SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functions

Hoang-Thang Ta

cs.LGJun 30, 2026

SechKAN: Kolmogorov-Arnold Networks with Hyperbolic Secant Functions

Authors: Hoang-Thang Ta

Organizations: University of Information Technology, Ho Chi Minh City, Vietnam · Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam

Abstract

In recent years KolmogorovArnold Networks KANs have attracted increasing attention due to their effectiveness in machine learning and scientific computing offering a new paradigm for neural network design In this paper we present SechKAN a novel KAN based on hyperbolic secant sech functions The hyperbolic secant basis is adopted for its smooth bellshaped form localized responses and wellbehaved gradients We employ a 1D linear projection to reduce the number of parameters allowing SechKAN to maintain a model size comparable to that of multilayer perceptrons MLPs Experimental results show the effectiveness of SechKAN on function fitting PDE surrogate modeling and image classification benchmarks including MNIST FashionMNIST CIFAR10 and CIFAR100 On function fitting SechKAN achieves performance comparable to both MLPs and representative KAN variants On PDE surrogate modeling it outperforms MLPs and achieves competitive or better performance than representative KAN variants On image classification benchmarks SechKAN achieves the best performance among the evaluated KAN variants while remaining competitive with MLPs using a comparable number of parameters However SechKAN still incurs higher computational cost than MLPs and some KAN variants Our source code is publicly available at https://github.com/hoangthangta/All-KAN.

Figures & tables

Figure 1: Left: The structure of KAN(2,3,1). Right: Calculate

\phi_{1,1,1}

by using control points and B-splines.

G

and

k

represent the grid size and the spline order, while the number of B-splines,

n

, is given by

G+k=3+3=6

. The function

\phi

is calculated differently in theory and in implementation, where it is multiplied by its weight matrix and then added to a linear projection. The figure was modified from Ta et al. [34] .

Variant	Inner Function	Base Function
EfficientKAN [ 5 ]	$\phi(x)=w_{b}b(x)+w_{s}B(x)$	$\mathit{B}(x)=\sum_{i}{c_{i}B_{i}(x)},\quad c_{i}=$ spline coefficient
		$B_{i}(x)\equiv B_{i,k}(x)\text{ for chosen order }k$
		$B_{i,0}(x)=\begin{cases}1,&t_{i}\leq x<t_{i+1}\\ 0,&\text{otherwise}\end{cases}$
		$B_{i,k}(x)=\frac{x-t_{i}}{t_{i+k}-t_{i}}B_{i,k-1}(x)+\frac{t_{i+k+1}-x}{t_{i+k+1}-t_{i+1}}B_{i+1,k-1}(x)$
FastKAN [ 24 ]	$\phi(x)=w_{b}b(x)+w_{s}\mathit{RBF}(x)$	$\mathit{RBF}(x)=\exp\!\left(-\tfrac{\\|x-d\\|^{2}}{2h^{2}}\right),$
		$d=$ center, $h=$ function width or spread

Table 1: Summary of KAN variants, their defining base and inner functions.

Figure 2: SechKAN with multiple layers, and the architecture of a single SechKAN layer using sech basis functions.

Basis Function (KAN Variant)	Basis Characteristics	Parameter Reduction
B-spline (EfficientKAN [ 5 ] )	Piecewise polynomial, compact support, piecewise smooth, efficient B-spline implementation	–
Gaussian RBF (FastKAN [ 24 ] )	Infinitely smooth, Gaussian decay, smooth bounded gradients, analytic closed form	–
RSWAF (FasterKAN [ 10 ] )	Infinitely smooth, tanh-based localization, smooth bounded gradients, analytic closed form	–
ReLU-derived basis (ReLU-KAN [ 29 ] )	Piecewise quartic polynomial, compact support, piecewise smooth, analytic closed form	–
Hyperbolic secant or sech (SechKAN, Ours)	Infinitely smooth, exponential decay, smooth bounded gradients, analytic closed form	1D linear projection
B-spline and Gaussian RBF (PRKAN [ 35 ] )	Original B-spline and Gaussian RBF basis with dimensionality reduction	Attention, Conv, Conv+Pool, Dim-Sum, Feature Weight Vector

Table 2: Comparison of SechKAN and KAN variants based on their basis characteristics and parameter-reduction strategies. ’–’ indicates that no parameter reduction method is used.

Figure 3: Predictions and absolute error maps of different models compared with the ground truth during training on $f(x_{1},x_{2})=\sin(3\pi x_{1}^{2})\cos(3\pi x_{2})$ . From visual inspection, BSRBF-KAN, EfficientKAN, and FastKAN closely match the ground truth, followed by ReLU-KAN and SechKAN.

Functions	Other KANs	SechKAN
$f_{1}(x)=\exp(-4x^{2})\sin(6\pi x)$	[1,1]	[1,1]
$f_{2}(x)=\sin(4\pi x^{2})\cos(2\pi x)$	[1,1]	[1,1]
$f_{3}(x_{1},x_{2})=\sin(3\pi x_{1}^{2})\cos(3\pi x_{2})$	[2,4,4,1]	[2,14,14,1]
$f_{4}(x_{1},x_{2},x_{3})=\exp\!\left(-3(x_{1}^{2}+x_{2}^{2})\right)\sin\!\left(4\pi x_{3}+x_{1}x_{2}\right)$	[3,4,4,1]	[3,16,16,1]
$\begin{aligned} f_{5}(x_{1},x_{2},x_{3},x_{4})&=\sin\!\big(3\pi(x_{1}x_{2}+x_{3})\big)\\ &\quad\times\cos\!\big(2\pi(x_{2}x_{4})\big)\end{aligned}$	[4,8,8,1]	[4,32,32,1]
$f_{6}(x_{1},x_{2},x_{3})=x_{1}\left(1+x_{2}\cos x_{3}\right)$	[3,4,4,1]	[3,16,16,1]

Table 3: Functions and network structures used in the experiments. The network structures were designed so that all models have approximately the same number of trainable parameters. “Other KANs” refers to BSRBF-KAN, EfficientKAN, and FastKAN. ReLU-KAN uses the same network structures except for

f_{10}

, where the architecture is

[5,9,9,1]

Model	Function	Used Params	Loss (MSE) $\downarrow$	Time (s) $\downarrow$
BSRBF-KAN	Avg	648	$3.22\times 10^{-2}$	6.89
EfficientKAN	Avg	648	$1.93\times 10^{-2}$	6.69
FastKAN	Avg	658	$4.33\times 10^{-2}$	5.65
ReLU-KAN	Avg	643	$3.85\times 10^{-2}$	5.73
SechKAN	Avg	651	$\mathbf{1.75\times 10^{-2}}$	6.74

Table 4: Average performance comparison of models on 10 functions (

f_{1}

–

f_{1}0

) and 10 seeds. For each model, the number of parameters used, the MSE loss, and the running time are computed as the average over 100 runs (10 functions × 10 seeds).

Figure 4: Performance comparison of MSE loss and runtime for different models on 10 functions (

f_{1}

–

f_{1}0

) over 10 seeds (0–9). The reported MSE values are averaged across seeds for each function and model. (Top) MSE of the models for each function (log scale). (Bottom) Relationship between log-scale MSE and runtime across all models and functions.

Comparison	Mean Difference $\downarrow$	$p$ -value $\downarrow$	n Functions
SechKAN vs FastKAN	-0.025807	0.152600	10
SechKAN vs ReLU-KAN	-0.021036	0.110100	10
SechKAN vs BSRBF-KAN	-0.014759	0.152600	10
SechKAN vs EfficientKAN	-0.001836	0.747600	10

Table 5: Per-function permutation test results comparing SechKAN with baseline models (

n=10

functions). The permutation test is performed on the mean MSE of each function. Mean difference is computed as SechKAN minus the baseline.

Function	Comparison	Mean Difference $\downarrow$	$p$ -value	n Seeds
$f_{1}$	SechKAN vs EfficientKAN	0.001945	0.1375	10
	SechKAN vs FastKAN	-0.095961	0.0015	10
	SechKAN vs BSRBF-KAN	-0.028931	0.0015	10
	SechKAN vs ReLU-KAN	-0.063311	0.0015	10
$f_{2}$	SechKAN vs EfficientKAN	-0.003059	0.2158	10
	SechKAN vs FastKAN	-0.053119	0.0015	10

Table 6: Seed-level permutation test results for each function (

n=10

seeds per function). The permutation test is performed on paired MSE differences over the 10 random seeds. Mean difference is computed as SechKAN minus the baseline.

Figure 5: Visualization of the generated Navier–Stokes and Shallow Water datasets at different time steps. The first row shows the evolution of velocity magnitude in the Navier–-Stokes simulation, where small-scale turbulent structures gradually merge into larger coherent flow patterns. The second row shows the evolution of water surface height in the Shallow Water simulation, exhibiting wave propagation and layer dynamics over time.

Model	Val. Rel. L2 $\downarrow$	Test Rel. L2 $\downarrow$	Runtime (s) $\downarrow$	Used Params
BSRBF-KAN	0.09664 $\pm$ 0.00728	0.30327 $\pm$ 0.01853	422.67 $\pm$ 3.29	39744
EfficientKAN	0.08496 $\pm$ 0.00345	0.26524 $\pm$ 0.01692	388.91 $\pm$ 2.15	39630
FastKAN	0.30503 $\pm$ 0.01253	0.72194 $\pm$ 0.02485	86.05 $\pm$ 1.42	38684
FasterKAN	0.26852 $\pm$ 0.06415	0.55705 $\pm$ 0.13917	179.68 $\pm$ 51.91	39152
MLP	0.69984 $\pm$ 0.00096	0.72133 $\pm$ 0.00361	130.69 $\pm$ 21.11	39396
ReLU-KAN	0.78003 $\pm$ 0.01123	0.89849 $\pm$ 0.03682	139.67 $\pm$ 39.35	39782

Table 7: Performance comparison of SechKAN variants and baseline models on the Navier–Stokes dataset. SechKAN uses num_grids = 4 . Results are reported as mean

\pm

standard deviation across 5 random seeds.

Model	Val. Rel. L2 $\downarrow$	Test Rel. L2 $\downarrow$	Runtime (s) $\downarrow$	Used Params
BSRBF-KAN	0.15825 $\pm$ 0.01704	0.41038 $\pm$ 0.05549	215.60 $\pm$ 119.96	40320
EfficientKAN	0.20637 $\pm$ 0.01049	0.57372 $\pm$ 0.03928	143.23 $\pm$ 40.57	40230
FastKAN	0.21505 $\pm$ 0.01030	0.59755 $\pm$ 0.04715	116.11 $\pm$ 40.31	39252
FasterKAN	0.22423 $\pm$ 0.02903	0.39949 $\pm$ 0.03972	178.42 $\pm$ 65.66	39688
MLP	0.41146 $\pm$ 0.00383	0.46458 $\pm$ 0.00813	108.28 $\pm$ 69.98	38016
ReLU-KAN	0.41802 $\pm$ 0.01192	0.61521 $\pm$ 0.00488	700.64 $\pm$ 92.31	40311

Table 8: Performance comparison of SechKAN and baseline models on the Shallow Water dataset. SechKAN uses num_grids = 8 and LayerNorm at Norm1. Results are reported as mean

\pm

standard deviation over 5 random seeds.

Dataset	Model	Network structure	Used Params	MFLOPs $\downarrow$	Peak Mem. (MB) $\downarrow$
MNIST Fashion-MNIST	SechKAN	(784, 64, 10)	52,608	0.096	23.84
	MLP	(784, 64, 10)	50,816	0.051	17.42
	CNN	2 Conv layers + 2 MLP layers	52,138	0.378	19.87
	SechKAN-CNN	2 Conv layers + 2 SechKAN layers	52,160	0.418	21.09
CIFAR-10	SechKAN	(3072, 64, 10)	203,632	0.501	31.84
	MLP	(3072, 64, 10)	197,248	0.198	20.21

Table 9: Model configurations, parameter counts, MFLOPs, and peak GPU memory used in the image classification experiments.

Dataset	Network	Train. Acc. $\uparrow$	Val. Acc. $\uparrow$	Val. F1 $\uparrow$	Runtime (s) $\downarrow$
MNIST	MLP	$97.97\pm 0.02$	$97.28\pm 0.05$	$97.24\pm 0.05$	$69.59\pm 5.22$
	SechKAN	$99.52\pm 0.07$	$97.68\pm 0.08$	$97.65\pm 0.08$	$70.73\pm 2.32$
	SechKAN-CNN	$99.97\pm 0.01$	$98.94\pm 0.04$	$98.94\pm 0.04$	$179.76\pm 13.41$
	CNN	$99.95\pm 0.00$	$\mathbf{99.11\pm 0.02}$	$\mathbf{99.10\pm 0.02}$	$132.52\pm 5.42$
Fashion-MNIST	MLP	$90.57\pm 0.01$	$87.93\pm 0.04$	$87.85\pm 0.04$	$67.69\pm 5.63$
	SechKAN	$92.82\pm 0.06$	$89.03\pm 0.07$	$88.95\pm 0.07$	$70.69\pm 4.25$

Table 10: Performance comparison of MLP, SechKAN, CNN, and SechKAN-CNN across datasets. Results are averaged over five runs, with one run per seed (0, 1, 2, 3, and 4).

Figure 6: Forward pass time, backward pass time, and training throughput of MLP, SechKAN, CNN, and SechKAN-CNN across different datasets.

Figure 7: Validation accuracy per second and per parameter, normalized to [0, 1], reflecting training efficiency. A value of 1 indicates the best result, while 0 indicates the worst.

Model	Basis	1D Projection	Loss $\downarrow$	Params	Runtime (s) $\downarrow$
EfficientKAN	B-spline	$\times$	$\mathbf{2.03\times 10^{-2}\pm 5.48\times 10^{-2}}$	648	7.02 $\pm$ 2.43
Modified EfficientKAN	B-spline	$\checkmark$	$2.76\times 10^{-2}\pm 7.55\times 10^{-2}$	647	7.28 $\pm$ 2.52
Modified SechKAN	sech	$\times$	$5.26\times 10^{-2}\pm 8.24\times 10^{-2}$	653	6.01 $\pm$ 1.84
SechKAN (Proposed)	sech	$\checkmark$	$3.94\times 10^{-2}\pm 7.60\times 10^{-2}$	660	6.22 $\pm$ 1.85

Table 11: Ablation study of the sech basis function and 1D projection on the function-fitting benchmarks. Results are averaged over 10 functions and 3 random seeds (0, 1, and 2). Lower loss and runtime indicate better performance. All models are configured to have approximately the same number of trainable parameters, and no normalization layers are used.

Dataset	Model	Basis	1D Projection	Val. Acc. $\uparrow$	Runtime (s) $\downarrow$
MNIST	EfficientKAN	B-spline	$\times$	$92.09\pm 0.22$	$86.37\pm 1.52$
	Modified EfficientKAN	B-spline	$\checkmark$	$96.91\pm 0.06$	$86.15\pm 2.65$
	Modified SechKAN	sech	$\times$	$95.52\pm 0.09$	$\mathbf{71.08\pm 2.26}$
	SechKAN (Proposed)	sech	$\checkmark$	$\mathbf{97.12\pm 0.21}$	$71.69\pm 3.84$
Fashion-MNIST	EfficientKAN	B-spline	$\times$	$85.94\pm 0.06$	$88.21\pm 1.89$
	Modified EfficientKAN	B-spline	$\checkmark$	$88.01\pm 0.12$	$86.25\pm 4.05$

Table 12: Ablation study of the sech and B-spline basis functions and the 1D projection on image classification datasets. Results are averaged over 3 random seeds (0, 1, and 2). Higher validation accuracy and lower training time indicate better performance. All models are configured to have approximately the same number of trainable parameters.

Figure 8: SechKAN results with different numbers of grids, averaged over two seeds (0 and 1). (Left) MSE loss on 10 function-fitting tasks (

f_{1}

–

f_{10}

)

Figure 9: Heatmap of SechKAN validation accuracies, scaled from 0 to 1, with different numbers of grids on image classification datasets, averaged over two seeds (0 and 1). A value of 1 indicates the best result, while 0 indicates the worst.

Figure 10: Heatmap of SechKAN results with different activation functions, averaged over two seeds (0 and 1). MSE loss and relative L2 error are normalized to $[0,1]$ for each function or PDE problem (Navier–Stokes and Shallow Water). A value of 1 indicates the best result, while 0 indicates the worst.

Figure 11: Heatmap of SechKAN validation accuracies with different activation functions on image classification datasets, averaged over two seeds (0 and 1). Validation accuracies are normalized to $[0,1]$ . A value of 1 indicates the best result, while 0 indicates the worst.

Figure 12: Heatmap of SechKAN results averaged over two seeds (0 and 1) under different data normalization methods: no normalization, Norm1 or Norm2 positions with Min–Max, LayerNorm, BatchNorm, and RMSNorm. MSE loss and relative L2 error are normalized to $[0,1]$ for each function or PDE problem (Navier–Stokes and Shallow Water). A value of 1 indicates the best result, while 0 indicates the worst.

Figure 13: Heatmap of SechKAN validation accuracies on image classification datasets averaged over two seeds (0 and 1) for different data normalization methods: no normalization, and Min–Max, LayerNorm, BatchNorm, or RMSNorm applied at the Norm1 or Norm2 position. A value of 1 indicates the best result, whereas 0 indicates the worst result.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Function	Formula
sech	$\displaystyle\mathrm{sech}(x)=\frac{1}{\cosh(x)}$
Gaussian Radial Basis Function (Gaussian RBF)	$\displaystyle\psi(x)=e^{-x^{2}/2}$
Derivative of Gaussian	$\displaystyle\psi(x)=-x\,e^{-x^{2}/2}$
Mexican hat wavelet	$\displaystyle\psi(x)=\frac{2}{\sqrt{3}\pi^{1/4}}(1-x^{2})e^{-x^{2}/2}$
Morlet wavelet	$\displaystyle\psi(x)=\cos(\omega x)e^{-x^{2}/2},\quad\omega=5.0$
Cubic B-spline	$\displaystyle B(x)=\begin{cases}\displaystyle\frac{\mkern 4.0mu4-6\|x\|^{2}+3\|x\|^{3}\mkern 4.0mu}{6},&\|x\|<1,\\[7.74998pt] \displaystyle\frac{(2-\|x\|)^{3}}{6},&1\leq\|x\|<2,\\ 0,&\text{otherwise}.\end{cases}$

Appendix

Table 13: Some popular basis functions used in KANs.

Figure 14: Comparison of basis functions: (a) basis function shapes; (b) interpolation of four data points using each basis function.

Basis Function	Runtime (ms) $\downarrow$
Sech	$\mathbf{0.040\pm 0.127}$
Gaussian RBF	$0.050\pm 0.203$
Derivative of Gaussian	$0.051\pm 0.212$
Mexican hat wavelet	$0.146\pm 0.599$
Morlet wavelet ( $\omega=5$ )	$0.059\pm 0.156$
Cubic B-spline	$0.229\pm 0.449$

Appendix

Table 14: Evaluation runtime of basis functions implemented using native PyTorch CUDA operators. An input tensor of size

1000

is evaluated for each basis function. Results are averaged over

1{,}000{,}000

runs (mean

\pm

standard deviation in milliseconds).

Basis Function	Runtime (ms) $\downarrow$
B-spline (EfficientKAN)	$0.367\pm 0.017$
Bare sech (SechKAN)	$\mathbf{0.031\pm 0.002}$
Parametric sech (SechKAN)	$0.081\pm 0.002$
Gaussian RBF (FastKAN)	$0.056\pm 0.004$
RSWAF (FasterKAN)	$0.052\pm 0.004$

Appendix

Table 15: Forward-pass runtime of representative KAN basis implementations in PyTorch (CUDA). All implementations use eight basis functions and are evaluated on an input tensor of size

(64,784)

. Runtime is reported as the mean

\pm

standard deviation (ms) over

20{,}000

forward passes after

100

warm-up iterations.

Model	Function	Used Params	Loss (MSE) $\downarrow$	Runtime (s) $\downarrow$
BSRBF-KAN	$f_{1}$	22	$3.33\times 10^{-2}\pm 2.06\times 10^{-3}$	$2.91\pm 0.11$
FastKAN	$f_{1}$	22	$1.00\times 10^{-1}\pm 1.77\times 10^{-3}$	$2.70\pm 0.09$
ReLU-KAN	$f_{1}$	22	$6.76\times 10^{-2}\pm 1.22\times 10^{-2}$	$2.80\pm 0.12$
SechKAN	$f_{1}$	22	$4.32\times 10^{-3}\pm 3.72\times 10^{-3}$	$3.29\pm 0.15$
EfficientKAN	$f_{1}$	22	$\mathbf{2.37\times 10^{-3}\pm 8.43\times 10^{-6}}$	$2.97\pm 0.05$
BSRBF-KAN	$f_{2}$	22	$4.74\times 10^{-2}\pm 1.04\times 10^{-3}$	$2.92\pm 0.10$

Appendix

Table 16: Performance comparison of different models on functions

f_{1}

–

f_{1}0

(mean

\pm

standard deviation over 10 seeds).

Model	Val. Rel. L2 $\downarrow$	Test Rel. L2 $\downarrow$	Runtime (s) $\downarrow$	Used Params
SechKAN1	0.14179 $\pm$ 0.01390	0.31438 $\pm$ 0.02998	290.13 $\pm$ 2.10	38267
SechKAN2	0.22064 $\pm$ 0.21106	0.39155 $\pm$ 0.20445	298.70 $\pm$ 9.58	39035
SechKAN3	0.20465 $\pm$ 0.04908	0.33635 $\pm$ 0.04548	316.47 $\pm$ 67.48	38267
SechKAN4	0.15595 $\pm$ 0.03044	0.31355 $\pm$ 0.03014	229.53 $\pm$ 50.08	38243
SechKAN5 ( Table 7 )	0.13664 $\pm$ 0.01529	0.27695 $\pm$ 0.00954	266.04 $\pm$ 1.87	39011
SechKAN6	0.33738 $\pm$ 0.18304	0.41670 $\pm$ 0.13447	197.55 $\pm$ 64.63	38243

Appendix

Table 17: Performance comparison of SechKAN variants on the Navier–Stokes dataset (mean

\pm

std over 5 seeds).

Model	Val. Rel. L2 $\downarrow$	Test Rel. L2 $\downarrow$	Runtime (s) $\downarrow$	Used Params
SechKAN1 ( Table 8 )	0.12637 $\pm$ 0.00567	0.29879 $\pm$ 0.01425	196.59 $\pm$ 18.88	39228
SechKAN2	0.21322 $\pm$ 0.10831	0.34455 $\pm$ 0.09023	209.61 $\pm$ 40.14	39204
SechKAN3	0.29362 $\pm$ 0.12529	0.39399 $\pm$ 0.10176	87.34 $\pm$ 19.05	38436
SechKAN4	0.18429 $\pm$ 0.11008	0.33093 $\pm$ 0.10245	167.14 $\pm$ 66.28	38436
Average	0.20438	0.34207	165.17	38826

Appendix

Table 18: Performance comparison of SechKAN variants on the Shallow Water dataset (mean

\pm

std over 5 seeds).

Dataset	Network	Acc / Time (s) $\uparrow$	Acc / Param $\uparrow$
MNIST	MLP	$\mathbf{1.40\times 10^{-2}}$	$\mathbf{1.91\times 10^{-3}}$
	SechKAN	$1.38\times 10^{-2}$	$1.86\times 10^{-3}$
	SechKAN-CNN	$5.50\times 10^{-3}$	$1.90\times 10^{-3}$
	CNN	$7.48\times 10^{-3}$	$1.90\times 10^{-3}$
Fashion-MNIST	MLP	$\mathbf{1.30\times 10^{-2}}$	$1.73\times 10^{-3}$
	SechKAN	$1.26\times 10^{-2}$	$1.69\times 10^{-3}$

Appendix

Table 19: Efficiency comparison (validation accuracy per second and per parameter) of MLP, SechKAN, CNN, and SechKAN-CNN across datasets.

Dataset	Model	Network structure	Used Params	MFLOPs $\downarrow$	Peak Mem. (MB) $\downarrow$
MNIST Fashion-MNIST	SechKAN	(784, 64, 10)	52,608	0.096	23.84
	BSRBF-KAN	(784,7,10)	51,604	0.324	41.55
	FastKAN	(784,7,10)	51,621	0.108	26.59
	FasterKAN	(784,8,10)	52,400	0.086	26.64
	EfficientKAN	(784,7,10)	50,022	0.273	24.36
CIFAR-10	SechKAN	(3072, 64, 10)	203,632	0.501	31.84

Appendix

Table 20: Model configurations, parameter counts, MFLOPs, and peak GPU memory of different KAN variants in image classification experiments.

Dataset	Network	Train. Acc. $\uparrow$	Val. Acc. $\uparrow$	Val. F1 $\uparrow$	Runtime (s) $\downarrow$
MNIST	BSRBF-KAN	$94.47\pm 0.13$	$92.54\pm 0.08$	$92.38\pm 0.08$	$84.22\pm 2.32$
	EfficientKAN	$92.88\pm 0.19$	$92.27\pm 0.17$	$92.14\pm 0.18$	$86.26\pm 1.85$
	FastKAN	$94.79\pm 0.12$	$93.23\pm 0.14$	$93.08\pm 0.14$	$71.69\pm 2.24$
	FasterKAN	$93.06\pm 0.09$	$92.70\pm 0.08$	$92.53\pm 0.09$	$\mathbf{67.86\pm 2.26}$
	MLP	$97.97\pm 0.02$	$97.28\pm 0.05$	$97.24\pm 0.05$	$69.59\pm 5.22$
	SechKAN	$\mathbf{99.52\pm 0.07}$	$\mathbf{97.68\pm 0.08}$	$\mathbf{97.65\pm 0.08}$	$70.73\pm 2.32$

Appendix

Table 21: Performance comparison of KAN variants across datasets. Results are averaged over 5 runs, with one run per seed (0, 1, 2, 3, and 4).

Figure 15: Forward pass time, backward pass time, and training throughput of SechKAN, BSRBF-KAN, FastKAN, FasterKAN, and EfficientKAN across different datasets.

Model	Norm1	Norm2	Width	Best Loss	Final Grad.	Max Grad.	Grad. Std.
Model1	None	None	No	$1.61\times 10^{-4}$	$1.23\times 10^{-1}$	$1.92$	$2.01\times 10^{-1}$
Model2	None	None	Yes	$2.39\times 10^{-5}$	$1.82\times 10^{-3}$	$3.87$	$3.44\times 10^{-1}$
Model3	LayerNorm	None	No	$4.17\times 10^{-7}$	$4.54\times 10^{-2}$	$1.41$	$1.44\times 10^{-1}$
Model4	BatchNorm	None	No	$4.44\times 10^{-8}$	$1.24\times 10^{-2}$	$9.07\times 10^{-1}$	$3.88\times 10^{-2}$
Model5	RMSNorm	None	No	$1.57\times 10^{-6}$	$7.56\times 10^{-2}$	$1.33$	$1.54\times 10^{-1}$
Model6	Min–Max	None	No	$9.56\times 10^{-4}$	$5.72\times 10^{-2}$	$11.05$	$9.93\times 10^{-1}$

Appendix

Table 22: Configuration and empirical gradient statistics of the 18 SechKAN variants used for gradient flow analysis. Norm1 and Norm2 denote the two normalization positions illustrated in Figure 2 .

Figure 16: Training loss (solid blue) and L2 gradient norm (dashed red) during 5,000 training epochs for the 18 SechKAN variants. The variants differ in normalization placement (Norm1 or Norm2) and the use of a learnable basis width, as summarized in Table 22 .

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Sep 22, 2026cs.LG

Evaluating the Effectiveness of SechKAN on 1D Data

The connection between the Kolmogorov-Arnold representation theorem (KART) and neural network design has led to the development of Kolmogorov-Arnold Networks (KANs), with applications ranging from STEM problems to AI tasks. In this paper, we investigate the effectiveness of a KAN variant, SechKAN, which relies on hyperbolic secant (sech) functions as basis functions, with a 1D projection to reduce the number of parameters to a level comparable to MLPs. We evaluate SechKAN on three 1D classification datasets: UCI Human Activity Recognition (UCI HAR), ElectricDevices, and Crop, and compare it with several effective networks, including EfficientKAN, MLP, CNN1D, ResNet1D, and DSCNN1D, using approximately comparable parameter budgets. The results indicate that SechKAN achieves competitive performance across the three datasets, with particularly strong performance on Crop. Ablation studies further show that grid size and normalization affect performance, suggesting that SechKAN's effectiveness depends on the dataset and architectural choices. Our source code and experimental implementation are publicly available at: https://github.com/hoangthangta/SechKAN_1D.

Hoang-Thang Ta

University of Information Technology, Ho Chi Minh City, Vietnam · Vietnam National University Ho Chi Minh City, Ho Chi Minh City, Vietnam

Kolmogorov-Arnold Networks Human Activity Recognition Artificial Intelligence Systems

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Sep 1, 2026cs.LG

RecKAN: Kolmogorov-Arnold Networks with a Learnable Recursive Polynomial Basis

Kolmogorov--Arnold Networks (KANs) replace the fixed scalar weights of a standard network with learnable univariate functions on each edge, but existing variants still fix the \emph{basis} that those functions are built from: B-splines, Chebyshev polynomials, wavelets, or Jacobi polynomials, and learn only the combination weights over it. We introduce RecKAN, which instead defines the basis itself by a second order polynomial recurrence,

R_{n+1}(x) = (ax^2+bx+c)R_n(x) + (dx+e)R_{n-1}(x)

, whose five coefficients are learned jointly with the network. We show this recurrence recovers several classical polynomial families including both kinds of Chebyshev polynomials, Fibonacci, Pell, and Jacobsthal polynomials as special cases, and prove that its degree grows linearly in

n

exactly on the sub-family containing all of them, giving a concrete sense in which the learned basis can move beyond any fixed classical choice. Across multiple benchmark datasets spanning image, text, biomedical time series classification, and time series forecasting, RecKAN outperforms three parameter-matched KAN baselines (Chebyshev, Jacobi, and spline based) on all classification tasks and achieves the lowest MSE on the ETTh1 forecasting benchmark. Additionally, when used as a classifier head with a convolutional backbone, RecKAN achieves higher accuracy than standard MLP heads on Fashion MNIST, CIFAR-10, and SVHN. On a synthetic function fitting benchmark it tracks a sharply oscillatory target that a parameter comparable MLP under fits. We further show that the learned recurrence coefficients are interpretable: on the task requiring the most local structure, training moves the basis away from the linear degree growth regime that contains every classical family we identify, consistent with our theoretical analysis of what that structural shift enables.

Amirhosein Azarpour

Department of Computer Science Shahid Beheshti University Tehran, Iran

Kolmogorov-Arnold Networks Basis Functions Recurrent Neural Networks Polynomials

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Jun 25, 2026cs.LG

Kolmogorov Arnold networks (KAN) for aerodynamic prediction: a comparison with MLPs and GNNs

Kolmogorov Arnold networks (KAN) have recently been introduced as a (deep) neural network architecture whose trainable parameters adapt the activation functions, instead of the coefficients of the affine transformations at the core of traditional architectures such as deep multilayer perceptrons (MLPs). This architecture builds on the Kolmogorov-Arnold theorem, which endows it with universal approximation properties. While the advent of KANs has been received with excitement, there is a current debate about the possible KAN supremacy over deep multilayer perceptrons (MLPs) for classic fields such as symbolic regression, generic-purpose machine learning, natural language processing or computer vision. Here we assess the performance of KANs --and its nuanced comparison against MLPs and graph neural networks (GNNs)-- in the realm of fluid dynamics surrogate modelling. To that aim, we consider the task of predicting the surface pressure distribution over subsonic and transonic airfoils, a canonical task in aerodynamics. Our results show that KAN models show good performance in predicting the whole pressure coefficients and is able to interpolate across Mach numbers and angles of attack, however its performance is comparable --marginally inferior-- to a suitably trained MLP, where best performance is achieved by a GNN at the expense or requiring lengthier training. While the optimal KAN model have typically much lower complexity than MLP and GNN --hence resulting in faster training--, we find that KANs suffer from training instabilities, and their performance is highly dependent on a proper hyperparameter optimisation.

Miguel Jaraiz, Fermin Gutierrez, Pablo Yeste +4

ETSIAE-UPM-School of Aeronautics, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros 3, E-28040 Madrid, Spain · Center for Computational Simulation, Universidad Politécnica de Madrid, Campus de Montegancedo, Boadilla del Monte, 28660 Madrid, Spain · Institute for Cross-Disciplinary Physics and Complex Systems (IFISC, CSIC-UIB), 07122 Palma de Mallorca (Spain)

Kolmogorov-Arnold Networks Surrogate Models Aerodynamics Multilayer Perceptrons Graph Neural Networks Comparison

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