CKAN: Convolutional Kolmogorov-Arnold Networks

• CKAN is a family of architectures that extends traditional convolution by replacing fixed scalar weights with learnable univariate nonlinear functions, such as splines.
• It is applied across domains like image classification, segmentation, option pricing, and astrophysical inference, often achieving competitive performance with fewer parameters.
• Despite its innovative design, CKAN faces practical challenges in training speed, memory usage, and robustness, sparking debates on theoretical fidelity and efficiency.

Convolutional Kolmogorov-Arnold Network (CKAN) is a family of architectures that extends Kolmogorov-Arnold Networks (KANs) into convolutional settings by replacing fixed scalar kernel weights, or parts of the conventional CNN pipeline, with learnable univariate functions, typically spline-based. In the canonical 2024 formulation, a CKAN convolutional kernel is a matrix of functions rather than a matrix of numbers, so the local mapping is nonlinear inside the convolution itself rather than only after a linear convolution followed by a fixed activation (Bodner et al., 2024). Subsequent work uses the label more broadly for hybrid systems that combine CNN feature extraction with KAN classifier heads, fuzzy pooling, attention modules, or specialized operator-learning sub-networks (Igali et al., 2024, Yunusa et al., 2024, Wu et al., 9 Oct 2025). This suggests that CKAN is best understood as an active design space rather than a single settled architecture.

1. Theoretical basis and canonical formulation

CKAN inherits its conceptual basis from the Kolmogorov-Arnold representation theorem, written in several papers as

$$f(x_1,\dots,x_n)=\sum_{q=1}^{2n+1}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right).$$

In KANs, the central departure from MLPs is that learnable univariate functions are placed on edges rather than fixed activations on nodes, and these functions are commonly parameterized with splines and a base activation such as SiLU (Bodner et al., 2024).

In the original convolutional adaptation, a CKAN kernel entry is a learnable nonlinear function $\phi_{kl}$, and the convolution at spatial position $(i,j)$ becomes

$$(\text{Image} * K)_{i,j}=\sum_{k=1}^{N}\sum_{l=1}^{M}\phi_{kl}(a_{i+k,j+l}),$$

instead of a standard dot product with scalar weights (Bodner et al., 2024). A more general convolutional expression used in a later design study is

$$x_{ij}=\sum_{d=1}^{c}\sum_{a=0}^{k-1}\sum_{b=0}^{k-1}\varphi_{a,b,d}(y_{d,i+a,j+b}),$$

where $\varphi_{a,b,d}$ is a trainable univariate nonlinear function for each relative kernel position and channel (Drokin, 2024).

A common kernel parameterization is

$$\phi(x)=w_1\cdot \text{spline}(x)+w_2\cdot \text{silu}(x),$$

with the spline expanded as a B-spline basis and $w_1,w_2$ trained jointly (Bodner et al., 2024). The crucial architectural implication is that convolution is no longer a linear local operator followed by a separate nonlinearity; it is already nonlinear at each kernel position.

2. Architectural realizations and design variants

The earliest CKAN work presents a direct substitution strategy: conventional convolutional kernels are replaced by learnable spline-based functions, and the resulting networks are evaluated as CNN alternatives (Bodner et al., 2024). In this formulation, each kernel location performs its own learned scalar-to-scalar transform before summation, which is the most literal convolutional generalization of KAN.

A second line of work uses CKAN more loosely for hybrid pipelines. One example combines three components: convolutional neural networks for feature extraction, Type-1 Fuzzy Pooling for subsampling, and a KAN classification head for final decision-making. In that pipeline, the image passes through convolutional layers, fuzzy pooling replaces max or average pooling, and the flattened features are sent to a KAN instead of an MLP classifier. The fuzzy pooling stage fuzzifies each patch into three fuzzy sets by triangular membership functions, aggregates by fuzzy algebraic sum, selects the most certain set via

$v^*=\arg\max_v s_{\pi_v}^n,$

and then defuzzifies by a center-of-gravity weighted average (Igali et al., 2024).

A broader design study extends the convolutional KAN idea beyond splines. It examines splines, radial basis functions, wavelet transforms, and several polynomial families including Legendre, Chebyshev, and Gram polynomials, and also proposes bottleneck convolutional KAN layers, a Mixture-of-Experts bottleneck, parameter-efficient finetuning, and KAN convolutional versions of self-attention and focal modulation (Drokin, 2024). In that literature, CKAN is less a single layer type than a general recipe for embedding trainable univariate basis functions into local operators.

Later work also questions whether early CKAN implementations preserve the theorem’s intended structure. KAConvNet argues that existing attempts such as CKAN and KACN “merely replaced learnable activation functions with weights,” and proposes a KAConvLayer that applies inner functions to each pixel or channel and outer functions to aggregate them, with both parameterized as learnable piecewise-linear activation functions (“GLinear”) rather than B-splines (Liu et al., 25 Apr 2026). This is a critique internal to the literature rather than a settled consensus, but it shows that “theory-aligned” CKAN design is itself contested.

3. Empirical behavior in vision benchmarks

Published results on image classification and segmentation are heterogeneous. Early studies report parameter-efficient or competitive performance on small and medium-scale benchmarks, while later studies report degraded robustness, high memory use, and underperformance when CKAN replaces convolutional layers more aggressively or when the task becomes larger and noisier (Bodner et al., 2024, Igali et al., 2024, Cang et al., 2024).

Setting	CKAN-side result	Comparator
Fashion-MNIST small model	CKAN (Small): 94.9k params, 89.69%, 1.83 min/epoch	CNN (Medium): 157k, 90.14%, 0.29 min/epoch
CIFAR-10	KAN + Fuzzy Pooling: 67.06%	CNN+MLP+Max Pooling: 66.69%
FashionMNIST	KAN + Fuzzy Pooling: 89.88%	CNN+MLP+Max Pooling: 89.81%
MNIST	KAN + Fuzzy Pooling: 98.91%	CNN+MLP+Max Pooling: 98.89%
CIFAR-100	MobileNet + CKAN (First layer): 58.27; MobileNet + CKAN (Last layer): 58.64	MobileNet (Baseline): 61.14
PASCAL VOC2012 segmentation	UNet + CKAN: 59.13 mIoU	UNet (Baseline): 63.28 mIoU

The original CKAN paper reports competitive accuracy on Fashion-MNIST with up to 50% fewer parameters compared to baseline classic convolutions, and gives a concrete Fashion-MNIST example in which CKAN (Small) achieves 89.69% with 94.9k parameters while CNN (Medium) reaches 90.14% with 157k parameters (Bodner et al., 2024). The fuzzy-pooling hybrid reports 67.06% on CIFAR-10, 89.88% on FashionMNIST, and 98.91% on MNIST, each slightly above or close to the corresponding CNN+MLP+Max Pooling baselines listed in the paper (Igali et al., 2024).

By contrast, a later computer-vision study finds that CKAN underperforms traditional CNN baselines in both image classification and semantic segmentation. On CIFAR-100, MobileNet scores 61.14, whereas MobileNet + CKAN in the first or last layer reaches 58.27 and 58.64. On PASCAL VOC2012, UNet records 63.28 mIoU while UNet + CKAN records 59.13. The same study reports that CKAN consistently underperforms CNNs across dataset sizes and is highly sensitive to label noise (Cang et al., 2024).

The broader Kolmogorov-Arnold convolution literature reports more favorable outcomes for alternative bases and architectures. A design study reports state-of-the-art results on BUSI, GlaS, and CVC segmentation datasets with U-Net-like architectures using KAN convolutions, and empirical evaluations spanning MNIST, CIFAR10, CIFAR100, Tiny ImageNet, ImageNet1k, and HAM10000 for image classification (Drokin, 2024). KAConvNet reports 80.1% Top-1 accuracy for KAConvNet-L on ImageNet-1K and 70.58 mIoU on Cityscapes, while explicitly positioning itself as an improvement over prior methods combining KAN and convolution (Liu et al., 25 Apr 2026). This suggests that reported performance depends strongly on the exact convolutional formulation, the nonlinearity basis, and whether KAN principles are used in the convolutional body, the classifier head, or both.

4. Efficiency, regularization, and robustness

Efficiency is the dominant practical objection to CKAN. The original CKAN work already notes slower training, reporting that current implementations are slower, with 6x longer per epoch in some cases, even when parameter counts are favorable (Bodner et al., 2024). A later scrutiny on ImageNet-1k, MNIST, and the MoA tabular dataset makes the bottleneck explicit: CKANs are reported to perform fair yet slower than CNNs on small datasets and to be “not nearly comparable” as the dataset gets larger and more complex (Dahal et al., 27 Jan 2025).

That study reports the following contrasts. AlexNet KAN has 1.61B FLOPS versus 0.71B for AlexNet, 0.0074s inference time per sample versus 0.0018s, and 48 days to train 100 epochs versus 3 days, while also underperforming in Top-1 and Top-5 accuracy: 42.8% and 67.7% for AlexNet KAN versus 56.6% and 79.1% for AlexNet. On MNIST, LeNet KAN reaches 98.8% against 98.9% for LeNet, but still requires 0.0030s inference time versus 0.0007s (Dahal et al., 27 Jan 2025). These results support the claim that the main bottleneck is not only parameter count but also the cost of evaluating many nonlinear kernel functions on hardware optimized for linear algebra.

Robustness is a second recurring issue. One vision study attributes CKAN’s weak practical performance to low generalization ability, noise sensitivity, high memory footprint, slow computation, and overfitting induced by the flexibility of spline activations (Cang et al., 2024). In that same study, smoothness regularization penalizing spline curvature and Segment Deactivation, which replaces a spline with a linear function joining its endpoints with probability $p$, improve KAN when used as a final classifier layer, but do not rescue CKAN when used inside the convolutional body. On CIFAR-10, the paper reports 71.96% for CNN+MLP, 72.58% for CNN+KAN, 73.04% for CNN+KAN+SmoothReg, 73.12% for CNN+KAN+Segment Deactivation, and 73.30% for CNN+KAN+Both (Cang et al., 2024).

Regularization has also been studied in broader KAN-convolution work. A design study assesses $\phi_{kl}$0 and $\phi_{kl}$1 penalties, several forms of dropout, additive Gaussian noise injection, label smoothing, and weight decay, and states that additive noise can be as effective or better than dropout, with “Full” dropout or noise injection recommended for KAGN models (Drokin, 2024). A separate continual-learning study reports that KAN-based convolutional models did not outperform standard CNNs in class-incremental learning on MNIST, that all models suffered notable catastrophic forgetting, and that batch normalization was essential for convergence in both KAN and traditional convolutional models (Cacciatore et al., 2024).

5. Domain-specific applications

CKAN has been adapted to several domains outside conventional image classification. In option pricing, Conv-KAN is introduced alongside KAN, Conv-LSTM, and LSTM under a Black-Scholes-Merton setting with an additional variable. The paper states that this is the first application of (Conv-)KANs to option pricing and reports the following test-set metrics: Conv-KAN obtains MSE 0.00790, RMSE 0.08890, and MAE 0.07098, outperforming B-S, B-S-M, LSTM, Conv-LSTM, and KAN on those three metrics, while MAPE remains high for the convolutional models (Li et al., 2024).

In crop field detection, KonvLiNA integrates cKAN with Nyström attention. The cKAN module is deployed inside a cKSPP component for multi-scale feature extraction, where pooled feature maps are passed through a $\phi_{kl}$2 cKAN layer before fusion with an attention-based upsampling branch. On the rice crop dataset, the framework reports 0.415 AP and 0.459 AR with the Swin-L backbone, and the paper states that it surpasses YOLOv8 by 3.5% AP and 4.37% AR; the inclusion of cKSPP is reported to yield a dramatic AP boost of +15.6% in ablations (Yunusa et al., 2024).

In astrophysical inference, a CKAN is used to distinguish self-interacting dark matter from cold dark matter in galaxy clusters. The reported architecture uses three parallel subnetworks, one per channel, with convolutional KAN layers and fully connected KAN layers, followed by summed outputs and a softmax prediction. The paper reports 14,208 parameters and 80% accuracy for CKAN, compared with 13,480,419 parameters and $\phi_{kl}$3 accuracy for an InceptionCNN, and estimates that the minimum cross-section required to reliably identify SIDM falls between $\phi_{kl}$4 and $\phi_{kl}$5 (Huang et al., 8 Sep 2025).

In single-image super-resolution, SRGAN-CKAN reformulates convolution as a nonlinear patch-based transformation. The input is unfolded into local patches, each patch is processed by a KAN-based mapping, and the outputs are folded back into the spatial grid. The paper reports that chunked processing allows $\phi_{kl}$6 images to fit into an 8GB GPU with batch size 1, and gives quantitative results in which SRGAN-CKAN achieves PSNR-Y 24.90, MS-SSIM-Y 0.9422, and LPIPS 0.1947, compared with SRGAN (conv) at 20.07, 0.8550, and 0.3226 (Navaro-Aviña et al., 2 May 2026).

6. Terminological ambiguity, criticisms, and later developments

The label “CKAN” is not entirely stable across the literature. In most 2024 vision papers it denotes Convolutional Kolmogorov-Arnold Networks, but in PO-CKAN it denotes Chunkwise Rational Kolmogorov-Arnold Networks used as branch and trunk sub-networks inside a physics-informed DeepONet-style operator-learning framework (Wu et al., 9 Oct 2025). That paper replaces edge-wise spline activations with chunkwise shared rational functions,

$\phi_{kl}$7

and states that chunkwise sharing reduces the prohibitive parameter and computation cost of standard KANs. On Burgers’ equation with $\phi_{kl}$8, it reports an approximately 48% reduction in mean relative $\phi_{kl}$9 error compared to PI-DeepONet (Wu et al., 9 Oct 2025). This use is architecturally related to KAN efficiency engineering, but it is not a convolutional vision model in the standard CKAN sense.

A further controversy concerns theoretical fidelity. KAConvNet argues that prior KAN-based convolution methods, including CKAN and KACN, undermine the Kolmogorov-Arnold theoretical foundation by simplifying the placement and sharing of activation functions, and it proposes a new KAConvLayer intended to align more directly with the theorem (Liu et al., 25 Apr 2026). This criticism coexists with earlier positive empirical reports on CKAN-like models (Bodner et al., 2024, Igali et al., 2024), and with broader design studies showing strong results for non-spline bases and bottleneckized variants (Drokin, 2024). A plausible implication is that the field is converging not on a single “correct” CKAN, but on a spectrum of KAN-inspired local operators with different trade-offs in theorem alignment, interpretability, parameterization, and hardware efficiency.

Taken together, the literature presents CKAN as a technically fertile but unsettled paradigm. Its core idea—replacing linear local mappings with learnable univariate functional operators—has produced competitive small-scale vision models, specialized domain systems, and several downstream variants. At the same time, large-scale efficiency, robustness to noise, and agreement on what constitutes a theorem-faithful convolutional KAN remain open points of contention (Cang et al., 2024, Dahal et al., 27 Jan 2025, Liu et al., 25 Apr 2026).