DualFlexKAN: Dual-stage Kolmogorov-Arnold Networks with Independent Function Control

Andrés Ortiz; Nicolás J. Gallego-Molina; Carmen Jiménez-Mesa; Juan M. Górriz; Javier Ramírez

cs.LGMar 9, 2026

DualFlexKAN: Dual-stage Kolmogorov-Arnold Networks with Independent Function Control

Authors: Andrés Ortiz, Nicolás J. Gallego-Molina, Carmen Jiménez-Mesa, Juan M. Górriz, Javier Ramírez

Organizations: Department of Communications Engineering University of Malaga Malaga, Spain · DaSCI Andalusian Institute of Data Science and Computational Intelligence University of Granada Granada, Spain · Department of signal Theory, Communications and Telematics University of Granada Granada, Spain

Abstract

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.

Figures & tables

Figure 1: Classical MLP architecture

Figure 2: Classical KAN architecture

Figure 3: DualFlexKAN architecture. Some configuration options are shown in the Figure.

Basis Type	Definition	Domain	Weight $w(x)$	Learnable Parameters
Standard Polynomial	$\mathcal{B}_{k}(x)=x^{k-1}$	$\mathbb{R}$	N/A	$\{c_{k}\}_{k=1}^{m}$
Legendre	$P_{n}(x)$ via recurrence: $P_{0}=1$ , $P_{1}=x$	$[-1,1]$	$w(x)=1$	$\{c_{k}\}_{k=1}^{m}$
Chebyshev	$T_{n}(x)$ via recurrence: $T_{0}=1$ , $T_{1}=x$	$[-1,1]$	$w(x)=\frac{1}{\sqrt{1-x^{2}}}$	$\{c_{k}\}_{k=1}^{m}$
Gegenbauer	$C_{n}^{(\alpha)}(x)$ parameterized by $\alpha>-\frac{1}{2}$	$[-1,1]$	$w(x)=(1-x^{2})^{\alpha-\frac{1}{2}}$	$\{c_{k}\}_{k=1}^{m}$ , parameter $\alpha$ (optional)
Jacobi	$P_{n}^{(\alpha,\beta)}(x)$ parameterized by $\alpha,\beta>-1$	$[-1,1]$	$w(x)=(1-x)^{\alpha}(1+x)^{\beta}$	$\{c_{k}\}_{k=1}^{m}$ , parameters $\alpha,\beta$ (optional)
B-Splines	Cox-de Boor recursion with knot sequence $\{t_{i}\}$	Defined by knots	N/A	$\{c_{k}\}_{k=1}^{m}$ , knot positions $\{t_{i}\}$ (optional)

Table 1: Basis Function Families in DualFlexKAN

Dataset	Feat.	Samples	Characteristic Challenge
Real World / UCI
Yacht Hydrodynamics	6	308	Fluid dynamics, low $N$
Servo	4	167	Mechanical system, categorical
Boston Housing	13	506	Heterogeneous features
Auto MPG	7	392	Fuel efficiency
Diabetes	10	442	Medical regression

Table 2: Summary of Datasets used in the comparative analysis.

Figure 4: Parameter Efficiency: DualFlexKAN reduces parameter count by orders of magnitude compared to vanilla KAN, matching MLP efficiency.

Figure 5: Training time (s) for the different architectures benchmarked in this work.

Figure 6: Global Model Complexity: Comparison of the Effective Parameter count (log scale) required to maintain 90% accuracy across all datasets, including complex physical interactions [ 29 ] . DFKAN (green) demonstrates superior sparsity, achieving the same performance as MLPs (red) with nearly two orders of magnitude fewer parameters, and consistently outperforming Classic KAN (blue) by avoiding the parameter explosion caused by edge-based functions [ 12 ] .

Figure 7: Goodness of fit: $R^{2}$ scores demonstrating the superior stability of DualFlexKAN on real-world data compared to the volatile performance of Classical KAN.

Figure 8: Approximation Accuracy: Mean Squared Error (MSE). DualFlexKAN outperforms Classical KAN on smooth physics-informed tasks.

Figure 9: Visual Interpretability: Decomposition of the learned function $y=e^{-x^{2}}+0.5\sin(3x)$ . The colored lines represent the activation outputs of individual internal neurons using B-Spline bases [ 35 ] . DFKAN autonomously learns to approximate the constituent sinusoidal and Gaussian modes required to reconstruct the target signal (grey).

Figure 10: Symbolic Discovery and Robustness: Comparison of model behavior under noise mimicking physical experimental data [ 29 ] . The Classic KAN (dashed blue) suffers from overfitting, capturing noise artifacts. In contrast, DualFlexKAN (solid green) acts as a structural regularizer, recovering the smooth underlying physical law ( $2x^{2}-x+0.5$ ) and allowing for the extraction of a clean symbolic formula.

Figure 11: Intrinsic Feature Selection. Learned attention weights ( $\alpha$ ) on the Friedman #1 dataset [ 28 ] . The model automatically prioritizes the main relevant features ( $x_{1},x_{2},x_{3},x_{5}$ ) involved in the target function. While most noise variables are effectively suppressed, finite-sample spurious correlations cause a slight overlap between $x_{4}$ and $x_{7}$ . Overall, this demonstrates built-in interpretability without requiring external explainers [ 37 ] .

Figure 12: Manifold and Gradient Analysis: Comparison of learned surfaces for $z=\sin(2x)\cos(2y)$ . Top Row: Predicted function values. Bottom Row: Magnitude of the spatial gradients $|\nabla Z|$ . Classic KAN fails to converge (high MSE), exhibiting visible grid-like artifacts in both the function and gradient maps. MLP captures the overall gradient pattern but with reduced sharpness at high-frequency transition regions due to spectral bias [ 30 ] . DualFlexKAN achieves the lowest MSE and accurately reconstructs the core differential topology. Note that its spatial gradient exhibits slight geometric artifacts reflecting the footprint of the underlying basis functions, but it significantly outperforms both baselines in preserving high-frequency sharpness and structural stability.

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FEKAN: Feature-Enriched Kolmogorov-Arnold Networks

Sidharth S. Menon, Ameya D. Jagtap

Aerospace Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA.

Kolmogorov-Arnold Networks Neural Network Approximation Theory

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Kolmogorov-Arnold Networks (KANs) have recently emerged as a compelling alternative to multilayer perceptrons, offering enhanced interpretability via functional decomposition. However, existing KAN architectures, including spline-, wavelet-, radial-basis variants, etc., suffer from high computational cost and slow convergence, limiting scalability and practical applicability. Here, we introduce Feature-Enriched Kolmogorov-Arnold Networks (FEKAN), a simple yet effective extension that preserves all the advantages of KAN while improving computational efficiency and predictive accuracy through feature enrichment, without increasing the number of trainable parameters. By incorporating these additional features, FEKAN accelerates convergence, increases representation capacity, and substantially mitigates the computational overhead characteristic of state-of-the-art KAN architectures. We investigate FEKAN across a comprehensive set of benchmarks, including function-approximation tasks, physics-informed formulations for diverse partial differential equations (PDEs), and neural operator settings that map between input and output function spaces. For function approximation, we systematically compare FEKAN against a broad family of KAN variants, FastKAN, WavKAN, ReLUKAN, HRKAN, ChebyshevKAN, RBFKAN, and the original SplineKAN. Across all tasks, FEKAN demonstrates substantially faster convergence and consistently higher approximation accuracy than the underlying baseline architectures. We also establish the theoretical foundations for FEKAN, showing its superior representation capacity compared to KAN, which contributes to improved accuracy and efficiency.

Abstract

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