A Unified Benchmark of Physics-Informed Neural Networks and Kolmogorov-Arnold Networks for Ordinary and Partial Differential Equations

TL;DR

This paper compares traditional physics-informed neural networks with Kolmogorov-Arnold networks, demonstrating that the latter offer superior accuracy, faster convergence, and better gradient estimation for solving differential equations.

Contribution

It provides a systematic comparison between PINNs and KAN-based PIKANs, showing the latter's advantages in accuracy and efficiency across various differential equations.

Findings

PIKANs outperform PINNs in solution accuracy.

PIKANs converge faster than PINNs.

PIKANs provide more accurate gradient estimates.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful mesh-free framework for solving ordinary and partial differential equations by embedding the governing physical laws directly into the loss function. However, their classical formulation relies on multilayer perceptrons (MLPs), whose fixed activation functions and global approximation biases limit performance in problems with oscillatory behavior, multiscale dynamics, or sharp gradients. In parallel, Kolmogorov-Arnold Networks (KANs) have been introduced as a functionally adaptive architecture based on learnable univariate transformations along each edge, providing richer local approximations and improved expressivity. This work presents a systematic and controlled comparison between standard MLP-based PINNs and their KAN-based counterparts, Physics-Informed Kolmogorov-Arnold Networks (PIKANs), using identical physics-informed formulations and matched parameter budgets to isolate the architectural effect. Both models are evaluated across a representative collection of ODEs and PDEs, including cases with known analytical solutions that allow direct assessment of gradient reconstruction accuracy. The results show that PIKANs consistently achieve more accurate solutions, converge in fewer iterations, and yield superior gradient estimates, highlighting their advantage for physics-informed learning. These findings underline the potential of KAN-based architectures as a next-generation approach for scientific machine learning and provide rigorous evidence to guide model selection in differential equation solving.