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This paper introduces FHKAN, a novel neural architecture that replaces standard activation functions with adaptive fractional-order Hermite bases to handle unbounded pre-activations without saturation. The authors demonstrate superior performance across synthetic regression, image tasks, and physics-informed differential equations compared to existing KAN variants and fixed activations. This work advances the theoretical foundation of Kolmogorov-Arnold Networks by addressing spectral limitations inherent in bounded polynomial approximations.
FHKAN: Fractional Hermite Kolmogorov-Arnold Networks for Data-Driven and Physics-Informed Learning
arXiv:2610.25148v1 Announce Type: new Abstract: Kolmogorov-Arnold Networks (KANs) replace fixed scalar weights by learnable univariate maps, yet most spectral KANs still rely on Chebyshev polynomials that live on a bounded interval and therefore require a saturating squashing of the pre-activation. We introduce the Fractional Hermite Kolmogorov-Arnold Network (FHKAN), a neural architecture that equips KANs with an adaptive, trainable fractional-order generalized Hermite basis. With a nonnegative shape parameter, the underlying generalized Hermite functions are orthonormal on the entire real line and uniformly bounded under a Gaussian envelope, so they act directly on unbounded pre-activations without any saturating map. We derive closed-form derivatives of the generalized, fractional, and scaled basis away from the origin and determine its order of differentiability at the origin, where the fractional power map creates a singular point. On synthetic regression, FHKAN attains the lowest test error at every network depth among nine fixed activations and the spline-based KAN. On image classification, image denoising, and sentiment analysis, the best-performing fractional Hermite block yields the highest accuracy or reconstruction quality. On physics-informed problems, FHKAN reaches mean absolute errors of order 10^-6 to 10^-5 on the Lane-Emden equation, residuals of order 10^-4 on the Burgers equation, and absolute errors below 10^-4 on a fractional delay differential equation.
Abstract:Kolmogorov-Arnold Networks (KANs) replace fixed scalar weights by learnable univariate maps, yet most spectral KANs still rely on Chebyshev polynomials that live on a bounded interval and therefore require a saturating squashing of the pre-activation. We introduce the Fractional Hermite Kolmogorov-Arnold Network (FHKAN), a neural architecture that equips KANs with an adaptive, trainable fractional-order generalized Hermite basis. With a nonnegative shape parameter, the underlying generalized Hermite functions are orthonormal on the entire real line and uniformly bounded under a Gaussian envelope, so they act directly on unbounded pre-activations without any saturating map. We derive closed-form derivatives of the generalized, fractional, and scaled basis away from the origin and determine its order of differentiability at the origin, where the fractional power map creates a singular point. On synthetic regression, FHKAN attains the lowest test error at every network depth among nine fixed activations and the spline-based KAN. On image classification, image denoising, and sentiment analysis, the best-performing fractional Hermite block yields the highest accuracy or reconstruction quality. On physics-informed problems, FHKAN reaches mean absolute errors of order 10^-6 to 10^-5 on the Lane-Emden equation, residuals of order 10^-4 on the Burgers equation, and absolute errors below 10^-4 on a fractional delay differential equation.
Machine Learning (cs.LG)
arXiv:2610.25148 [cs.LG]
(or arXiv:2610.25148v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2610.25148 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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