CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains

Abstract

Physics-Informed Kolmogorov-Arnold Networks have recently emerged as an effective class of neural solvers for partial differential equations, combining the expressive power of spline-based Kolmogorov-Arnold representations with physics-informed learning. However, existing PIKAN formulations are primarily developed for Cartesian domains and cannot naturally accommodate the geometric complexity introduced by curvilinear domains. In this work, we propose Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks (CT-PIKAN), a geometry-aware framework for solving PDEs on arbitrarily shaped domains through coordinate transformation. A smooth mapping transforms the physical domain into a regular computational domain, while the transformed governing equations are enforced directly within the physics-informed loss. Unlike conventional transformed PINN approaches that require manually derived metric coefficients, CT-PIKAN employs automatic differentiation to evaluate Jacobians, metric tensors, and transformed differential operators directly from the coordinate mapping, eliminating analytical derivations and improving implementation flexibility. To establish the proposed framework, a data-free B-spline-based PIKAN is first constructed and validated on the two-dimensional advection equation. The CT-PIKAN methodology is subsequently assessed on representative elliptic, parabolic, and hyperbolic benchmark problems, including the Poisson, heat, and advection equations formulated in polar and wavy curvilinear coordinates. The proposed framework provides a general and extensible methodology for integrating differential geometry with physics-informed Kolmogorov-Arnold networks, enabling efficient and accurate PDE solutions on complex domains.