Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction

Kolmogorov-Arnold Networks (KANs) are a recent neural network architecture offering an alternative to Multilayer Perceptrons (MLPs) with improved explainability and expressibility. However, KANs are significantly slower than MLPs due to the recursive nature of B-spline function computations, limiting their application. This work addresses these issues by proposing a novel base-spline Linear-Time B-splines Kolmogorov-Arnold Network (LTBs-KAN) with linear complexity. Unlike previous methods that rely on the Boor-Mansfield-Cox spline algorithm or other computationally intensive mathematical functions, our approach significantly reduces the computational burden. Additionally, we further reduce model's parameter through product-of-sums matrix factorization in the forward pass without sacrificing performance. Experiments on MNIST, Fashion-MNIST and CIFAR-10 demonstrate that LTBs-KAN achieves good time complexity and parameter reduction, when used as building architectural blocks, compared to other KAN implementations.

Algorithmic speedup of training common neural architectures is made difficult by the lack of structure guaranteed by the function compositions inherent to such networks. In contrast to multilayer perceptrons (MLPs), Kolmogorov-Arnold networks (KANs) provide more structure by expanding learned activations in a specified basis. This paper exploits this structure to develop practical algorithms and theoretical insights, yielding training speedup via multilevel training for KANs. To do so, we first establish an equivalence between KANs with spline basis functions and multichannel MLPs with power ReLU activations through a linear change of basis. We then analyze how this change of basis affects the geometry of gradient-based optimization with respect to spline knots. The KANs change-of-basis motivates a multilevel training approach, where we train a sequence of KANs naturally defined through a uniform refinement of spline knots with analytic geometric interpolation operators between models. The interpolation scheme enables a ``properly nested hierarchy'' of architectures, ensuring that interpolation to a fine model preserves the progress made on coarse models, while the compact support of spline basis functions ensures complementary optimization on subsequent levels. Numerical experiments demonstrate that our multilevel training approach can achieve orders of magnitude improvement in accuracy over conventional methods to train comparable KANs or MLPs, particularly for physics informed neural networks. Finally, this work demonstrates how principled design of neural networks can lead to exploitable structure, and in this case, multilevel algorithms that can dramatically improve training performance.

Multi-Layer Perceptrons (MLPs) rely on pre-defined, fixed activation functions, imposing a static inductive bias that forces the network to approximate complex topologies solely through increased depth and width. Kolmogorov-Arnold Networks (KANs) address this limitation through edge-centric learnable functions, yet their formulation suffers from quadratic parameter scaling and architectural rigidity that hinders the effective integration of standard regularization techniques. This paper introduces the DualFlexKAN (DFKAN), a flexible architecture featuring a dual-stage mechanism that independently controls pre-linear input transformations and post-linear output activations. This decoupling enables hybrid networks that optimize the trade-off between expressiveness and computational cost. Unlike standard formulations, DFKAN supports diverse basis function families, including orthogonal polynomials, B-splines, and radial basis functions, integrated with configurable regularization strategies that stabilize training dynamics. Comprehensive evaluations across regression benchmarks, physics-informed tasks, and function approximation demonstrate that DFKAN outperforms both MLPs and conventional KANs in accuracy, convergence speed, and gradient fidelity. The proposed hybrid configurations achieve superior performance with one to two orders of magnitude fewer parameters than standard KANs, effectively mitigating the parameter explosion problem while preserving KAN-style expressiveness. DFKAN provides a principled, scalable framework for incorporating adaptive non-linearities, proving particularly advantageous for data-efficient learning and interpretable function discovery in scientific applications.