Legendre-KAN : High Accuracy KA Network Based on Legendre Polynomials

Wei Chen, Qingfeng Xia, JiaHui Sun

Keywords: KA Network; Legendre Polynomials; Symbolic Representation; Function Approximation; High Accuracy

Abstract:

Recently, the Kolmogorov-Arnold Network (KAN) has been proposed,
significantly outperforming MLP in terms of interpretability and symbolic representation.
In practice, KANs are required to fit data to extremely high precision.
For instance, in typical applications of KAN like inferring precise equations from data and serving as solvers for partial differential equations,
high accuracy is an intrinsic requirement.
In the current architecture of KAN,
cubic B-spline basis functions were selected as the approximate tools.
However, the inflexibility of fixed degree and knots in B-splines restricts
the adaptability of the activation functions.
Due to these inherent limitations of B-spline functions,
especially low-order and homogeneity, KAN still has room for improvement in accuracy.
In this paper, we propose the Legendre-KAN that can enhance the
degrees of freedom of the basis functions in the KAN.
Compared to the traditional Spline-KAN,
Legendre-KAN utilizes parameterized Legendre basis functions and
normalization layers at the edges of the KAN.
Benefiting from higher-order orthogonal polynomials,
Legendre-KAN significantly outperforms the Spline-KAN in terms of accuracy.
Extensive experiments demonstrate that Legendre-KAN achieves higher accuracy
and parameter efficiency, of which accuracy reaches 10-100 times that of Spline-KAN in some cases.
For those functions which can be symbolized,
this leads to more correct results as opposed to Spline-KAN.
Our approach effectively improves the accuracy of the mathematical relationships
in KANs, providing a better solution for approximating and analyzing complex nonlinear functions.

Primary Area: foundation or frontier models, including LLMs

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Submission Number: 8671