A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks (2406.02917v1)

Abstract: 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.

• The paper presents a fair and comprehensive comparison between MLP and KAN, emphasizing their effectiveness in modeling differential equations and operator networks.
• It employs rigorous analytical metrics to evaluate performance nuances and computational efficiency between both neural representations.
• The findings underscore significant implications for optimizing neural network approaches in addressing complex operator problems and differential equations.

Critical Review of "Title of Your Manuscript"

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Summary of Research

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Conclusion

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