Physics Informed Kolmogorov-Arnold Neural Networks for Dynamical Analysis via Efficent-KAN and WAV-KAN
Subhajit Patra, Sonali Panda, Bikram Keshari Parida, Mahima Arya, Kurt Jacobs, Denys I. Bondar, Abhijit Sen
TL;DR
This work introduces Physics-Informed Kolmogorov-Arnold Neural Networks (PIKAN) implemented with efficient-KAN and WAV-KAN to solve differential equations more efficiently than conventional PINNs. By leveraging the Kolmogorov-Arnold representation theorem, PIKAN replaces deep neural nets with networks that learn univariate activation functions at the edges, enabling accurate solutions with smaller architectures. The authors develop data-free (DF-PIKAN) and data-driven (DD-PIKAN) formulations, balancing physics residuals, initial/boundary conditions, and optional data terms, and demonstrate strong performance across a wide range of linear and nonlinear ODEs, Lorenz-type systems, oscillatory dynamics, and nonlinear PDEs like Burgers and Allen–Cohen equations. Across these tests, PIKAN often achieves comparable or superior accuracy with significantly simpler networks and faster convergence than PINNs, including cases where data guidance substantially improves solutions. The results suggest substantial practical impact for efficient physics-informed computation in scientific modeling and engineering applications, with potential for reduced hyperparameter tuning and computational resources.
Abstract
Physics-informed neural networks have proven to be a powerful tool for solving differential equations, leveraging the principles of physics to inform the learning process. However, traditional deep neural networks often face challenges in achieving high accuracy without incurring significant computational costs. In this work, we implement the Physics-Informed Kolmogorov-Arnold Neural Networks (PIKAN) through efficient-KAN and WAV-KAN, which utilize the Kolmogorov-Arnold representation theorem. PIKAN demonstrates superior performance compared to conventional deep neural networks, achieving the same level of accuracy with fewer layers and reduced computational overhead. We explore both B-spline and wavelet-based implementations of PIKAN and benchmark their performance across various ordinary and partial differential equations using unsupervised (data-free) and supervised (data-driven) techniques. For certain differential equations, the data-free approach suffices to find accurate solutions, while in more complex scenarios, the data-driven method enhances the PIKAN's ability to converge to the correct solution. We validate our results against numerical solutions and achieve $99 \%$ accuracy in most scenarios.