basis function expansion

To address the limited expressive capacity of Kolmogorov–Arnold Networks (KANs), this paper proposes FC-KAN—a novel KAN architecture that explicitly integrates elementary mathematical functions—including B-splines, Difference-of-Gaussians (DoG), wavelets, radial basis functions (RBFs), and polynomials—via low-dimensional, element-wise operations. We introduce a flexible multi-function composition mechanism encompassing summation, multiplication, quadratic/cubic representations, concatenation, and linear projection. Notably, we present two novel variants: DoG-B-spline and quadratic-linear hybrid functions—the first such incorporation in KAN literature. Extensive evaluation on MNIST and Fashion-MNIST across five independent trials demonstrates that FC-KAN achieves significantly higher average accuracy than standard MLPs and state-of-the-art KAN variants, including BSRBF-KAN, EfficientKAN, and FastKAN. These results empirically validate that explicit, mathematically grounded function composition enhances both the interpretability and modeling capability of neural networks.

This work addresses the poor interpretability and low parameter efficiency of traditional multilayer perceptrons (MLPs) in nonlinear modeling. Methodologically, grounded in the Kolmogorov–Arnold representation theorem, it replaces fixed activation functions with learnable piecewise spline functions, introducing a novel “learnable activations-as-weights” paradigm, and develops a symbol-numerical co-optimization framework with differentiable grid refinement. Contributions include: (i) the first comprehensive survey of Kolmogorov–Arnold Networks (KANs); (ii) theoretical and empirical identification of their structural advantages in function approximation, intrinsic interpretability, and parameter efficiency; and (iii) experimental validation showing KANs significantly outperform MLPs in fitting accuracy, generalization, and few-shot learning. To foster reproducibility and adoption, we publicly release a unified training framework, advancing the field of interpretable neural modeling.

To address the high computational overhead and poor hardware compatibility of Kolmogorov–Arnold Networks (KANs), this paper proposes SineKAN—a novel architecture that, for the first time, replaces conventional B-spline or Fourier basis functions in KANs with learnable sinusoidal activation functions at the edge level. This design preserves the theoretical expressivity guaranteed by the Kolmogorov–Arnold representation theorem while simultaneously enhancing periodic representation capability, gradient stability, and hardware efficiency. Experiments on visual benchmarks—including image classification—demonstrate that SineKAN matches or exceeds the accuracy of both B-spline and Fourier KANs, achieves significantly faster training convergence, and exhibits numerical precision scalability comparable to dense neural networks. By unifying interpretability, lightweight design, and hardware-aware computation, SineKAN establishes a new paradigm for efficient, theoretically grounded, and interpretable neural networks.

This work theoretically compares Kolmogorov–Arnold Networks (KANs) and multilayer perceptrons (MLPs) in terms of expressive power and spectral bias, specifically assessing KANs’ potential as MLP alternatives with improved efficiency in modeling high-frequency components. Method: Leveraging tools from approximation theory, spectral analysis, and spline interpolation, the study establishes rigorous theoretical characterizations of both architectures. Contribution/Results: We prove for the first time that any MLP can be exactly represented by a KAN with strictly lower parametric complexity. We reveal that KANs’ learnable spline grids inherently mitigate low-frequency bias—enhancing fidelity to high-frequency signal components. Theoretically, KANs match or exceed MLPs in expressivity; moreover, large-grid KANs achieve substantial parameter savings for specific target functions. Empirical validation confirms weaker spectral bias and significantly improved high-frequency approximation accuracy compared to MLPs.

This paper addresses the poor interpretability and limited accuracy of traditional multilayer perceptrons (MLPs) by proposing Kolmogorov–Arnold Networks (KANs), a novel neural architecture grounded in the Kolmogorov–Arnold representation theorem. Unlike MLPs—which employ fixed activation functions and linear weight layers—KANs parameterize learnable B-spline functions on **edges**, enabling flexible nonlinear modeling; nodes perform only summation, with no weights or activations. This design yields three key contributions: (1) Both theoretical analysis and empirical evaluation demonstrate superior neural scaling laws: small KANs significantly outperform large MLPs in data fitting and partial differential equation solving. (2) KANs enable direct parameter visualization and semantic interpretability, facilitating human–machine collaborative scientific discovery. (3) KANs constitute the first general-purpose neural network paradigm featuring edge-level learnable activations.

This work proposes a learnable framework for adaptive orthogonal bases that overcomes the rigidity of traditional fixed bases—such as Fourier or wavelet bases—in capturing data-specific structures. The target basis is treated as a point on the Lie manifold of the orthogonal group and is obtained by continuously evolving a reference basis along a path defined by an ordinary differential equation induced by a finite-rank skew-adjoint integral operator, parameterized via neural networks. Theoretically, it is shown that rank-2 generators suffice to densely approximate any orthogonal basis in the operator topology, ensuring both universality and flexibility. Experiments demonstrate successful adaptation of the Fourier basis into data-driven principal components, eigenfunctions of operators, and dynamic modes of physical systems, confirming the method’s effectiveness and broad applicability.

This work addresses the high parameter redundancy, poor scalability, and low training efficiency of Kolmogorov–Arnold Networks (KANs) by introducing hyperbolic geometry into the KAN framework for the first time. The proposed method embeds inputs into the bounded hyperbolic latent space of the Poincaré ball, performs KAN-style updates in the tangent space, and incorporates a low-rank prototype module to share function transformations across hidden dimensions. By integrating spline-based function learning, a radial coordinate structure, and a radius control mechanism, the approach enhances model interpretability and training stability. Empirical evaluations on eight benchmark datasets demonstrate that the method achieves predictive performance comparable to or better than existing approaches while substantially improving parameter efficiency.

This work addresses the inefficiency of Kolmogorov–Arnold Networks (KANs) arising from the recursive evaluation of B-spline basis functions, which hinders their practical deployment. To overcome this limitation, the authors propose Linear-Time B-spline KANs (LTBs-KAN), which achieve linear-time complexity in B-spline computation for the first time. The method further incorporates a product-sum matrix factorization strategy that substantially reduces both parameter count and computational overhead while preserving model expressiveness. Empirical evaluations on MNIST, Fashion-MNIST, and CIFAR-10 demonstrate that LTBs-KAN enables highly efficient forward inference and yields compact model sizes, achieving competitive accuracy with significantly enhanced practicality.

This study addresses the unclear neural scaling laws of Kolmogorov-Arnold Networks (KANs) under data scaling and the structural evolution mechanisms of their learnable activation functions. Building upon BSRBF, Gottlieb, and Faster-KAN architectures, this work systematically evaluates test loss variations with respect to dataset size and the complexity evolution of activation functions across MNIST classification and moiré magnetic texture scientific regression tasks. The contributions are threefold: revealing a dual-branch symmetry-breaking scaling behavior in KANs while quantifying critical exponent discrepancies; elucidating the data-driven evolutionary trajectory of activation functions toward stable symbolic forms; and providing an efficient application roadmap for balancing model expressivity against computational overhead.

本文提出RecKAN，通过学习二次多项式递归定义的基来改进Kolmogorov-Arnold网络，解决了现有方法中基函数固定的问题，在多个基准数据集上表现优异。