Kolmogorov-Arnold Networks Overview

• Kolmogorov–Arnold Networks (KANs) are neural architectures that replace fixed node activations with edge-based learnable univariate functions.
• KANs leverage the Kolmogorov–Arnold theorem to decompose multivariate functions, employing spline and other basis parameterizations for local, adaptable nonlinearity.
• Empirical studies demonstrate KANs’ competitive performance across tasks like time-series forecasting, image classification, and reinforcement learning, despite higher computational costs.

Kolmogorov–Arnold Networks (KANs) are feedforward neural architectures in which the primary nonlinearity is moved from nodes to edges: each connection carries a learnable univariate function, and node outputs are formed by additive aggregation of incoming transformed signals. They are motivated by the Kolmogorov–Arnold representation theorem, which states that continuous multivariate functions on compact domains can be represented as finite superpositions of univariate functions. In the original KAN formulation, linear weights are replaced by spline-parameterized univariate functions; later implementations retain the same edge-centric principle while varying the basis family, regularization, and computational realization (Liu et al., 2024, Noorizadegan et al., 28 Oct 2025).

1. Theorem, historical motivation, and conceptual position

The canonical motivation for KANs is the Kolmogorov–Arnold superposition form

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

or closely related index variants used across the literature. This theorem is not itself a training algorithm, but it provides the structural prior that multivariate dependence may be decomposed into sums and compositions of univariate mappings. KANs operationalize that prior by learning those univariate mappings directly rather than fixing activations in advance (Liu et al., 2024).

The original paper “KAN: Kolmogorov-Arnold Networks” formalized this idea as a practical alternative to multilayer perceptrons (MLPs), emphasizing that MLPs place fixed nonlinearities on nodes whereas KANs attach learnable functions to edges (Liu et al., 2024). Subsequent surveys and practitioner guides frame this as an “activate→mix” rather than “mix→activate” design: inputs are first transformed by edge-wise univariate functions and then summed at nodes, aligning the computation graph more closely with theorem-inspired superposition (Noorizadegan et al., 28 Oct 2025).

This does not imply that practical KANs are restricted to the theorem’s classical depth-2, width-$(2n+1)$ construction. The literature consistently treats modern KANs as deeper architectures with arbitrary widths and stacked layers, so the theorem functions primarily as architectural motivation rather than as a literal blueprint (Liu et al., 2024, Noorizadegan et al., 28 Oct 2025).

2. Canonical architecture and spline parameterization

A common KAN layer maps inputs $\{x_i\}$ to outputs $\{h_j\}$ through

$h_j=\sum_i s_{ij}(x_i)+b_j,$

where each $s_{ij}$ is a learnable univariate function associated with edge $i\to j$. In many implementations, these edge functions are parameterized by B-splines,

$$s(x;\boldsymbol{\theta})=\sum_{k=1}^{K}\theta_k\,B_k(x;\mathbf{t},d),$$

with knot vector $\mathbf{t}$, spline degree $d$, and learnable coefficients $(2n+1)$0. Because B-splines have local support, they provide localized nonlinear control without perturbing the entire input domain (Vaca-Rubio et al., 19 Oct 2025).

A widely used practical variant augments the spline with a residual base term,

$(2n+1)$1

with $(2n+1)$2 often chosen as $(2n+1)$3. This residual-spline form appears in several implementations because it improves optimization stability while preserving the core KAN principle that each edge carries a learnable univariate map (Liu et al., 2024, Gaonkar et al., 15 Jan 2026).

The architectural consequences differ sharply from MLPs. A standard dense MLP layer has $(2n+1)$4 scalar weights for input dimension $(2n+1)$5 and output dimension $(2n+1)$6; a KAN layer has $(2n+1)$7 basis coefficients when all $(2n+1)$8 atoms are active. For local bases such as B-splines, however, only about $(2n+1)$9 basis functions are nonzero at a given $\{x_i\}$0, so the effective forward and gradient cost becomes $\{x_i\}$1 rather than $\{x_i\}$2. This locality is one reason spline-based KANs remain numerically attractive despite their richer per-edge parameterization (Noorizadegan et al., 28 Oct 2025).

The choice of basis is central. The 2025 practitioner guide characterizes KANs as “basis-centric” architectures and surveys B-splines, Chebyshev and Jacobi polynomials, Gaussian radial basis functions (RBFs), Fourier series, and ReLU-power constructions, each with distinct smoothness, locality, and computational trade-offs (Noorizadegan et al., 28 Oct 2025).

3. Approximation theory, scaling laws, and relation to MLPs

The original KAN paper derives an approximation statement for functions that admit smooth KAN representations: if $\{x_i\}$3 has a KAN representation with sufficiently smooth univariate components, then a spline KAN with grid size $\{x_i\}$4 satisfies

$\{x_i\}$5

for $\{x_i\}$6. For cubic splines, this motivates the empirical observation that KAN test error can follow a substantially steeper neural scaling law than MLP baselines on toy regression and PDE tasks (Liu et al., 2024).

A later theoretical study places these claims in a broader function-space setting. Residual KANs with trainable cardinal B-spline activations are shown to approximate Besov functions $\{x_i\}$7 on bounded open domains, and even Ahlfors-regular fractal domains, at the optimal approximation rate with respect to weaker Besov norms $\{x_i\}$8 for $\{x_i\}$9. The same paper also gives a sample-complexity estimate for learning Besov-regular targets from noiseless data and explicitly incorporates residual or skip connections into the KAN architecture (Kratsios et al., 21 Apr 2025).

The relation to MLPs is not one of strict expressive superiority. The practitioner guide states that any $\{h_j\}$0 MLP can be parameterized as a KAN by approximating node activations with B-spline edge functions, and conversely that polynomial-type KANs can be rewritten as MLPs, though with substantially larger induced width. In its asymptotic comparison, the induced MLP parameter complexity is $\{h_j\}$1 versus $\{h_j\}$2 for the KAN of width $\{h_j\}$3, grid size $\{h_j\}$4, and depth $\{h_j\}$5, which is one formal explanation for KAN’s parameter-efficiency claims. The same source also notes that piecewise-linear KANs are functionally identical to ReLU MLPs (Noorizadegan et al., 28 Oct 2025).

This suggests that KANs are best viewed not as a new universality class, but as a different inductive bias: they represent nonlinearity through inspectable one-dimensional functions and often trade dense linear mixing for richer local function approximation.

4. Basis families, fast variants, and adaptive formulations

Although cubic B-splines remain the canonical basis, later work broadened the design space substantially. Chebyshev KANs use

$\{h_j\}$6

Jacobi KANs tune endpoint behavior through $\{h_j\}$7, RBF-KANs use Gaussian kernels

$\{h_j\}$8

and FourierKANs use sine and cosine bases for periodic structure (Noorizadegan et al., 28 Oct 2025).

A particularly influential efficiency result is “Kolmogorov-Arnold Networks are Radial Basis Function Networks,” which argues that the cubic B-splines used in early KANs can be well approximated by Gaussian RBFs. On an NVIDIA V100 benchmark for a $\{h_j\}$9 layer with 8 bases per input, FastKAN reported $h_j=\sum_i s_{ij}(x_i)+b_j,$0 forward time versus $h_j=\sum_i s_{ij}(x_i)+b_j,$1 for efficient-KAN, and $h_j=\sum_i s_{ij}(x_i)+b_j,$2 versus $h_j=\sum_i s_{ij}(x_i)+b_j,$3 for forward+backward, while showing equivalent MNIST validation behavior in the reported experiment (Li, 2024).

Another direction replaces the fixed “number of basis functions per edge” hyperparameter by a learned latent variable. InfinityKAN models each univariate function as a potentially infinite basis expansion, activates only a finite window of terms through a differentiable gating function, and learns the effective basis count via variational inference with Poisson latents. Empirically, it is competitive with fixed-basis KANs and often superior to matched-budget MLPs on several image and graph benchmarks, while introducing explicit machinery for adaptive basis-order selection (Alesiani et al., 3 Jul 2025).

By late 2025, the ecosystem had become broad enough to support specialized implementation families such as PyKAN, Efficient-KAN, JAX KAN, FourierKAN, FastKAN, ReLU-KAN, FBKAN, ChebyKAN, SincKAN, and wavelet KANs, reflecting a shift from a single spline architecture to a family of edge-function models differentiated primarily by basis choice and computational strategy (Noorizadegan et al., 28 Oct 2025).

5. Empirical performance across domains

Reported empirical results are heterogeneous but broad. In time-series forecasting for satellite traffic, a 4-depth KAN with architecture $h_j=\sum_i s_{ij}(x_i)+b_j,$4, cubic splines with $h_j=\sum_i s_{ij}(x_i)+b_j,$5 and $h_j=\sum_i s_{ij}(x_i)+b_j,$6, achieved MSE $h_j=\sum_i s_{ij}(x_i)+b_j,$7, RMSE $h_j=\sum_i s_{ij}(x_i)+b_j,$8, MAE $h_j=\sum_i s_{ij}(x_i)+b_j,$9, MAPE $s_{ij}$0, and 109k parameters, compared with an MLP $s_{ij}$1 at MSE $s_{ij}$2, RMSE $s_{ij}$3, MAE $s_{ij}$4, MAPE $s_{ij}$5, and 329k parameters (Vaca-Rubio et al., 2024).

That line of work was extended probabilistically in P-KAN, which replaces point forecasts by direct predictive distributions. On GEO satellite Physical Resource Block forecasting, the Student-$s_{ij}$6 P-KAN reported MSE $s_{ij}$7, MAE $s_{ij}$8, RMSE $s_{ij}$9, and CRPS $i\to j$0, while the Gaussian P-KAN reported stronger calibration and more conservative uncertainty. The paper emphasizes that P-KAN models used approximately 82k–90k parameters versus more than 240k for MLP baselines, and that Gaussian and Student-$i\to j$1 heads occupy different efficiency–risk operating points for dynamic thresholding (Vaca-Rubio et al., 19 Oct 2025).

In image classification, KANs have usually been deployed as heads on top of convolutional features rather than as full replacements for convolutional backbones. For metal surface defect classification, FourLayerConvKAN achieved $i\to j$2 on the NEU dataset and ThreeLayerConvTwoLayerKAN achieved $i\to j$3, with the authors noting that some KAN-augmented models reached higher accuracy than comparable CNN baselines with fewer parameters in some configurations, albeit with longer CPU training times (Krzywda et al., 10 Jan 2025). In autoencoding, KAN-based autoencoders were competitive on MNIST and stronger on SVHN and CIFAR-10 at several bottleneck sizes, though they often used far more parameters than the convolutional autoencoders to which they were compared (Moradi et al., 2024).

In reinforcement learning, replacing PPO actors or both actor and critic with KANs yielded competitive continuous-control performance. The KAN actor-only variant achieved the highest mean reward across six DeepMind Control Proprio robotics tasks, with average actor parameter count 348 versus 5348 for the 2×64 MLP actor. The trade-off was compute: on HalfCheetah, 1000 environment steps took 3.39 s for KAN versus 0.37 s for the MLP baseline (Kich et al., 2024).

In materials informatics, two strands emerged. For thermoelectric prediction, a KAN on CrystalFormer embeddings achieved test $i\to j$4, RMSE $i\to j$5 eV, and MAE $i\to j$6 eV for band gap, versus an MLP at $i\to j$7, RMSE $i\to j$8 eV, and MAE $i\to j$9 eV, while also enabling symbolic surrogates built from fitted edge functions (Fronzi et al., 3 Oct 2025). For a broader 33-property benchmark, KANs were worse, on par, or better than random forest about 35%, 60%, and 5% of the time, respectively; tuned KANs often achieved 10–20% lower errors than the standard KAN, and in the RPV embrittlement case a tuned KAN with 10,730 parameters achieved RMSE $$s(x;\boldsymbol{\theta})=\sum_{k=1}^{K}\theta_k\,B_k(x;\mathbf{t},d),$$0, close to a deep MLP NN at $$s(x;\boldsymbol{\theta})=\sum_{k=1}^{K}\theta_k\,B_k(x;\mathbf{t},d),$$1, while tiny KANs with 27–51 parameters were on par with hand-tuned domain models (Jacobs et al., 27 Jan 2026).

6. Interpretability, limitations, robustness, and open questions

KAN interpretability follows directly from its parameterization. Because each edge carries a learned univariate function, practitioners can inspect $$s(x;\boldsymbol{\theta})=\sum_{k=1}^{K}\theta_k\,B_k(x;\mathbf{t},d),$$2 directly, visualize monotonicity, saturation, local bumps, or periodicity, and in some studies fit these splines to simple symbolic expressions. The original KAN paper used pruning, sparsification, and symbolic “snapping” to recover compact laws in mathematics and physics, while later materials studies extracted low-complexity surrogates from fitted spline edges (Liu et al., 2024, Fronzi et al., 3 Oct 2025).

That interpretability does not remove substantial practical limitations. Training cost is a recurring concern: metal-defect experiments reported significantly longer CPU training times for KANs (Krzywda et al., 10 Jan 2025); RL experiments showed roughly one order of magnitude slower per-step computation despite much smaller actors (Kich et al., 2024); and the imbalanced tabular study measured baseline KANs at about 505 s training time and about 1.48 MB memory, versus MLPs clustered near 0.5–0.97 s and about 0.06 MB, concluding that KANs currently impose severe performance–resource trade-offs in that setting (Yadav et al., 18 Jul 2025).

Robustness results are also mixed. The adversarial study found that small and medium fully connected KANs are not consistently more robust than FCNNs, whereas large KANs are generally more robust under several white-box and query-based black-box attacks, though large fully connected KANs remained highly vulnerable to NES and robustness gains on CIFAR-10 and SVHN were limited or inconsistent (Alter et al., 2024). In imbalanced classification, KANs performed best on raw imbalanced tabular data but degraded under SMOTE–Tomek resampling and focal loss, while MLPs with imbalance handling became practically equivalent at far lower resource cost (Yadav et al., 18 Jul 2025).

Several open technical issues recur across the literature. Basis selection remains problem-dependent; knot placement and spline degree materially affect calibration and stability in probabilistic forecasting (Vaca-Rubio et al., 19 Oct 2025). Some implementations require grid extension or input normalization to prevent spline activations from falling outside their operating range; one comparative study notes that if inputs exceed the spline range and the grid is not refreshed, the layer can effectively degenerate toward a SiLU-only pathway, making the KAN behave more like an MLP with fixed activations (Gaonkar et al., 15 Jan 2026). More broadly, current reviews identify unresolved questions in scalability, stability, hyperparameter selection, theoretical generalization bounds with adaptive bases, and standardized benchmarking across domains (Noorizadegan et al., 28 Oct 2025, Faroughi et al., 30 Jul 2025).

A plausible implication is that KANs are becoming less a single architecture than a design space: edge-wise learnable univariate functions remain the defining principle, but practical success depends on choosing the right basis, regularization, and computational realization for the problem at hand.