FHKAN: Fractional Hermite Kolmogorov-Arnold Networks for Data-Driven and Physics-Informed Learning

Arsha Farnoush; Amirhossein Sadr; Aida Pakniyat; Dara Rahmati; Saeid Gorgin; Kourosh Parand

cs.LGOct 18, 2026

FHKAN: Fractional Hermite Kolmogorov-Arnold Networks for Data-Driven and Physics-Informed Learning

Authors: Arsha Farnoush, Amirhossein Sadr, Aida Pakniyat, Dara Rahmati, Saeid Gorgin, Kourosh Parand

Organizations: Department of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran · School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran · School of Physics, Engineering and Computer Science, University of Hertfordshire, Hatfield, UK · Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran ·

Abstract

Kolmogorov-Arnold Networks (KANs) replace fixed scalar weights by learnable univariate maps, yet most spectral KANs still rely on Chebyshev polynomials that live on a bounded interval and therefore require a saturating squashing of the pre-activation. We introduce the Fractional Hermite Kolmogorov-Arnold Network (FHKAN), a neural architecture that equips KANs with an adaptive, trainable fractional-order generalized Hermite basis. With a nonnegative shape parameter, the underlying generalized Hermite functions are orthonormal on the entire real line and uniformly bounded under a Gaussian envelope, so they act directly on unbounded pre-activations without any saturating map. We derive closed-form derivatives of the generalized, fractional, and scaled basis away from the origin and determine its order of differentiability at the origin, where the fractional power map creates a singular point. On synthetic regression, FHKAN attains the lowest test error at every network depth among nine fixed activations and the spline-based KAN. On image classification, image denoising, and sentiment analysis, the best-performing fractional Hermite block yields the highest accuracy or reconstruction quality. On physics-informed problems, FHKAN reaches mean absolute errors of order 10^-6 to 10^-5 on the Lane-Emden equation, residuals of order 10^-4 on the Burgers equation, and absolute errors below 10^-4 on a fractional delay differential equation.

Figures & tables

Basis family of the KAN variant	Natural domain	References
B-splines (original KAN)	bounded, local	[ 14 , 15 , 16 , 20 ]
Wavelets	bounded/local	[ 22 ]
Fourier bases	periodic	[ 23 ]
Radial basis functions	local	[ 24 ]
Temporal/recurrent bases	sequential	[ 25 , 19 ]
Smooth structural bases	bounded	[ 26 ]

Table 1: Summary of Kolmogorov–Arnold Network variants grouped by the univariate basis family used as edge functions, the natural domain on which each family is defined, and the corresponding references. Only the proposed fractional generalized Hermite basis operates natively on the entire real line.

Symbol	Type	Meaning
$\boldsymbol{\zeta},\ \zeta_{p}$	vector, scalar	input of the network and its $p$ -th component
$\chi,\ \hat{\chi}$	network map	target function and the map realized by the network
$\psi$	network map	task-dependent output map of a multi-layer perceptron (distinct from the Hermite functions $\psi_{n}^{(\mu)}$ )
$\Theta^{(i)},\ \mathbf{b}^{(i)}$	parameters	weight matrix and bias of layer $i$
$\theta_{p,q}$	parameter	expansion coefficient of order $q$ on input $p$
$\sigma$	univariate	fixed scalar activation, or the logistic sigmoid

Table 2: Summary of the notation. Lower-case symbols denote univariate functions and upper-case symbols denote network-level maps.

Panel of Figure 2 — Figure 2: Fractional Hermite functions defined in ( 13 ). They are bounded, decay at infinity, and stay defined on the whole real line for every admissible $\gamma$ , so they can be evaluated directly on the unbounded pre-activation of a linear layer.

Panel of Figure 3 — Figure 3: The fractional Hermite neural block defined in ( 14 ). A pre-activation $\zeta$ is rescaled by $\hat{\lambda}$ , passed through the odd fractional map of order $\hat{\gamma}$ , and expanded on the generalized Hermite function of order $q$ and shape $\hat{\mu}$ . The three parameters $\lambda$ , $\gamma$ , and $\mu$ (dashed) are trained end-to-end through their smooth reparametrizations.

Panel of Figure 4 — Figure 4: The two FHKAN topologies used in the experiments.

Panel of Figure 5 — Figure 5: Synthetic regression study. FHKAN is more accurate than standard activations and the vanilla KAN, its trainable parameters converge quickly, and its run time stays well below that of the spline-based KAN.

Act. func.	Number of hidden layers
Act. func.	1	3	5	7	9
Softplus	0.461 $\pm$ 0.046	0.448 $\pm$ 0.031	0.448 $\pm$ 0.017	0.418 $\pm$ 0.078	0.358 $\pm$ 0.107
SiLU	0.518 $\pm$ 0.125	0.388 $\pm$ 0.156	0.363 $\pm$ 0.344	0.162 $\pm$ 0.128	0.272 $\pm$ 0.380
SELU	0.677 $\pm$ 0.087	0.384 $\pm$ 0.151	0.147 $\pm$ 0.058	0.158 $\pm$ 0.075	0.127 $\pm$ 0.069
Leaky ReLU	0.489 $\pm$ 0.086	0.316 $\pm$ 0.140	0.214 $\pm$ 0.103	0.235 $\pm$ 0.067	0.189 $\pm$ 0.093
GELU	0.542 $\pm$ 0.134	0.165 $\pm$ 0.066	0.322 $\pm$ 0.616	0.075 $\pm$ 0.041	0.252 $\pm$ 0.618

Table 3: Test MAE across increasing depth for the synthetic regression experiment.

Act. func.	MAE	MSE	RMSE	Loss
ReLU	$0.520\pm 0.101$	$0.3425\pm 0.1291$	$0.575\pm 0.106$	$0.4035\pm 0.1113$
Tanh	$0.875\pm 0.012$	$0.9343\pm 0.0396$	$0.966\pm 0.021$	$0.9820\pm 0.0471$
KAN	$0.379\pm 0.051$	$0.0437\pm 0.0492$	$0.433\pm 0.049$	$0.1731\pm 0.0387$
FHKAN(5)	$0.074\pm 0.026$	$0.0127\pm 0.0078$	$0.107\pm 0.036$	$0.0038\pm 0.0024$
FHKAN(6)	$\mathbf{0.058\pm 0.017}$	$\mathbf{0.0086\pm 0.0059}$	$\mathbf{0.087\pm 0.032}$	$\mathbf{0.0025\pm 0.0016}$

Table 4: Single-layer regression metrics for the synthetic regression experiment.

Panel of Figure 6 — Figure 6: Convolutional architecture used for MNIST classification. The activation blocks are replaced by sigmoid, tanh, ReLU, or an fHNB.

Act. func.	Loss	Accuracy (%)
Sigmoid	$0.0611\pm 0.0031$	$98.092\pm 0.0814$
Tanh	$0.0322\pm 0.0018$	$98.904\pm 0.0572$
ReLU	$0.0256\pm 0.0013$	$99.140\pm 0.0386$
fHNB(2)	$0.0249\pm 0.0016$	$99.156\pm 0.0471$
fHNB(3)	$0.0227\pm 0.0018$	$99.211\pm 0.0762$
fHNB(4)	$\mathbf{0.0209}\pm 0.0012$	$\mathbf{99.272}\pm 0.0508$

Table 5: Loss and accuracy of a convolutional classifier on MNIST for different activation functions.

Panel of Figure 7 — Figure 7: Training curves on MNIST. The FHKAN curve reaches a lower validation loss and a higher validation accuracy than the sigmoid, tanh, and ReLU networks from the first epoch onward.

Panel of Figure 8 — Figure 8: Convolutional autoencoder used for Fashion-MNIST denoising. The activation is selected among sigmoid, tanh, ReLU, or an fHNB.

Act. func.	Pre-train			Denoising
Act. func.	MSE	PSNR	SSIM	MSE	PSNR	SSIM
Sigmoid	$0.018$ $\pm$ 0.0024	$18.03$ $\pm$ 0.64	$0.579$ $\pm$ 0.038	$0.018$ $\pm$ 0.0016	$17.81$ $\pm$ 0.29	$0.564$ $\pm$ 0.016
Tanh	$0.017$ $\pm$ 0.0017	$18.08$ $\pm$ 0.48	$0.583$ $\pm$ 0.029	$0.018$ $\pm$ 0.0013	$17.83$ $\pm$ 0.32	$0.562$ $\pm$ 0.023
ReLU	$0.017$ $\pm$ 0.0014	$18.12$ $\pm$ 0.26	$0.577$ $\pm$ 0.021	$0.019$ $\pm$ 0.0018	$17.54$ $\pm$ 0.21	$0.537$ $\pm$ 0.011
fHNB(5)	$0.013$ $\pm$ 0.0011	$19.24$ $\pm$ 0.27	$0.649$ $\pm$ 0.012	$0.017$ $\pm$ 0.0008	$18.17$ $\pm$ 0.25	$0.586$ $\pm$ 0.012
fHNB(6)	$\mathbf{0.012}$ $\pm$ 0.0009	$\mathbf{19.31}$ $\pm$ 0.28	$\mathbf{0.653}$ $\pm$ 0.016	$\mathbf{0.016}$ $\pm$ 0.0012	$\mathbf{18.24}$ $\pm$ 0.22	$\mathbf{0.589}$ $\pm$ 0.010

Table 6: Fashion-MNIST denoising quality. Higher PSNR and SSIM and lower MSE are better; the best mean in each column is shown in bold.

Panel of Figure 9 — Figure 9: One-dimensional convolutional network used for IMDB sentiment analysis. The activation is sigmoid, tanh, ReLU, or an fHNB.

Act. func.	Loss	Accuracy	Precision	Recall	$F_{1}$ -score	ROC-AUC
Sigmoid	$\mathbf{0.364}$ $\pm$ 0.028	$84.884$ $\pm$ 0.412	$84.473$ $\pm$ 1.274	$85.480$ $\pm$ 1.186	$84.974$ $\pm$ 0.871	$92.303$ $\pm$ 0.361
Tanh	$0.442$ $\pm$ 0.047	$84.672$ $\pm$ 0.693	$84.834$ $\pm$ 2.318	$84.440$ $\pm$ 2.147	$84.636$ $\pm$ 1.579	$92.264$ $\pm$ 0.448
ReLU	$0.416$ $\pm$ 0.051	$86.320$ $\pm$ 0.974	$86.554$ $\pm$ 1.836	$86.000$ $\pm$ 1.092	$86.276$ $\pm$ 1.065	$93.366$ $\pm$ 0.337
fHNB(1)	$0.407$ $\pm$ 0.058	$86.988$ $\pm$ 0.982	$87.140$ $\pm$ 2.541	$86.784$ $\pm$ 2.463	$86.961$ $\pm$ 1.769	$94.475$ $\pm$ 0.429
fHNB(2)	$0.379$ $\pm$ 0.033	$87.416$ $\pm$ 0.524	$87.154$ $\pm$ 1.128	$\mathbf{87.768}$ $\pm$ 0.947	$87.460$ $\pm$ 0.737	$94.681$ $\pm$ 0.392
fHNB(3)	$0.418$ $\pm$ 0.041	$87.156$ $\pm$ 0.638	$87.800$ $\pm$ 1.492	$86.304$ $\pm$ 1.173	$87.046$ $\pm$ 0.945	$94.495$ $\pm$ 0.351

Table 7: IMDB sentiment-classification metrics. The best mean in each column is shown in bold.

Method family	References
Meshless methods	[ 82 , 83 , 84 ]
Finite difference and multistep methods	[ 85 , 86 ]
Variational and perturbation iteration methods	[ 87 , 88 , 89 ]
Spectral and pseudo-spectral methods	[ 90 , 91 , 67 ]
Analytical and semi-analytical methods	[ 92 , 93 , 94 ]
Reproducing kernel Hilbert space methods	[ 95 , 96 , 97 ]

Table 8: Summary of the principal families of methods proposed in the literature for solving Lane–Emden type equations, compiled in the spirit of the comparative tables of [ 81 ] .

Figure 10: Mapped Hermite–Gauss collocation nodes (roots of

H_{N+1}

pushed through ( 11 )) and a uniform grid on a truncated interval

[0,16]

. Under ( 11 ) the origin is the image of

\omega=-\infty

. The nodes cluster near

\zeta=0

, where the coefficient

2/\zeta

is singular.

Figure 11: Lane–Emden profiles for

m=0,\dots,5

obtained with the parallel FHKAN (markers) against the reference solutions (solid lines). The domain is the semi-infinite interval

[0,+\infty)

. For

m=2,3,4

the reference is a high-order numerical solution.

$m$	Reference first root	FHKAN	Absolute error	MAE	MSE	Loss
0	$2.44948974$ (exact)	$2.44949712$ $\pm 1.8\times 10^{-6}$	$(7.37\pm 1.8)$ $\times 10^{-6}$	$(2.14\pm 0.38)$ $\times 10^{-6}$	$(6.81\pm 2.30)$ $\times 10^{-12}$	$(4.1\pm 0.8)$ $\times 10^{-8}$
1	$3.14159265$ (exact)	$3.14159083$ $\pm 4.7\times 10^{-7}$	$(1.82\pm 0.47)$ $\times 10^{-6}$	$(1.47\pm 0.29)$ $\times 10^{-6}$	$(3.22\pm 0.93)$ $\times 10^{-12}$	$(3.6\pm 0.7)$ $\times 10^{-8}$
2	$4.35287460$ (Horedt)	$4.35287902$ $\pm 1.1\times 10^{-6}$	$(4.42\pm 1.1)$ $\times 10^{-6}$	$(3.85\pm 0.81)$ $\times 10^{-6}$	$(2.41\pm 0.86)$ $\times 10^{-11}$	$(5.8\pm 1.2)$ $\times 10^{-8}$
3	$6.89684862$ (Horedt)	$6.89684723$ $\pm 3.6\times 10^{-7}$	$(1.39\pm 0.36)$ $\times 10^{-6}$	$(2.26\pm 0.47)$ $\times 10^{-6}$	$(8.05\pm 3.01)$ $\times 10^{-12}$	$(6.2\pm 1.4)$ $\times 10^{-8}$
4	$14.9715463$ (Horedt)	$14.9716277$ $\pm 1.9\times 10^{-5}$	$(8.14\pm 1.9)$ $\times 10^{-5}$	$(4.12\pm 0.96)$ $\times 10^{-5}$	$(2.73\pm 1.17)$ $\times 10^{-9}$	$(2.1\pm 0.4)$ $\times 10^{-7}$
5	$\infty$ (exact)	no finite root	—	$(6.51\pm 1.27)$ $\times 10^{-6}$	$(6.02\pm 2.19)$ $\times 10^{-11}$	$(7.4\pm 1.5)$ $\times 10^{-8}$

Table 9: First positive root of the Lane–Emden solution on

[0,+\infty)

, together with the pointwise mean absolute error, the mean squared error, and the collocation residual loss. For

m=0,1

the reference is exact; for

m=2,3,4

it is Horedt’s numerical table [ 80 ] ; for

m=5

there is no finite root. MAE and MSE are computed against the exact solution for

m=0,1,5

and against the Horedt/RK reference for

m=2,3,4

. The loss column is the collocation residual.

Panel of Figure 12 — Figure 12: Burgers equation solved with the parallel FHKAN for $(m_{0},m_{1},m_{2})=(1,0.01,0.1)$ (top) and $(0.1,10^{-4},0.5)$ (bottom). The computational domain is $(\zeta,\tau)\in[0,1]\times[0,1]$ ; collocation uses a $100\times 100$ tensor product of mapped Hermite–Gauss nodes.

Figure 13: Fractional delay differential equation solved with the parallel FHKAN: prediction (left), absolute error against the exact solution (center), and residual loss during training (right).

Figure 14: Parameter efficiency on the synthetic regression target, scored against the noiseless function. Markers compare a spline KAN, a Chebyshev KAN, and FHKAN(5) at hidden widths $h=5$ and $h=8$ . The marker color, read on the right-hand color bar, gives the mean training time. The Pareto frontier collects the models for which no other model is both more accurate and smaller.

Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) learn compositional functions through trainable one-dimensional edge transformations for approximation and interpretable modeling.

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FI-KAN: Fractal Interpolation Kolmogorov-Arnold Networks

Gnankan Landry Regis N'guessan

Axiom Research Group · Dept. of Applied Mathematics and Computational Science, NM-AIST, Arusha, Tanzania · AIMS Research and Innovation Centre, Kigali, Rwanda

Kolmogorov-Arnold Networks Neural Network Approximation Theory

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Kolmogorov-Arnold Networks (KAN) employ B-spline bases on a fixed grid, providing no intrinsic multi-scale decomposition for non-smooth function approximation. We introduce Fractal Interpolation KAN (FI-KAN), which incorporates learnable fractal interpolation function (FIF) bases from iterated function system (IFS) theory into KAN. Two variants are presented: Pure FI-KAN (Barnsley, 1986) replaces B-splines entirely with FIF bases; Hybrid FI-KAN (Navascues, 2005) retains the B-spline path and adds a learnable fractal correction. The IFS contraction parameters give each edge a differentiable fractal dimension that adapts to target regularity during training. On a Holder regularity benchmark (

α\in [0.2, 2.0]

), Hybrid FI-KAN outperforms KAN at every regularity level (1.3x to 33x). On fractal targets, FI-KAN achieves up to 6.3x MSE reduction over KAN, maintaining 4.7x advantage at 5 dB SNR. On non-smooth PDE solutions (scikit-fem), Hybrid FI-KAN achieves up to 79x improvement on rough-coefficient diffusion and 3.5x on L-shaped domain corner singularities. Pure FI-KAN's complementary behavior, dominating on rough targets while underperforming on smooth ones, provides controlled evidence that basis geometry must match target regularity. A fractal dimension regularizer provides interpretable complexity control whose learned values recover the true fractal dimension of each target. These results establish regularity-matched basis design as a principled strategy for neural function approximation.

Abstract

Kolmogorov-Arnold Networks

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