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Chebyshev Polynomial-Based Kolmogorov-Arnold Networks: An Efficient Architecture for Nonlinear Function Approximation
Sidharth SS, Keerthana AR, Gokul R, Anas KP
TL;DR
Complex nonlinear and high-dimensional function approximation remains challenging for MLPs because they can require many parameters and use fixed activation functions. The paper proposes Chebyshev KANs, which place learnable Chebyshev-polynomial functions on network edges, and reports strong performance across classification, function approximation, and fractal-generation tasks, including 98% MNIST test accuracy.
Problem
MLPs can require many parameters and fixed activation functions when approximating complex nonlinear functions, limiting parameter efficiency and interpretability.
Method
Chebyshev KANs use learnable functions parametrized by Chebyshev polynomials on network edges, combining KAN structure with Chebyshev approximation properties.
Results
98% test accuracy was achieved on digit classification, while experiments reported high-precision approximation of complex nonlinear and fractal functions with fewer parameters than traditional MLPs.
Takeaways & Limitations
Chebyshev KANs provide a flexible, parameter-efficient, and interpretable framework for nonlinear function approximation across scientific and engineering tasks.
Abstract
from arXiv · showhide
Accurate approximation of complex nonlinear functions is a fundamental challenge across many scientific and engineering domains. Traditional neural network architectures, such as Multi-Layer Perceptrons (MLPs), often struggle to efficiently capture intricate patterns and irregularities present in high-dimensional functions. This paper presents the Chebyshev Kolmogorov-Arnold Network (Chebyshev KAN), a new neural network architecture inspired by the Kolmogorov-Arnold representation theorem, incorporating the powerful approximation capabilities of Chebyshev polynomials. By utilizing learnable functions parametrized by Chebyshev polynomials on the network's edges, Chebyshev KANs enhance flexibility, efficiency, and interpretability in function approximation tasks. We demonstrate the efficacy of Chebyshev KANs through experiments on digit classification, synthetic function approximation, and fractal function generation, highlighting their superiority over traditional MLPs in terms of parameter efficiency and interpretability. Our comprehensive evaluation, including ablation studies, confirms the potential of Chebyshev KANs to address longstanding challenges in nonlinear function approximation, paving the way for further advancements in various scientific and engineering applications.
1 Introduction
The paper introduces Chebyshev KANs as an edge-based, learnable-function architecture motivated by the Kolmogorov-Arnold theorem and Chebyshev polynomial approximation. It targets more efficient and interpretable approximation of complex nonlinear and high-dimensional functions than conventional MLPs.
- Motivation: MLPs can approximate continuous functions but often require many parameters and offer limited interpretability because their activation functions are fixed.The paper contrasts fixed node activations in MLPs with learnable functions in KANs.
- Motivation: KANs sum incoming signals at nodes and apply nonlinear transformations through learnable functions, increasing flexibility and adaptability.This design places learnable nonlinear functions in the network structure rather than relying only on fixed activations.
- Architecture: Chebyshev KANs parametrize these learnable functions with Chebyshev polynomials, whose rapid convergence and numerical stability support efficient function approximation.The architecture uses learnable activation functions on edges, allowing responses to adjust to input data.
- Architecture: The resulting parametrization provides high approximation accuracy with fewer parameters and a compact representation of complex functions.The paper also reports that Chebyshev KANs surpass original KAN implementations in efficiency.
- Evaluation: The study combines theoretical formulations, ablation studies, and experiments on complex functions to evaluate Chebyshev KANs across scientific and engineering applications.The implementation and experiments are publicly available in a GitHub repository.
- Theoretical foundation: The Kolmogorov-Arnold theorem represents continuous multivariate functions on bounded domains through compositions of continuous univariate functions and linear operations.This decomposition motivates reducing multivariate-function modeling to combinations of one-variable functions.
3 Chebyshev Polynomials
Chebyshev polynomials are orthogonal polynomials used broadly in numerical analysis, approximation theory, and differential-equation solving. Their minimax property makes them important for polynomial approximation.
- Properties and applications: Chebyshev polynomials are orthogonal and are applied in numerical analysis, approximation theory, and solving differential equations.They are named after the mathematician Pafnuty Chebyshev.
- Properties and applications: Their minimax property minimizes the maximum error in polynomial approximation.
Definition and Equations
The paper introduces first- and second-kind Chebyshev polynomials as trigonometric-function-based polynomial families. It presents their definitions and explicit polynomial forms alongside visualizations of each family.
- Definitions: The first-kind T_n(x) and second-kind U_n(x) Chebyshev polynomials are defined in terms of trigonometric functions.
- First kind: The paper gives a definition and explicit polynomial form for the first-kind Chebyshev polynomials.
- Visualizations: Figures 2 and 3 visualize the first- and second-kind Chebyshev polynomial families, respectively.
- Second kind: The paper gives a definition and explicit polynomial forms for the second-kind Chebyshev polynomials.
Recurrence Relations
The paper identifies recurrence relations as a useful way to compute Chebyshev polynomials.
- Computation: The recurrence relations provide a computational method for generating Chebyshev polynomials.
Orthogonality
Chebyshev polynomials are orthogonal under specific weight functions on [−1, 1]. The supplied passages specify the weight for T_n(x) and the corresponding inner-product values.
- Chebyshev polynomials are orthogonal with respect to specific weight functions over [−1, 1].
- The inner product is zero when the polynomial indices differ, π/2 when equal and nonzero, and π for the zeroth polynomial.
Properties and Applications
Chebyshev polynomials support approximation through a minimax property and have explicitly characterized roots and extrema.
- Chebyshev polynomials minimize the maximum deviation from zero among same-degree polynomials with leading coefficient 1.
- The roots of T_n(x) are identified by a specific formula in the supplied passage.
- The extrema of T_n(x), where T_n(x)=±1, are identified by a specific formula in the supplied passage.
4 The Chebyshev Kolmogorov-Arnold Network
Chebyshev KAN represents nonlinear functions using learnable Chebyshev-polynomial transformations on network edges. Inputs are expanded into polynomial bases and combined with learned coefficients to produce outputs.
- 4.1 Chebyshev Polynomial Representation: Inputs are represented through Chebyshev polynomials, which form an orthogonal basis for function approximation.
- 4.4 Mathematical Explanation: The network learns a coefficient tensor C whose dimensions span input features, output features, and polynomial degree.
- 4.4 Mathematical Explanation: The output tensor y is produced by an Einstein summation combining polynomial bases T with learned coefficients C.
- 4.6 Function Approximation with Chebyshev Polynomials: Chebyshev KAN combines polynomial approximation with adaptive neural-network learning to provide a flexible mechanism for complex nonlinear functions.
- 4 The Chebyshev Kolmogorov-Arnold Network: Chebyshev KAN uses learnable edge functions parametrized by Chebyshev polynomials instead of fixed node activations.
- Advantages over MLPs: Compared with MLPs, Chebyshev KANs are described as more compact, potentially reducing parameters while maintaining or improving accuracy.
- Advantages over MLPs: The learnable edge functions can be visualized and analyzed, providing greater interpretability than traditional MLPs.
5 Experiments and Results
The evaluation tests Chebyshev KAN on MNIST digit classification and synthetic function approximation, with results reported against traditional MLPs. The model achieves strong classification and approximation performance in the supplied experiments.
- 5 Experiments and Results: The experiments cover MNIST digit classification, synthetic function approximation, and ablation studies of initialization, normalization, and polynomial choices.
- 5.1 Digit Classification on MNIST: MNIST contains 60,000 training images and 10,000 test images of 28×28 grayscale handwritten digits.
- 5.1.1 Experimental Setup: The MNIST Chebyshev KAN uses stacked Chebyshev KAN layers followed by fully connected layers, with flattened inputs normalized to [−1, 1].
- 5.1.2 Results: 98% test accuracy was achieved by the Chebyshev KAN model on MNIST.
- 5.2 Function Approximation: On f(x)=sin(x)+x^2, Chebyshev KAN achieved an MSE of 0.0012 and outperformed traditional MLPs in the reported approximation experiment.
6 Test on Fractal Function
The Chebyshev KAN was evaluated on a complex fractal-like function and trained against values across a 2D grid. It successfully approximated the function, while ablations showed that initialization, polynomial degree, normalization, and polynomial kind affected performance.
- Experimental setup: The model was trained to minimize MSE between predicted and actual fractal-function values across a 2D grid.The synthetic data were generated from a seed function followed by an iterative fractal-like transformation.
- Results: The Chebyshev KAN successfully approximated the fractal function, with substantially reduced final training loss.Figure 7 compares the original fractal function with the model’s predictions.
- Initialization ablation: Xavier initialization yielded the best overall results, although Lecun and uniform random initialization performed better during training.Orthogonal initialization was slightly lower, while normal initialization performed worst.
- Polynomial degree ablation: Increasing polynomial degree from 2 to 3 slightly improved accuracy, whereas degree 4 caused a significant performance drop and degree 5 remained below degree 3.The authors suggest degree 3 balances model complexity and generalization on MNIST.
- Normalization ablation: Standardization performed slightly better than tanh normalization and Min-Max Scaling, which achieved similar accuracy.The comparison evaluated input normalization techniques on MNIST.
- Polynomial-kind ablation: Second-kind Chebyshev polynomials U_n(x) achieved higher MNIST accuracy and lower function-approximation MSE, but required more parameters for similar accuracy.First-kind polynomials T_n(x) provided more stable and efficient approximations.
7 Conclusion
Chebyshev KAN combines the Kolmogorov-Arnold framework with Chebyshev polynomials to provide an efficient, flexible, and interpretable alternative to MLPs. Experiments across classification, nonlinear function approximation, and fractal generation support its effectiveness, while future work targets broader datasets and machine-learning frameworks.
- Chebyshev KAN offers a robust and efficient alternative to MLPs through parameter efficiency, dynamic activation functions, and interpretability.
- Across MNIST, function approximation, and fractal generation, Chebyshev KAN achieved high accuracy, captured complex nonlinear relationships, and handled intricate fractal patterns.
- Ablations identified Xavier initialization, polynomial degree, and normalization as important influences on model accuracy and stability.
- The architecture provides a transparent and interpretable framework for modeling complex functions, with future work aimed at more diverse datasets and broader machine-learning integration.