  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.
