0, 1]$. Each figure displays the outcomes of Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN (from top to bottom). From left to right in each row: the loss function, the predicted solution $u$, and the absolute error of the prediction are shown. The first plot in the top row shows the ground truth solution.
• Figure 4: Prediction results for the Helmholtz equation (Example \ref{['Exam.HelH']}) in the first scenario with parameters $(a_1, a_2, \kappa) = (0.25, 1.0, 1.0)$ on $\Omega = [-4,4]^2$. For each method (Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN, shown sequentially from top to bottom), the plots from left to right display the loss function, the predicted solution $u$, and the absolute error $|u_{\text{exact}} - u_{\text{predicted}}|$. The first plot in the top row shows the ground truth solution.
• Figure 5: Prediction results for the Helmholtz equation (Example \ref{['Exam.HelH']}) in the second scenario with parameters $(a_1, a_2, \kappa) = (1.0, 1.0, 1.0)$ on $\Omega = [-4,4]^2$. For each method (Scaled-cPIKAN, Scaled-PINN, cPIKAN, and PINN, shown sequentially from top to bottom), the plots from left to right display the loss function, the predicted solution $u$, and the absolute error $|u_{\text{exact}} - u_{\text{predicted}}|$. The first plot in the top row shows the ground truth solution.