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A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networks

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arxiv 2406.02917 v1 pith:YH6UOZYA submitted 2024-06-05 cs.LG physics.comp-ph

Khemraj Shukla , Juan Diego Toscano , Zhicheng Wang , Zongren Zou , George Em Karniadakis This is my paper · ORCID
classification cs.LGphysics.comp-ph
keywords networkskansoperatoralthoughdeepdifferentialequationsfair
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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.

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Cited by 5 Pith papers

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  1. Fast, accurate, and differentiable: a neural-network surrogate for NRSur7dq4 precessing binary black hole waveforms

    gr-qc 2026-07 accept novelty 6.0 of 10

    A piecewise MLP surrogate emulates NRSur7dq4 over its full domain at NR-faithful accuracy with ~1 ms GPU latency and a fully differentiable JAX likelihood pipeline.

  2. Kolmogorov-Arnold Network for Gene Regulatory Network Inference

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    scKAN uses Kolmogorov-Arnold networks in a one-vs-rest regression and treats model gradients as signed gene regulation strengths, outperforming baselines on several BEELINE benchmark tasks.

  3. Leveraging KANs for Expedient Training of Multichannel MLPs via Preconditioning and Geometric Refinement

    cs.LG 2025-05 conditional novelty 5.0 of 10

    Training in a B-spline KAN basis is equivalent to preconditioned gradient descent on a multichannel ReLU MLP, and geometric refinement plus trainable knots accelerate and improve training.

  4. TimeKAN: KAN-based Frequency Decomposition Learning Architecture for Long-term Time Series Forecasting

    cs.LG 2025-02 conditional novelty 5.0 of 10

    A frequency-decomposing KAN architecture achieves state-of-the-art or near-state-of-the-art long-term forecasting on five of six datasets with 12-38K parameters.

  5. PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

    cs.LG 2026-07 reject novelty 4.0 of 10

    PG-KINN pairs a KAN trial space with a Petrov–Galerkin test space for forward and inverse PDEs, but the inverse benchmark data is inconsistent with the governing equation and the accuracy claims are not supported by t...

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