A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks
This work addresses the need for fair comparisons in physics-informed machine learning, though it is incremental as it builds on existing KAN and MLP methods without introducing a new paradigm.
The study compared Kolmogorov-Arnold Networks (KANs) with MLPs for solving differential equations, finding that modified KANs with low-order orthogonal polynomials achieved performance comparable to PINNs and DeepONets but lacked robustness, sometimes diverging with different random seeds or higher-order polynomials.
Kolmogorov-Arnold Networks (KANs) were recently introduced as an alternative representation model to MLP. Herein, we employ KANs to construct physics-informed machine learning models (PIKANs) and deep operator models (DeepOKANs) for solving differential equations for forward and inverse problems. In particular, we compare them with physics-informed neural networks (PINNs) and deep operator networks (DeepONets), which are based on the standard MLP representation. We find that although the original KANs based on the B-splines parameterization lack accuracy and efficiency, modified versions based on low-order orthogonal polynomials have comparable performance to PINNs and DeepONet although they still lack robustness as they may diverge for different random seeds or higher order orthogonal polynomials. We visualize their corresponding loss landscapes and analyze their learning dynamics using information bottleneck theory. Our study follows the FAIR principles so that other researchers can use our benchmarks to further advance this emerging topic.