Spline-Based Kolmogorov-Arnold Networks

• Spline-based KANs are neural architectures where edges carry trainable univariate spline functions instead of scalar weights, motivated by the Kolmogorov-Arnold theorem.
• These networks integrate B-spline expansions with base activations to achieve improved function fitting in PDE-oriented and time-series applications.
• Recent advances address computational challenges with GPU-accelerated spline evaluation, effective grid management, and novel initialization strategies.

Searching arXiv for recent KAN papers to ground the article in current literature. Spline-based Kolmogorov-Arnold Networks (KANs) are neural architectures in which the trainable object on each connection is a univariate spline-parameterized function rather than a scalar weight, while nodes primarily aggregate incoming signals by summation. Motivated by the Kolmogorov-Arnold representation theorem, they relocate nonlinearity from nodes to edges and typically instantiate each edge function with a B-spline expansion, optionally combined with a base activation such as SiLU. In the original formulation by Liu et al., KANs were proposed as alternatives to multilayer perceptrons (MLPs), with claims of stronger accuracy-interpretability trade-offs in function fitting and PDE-oriented settings (Liu et al., 2024).

1. Theorem, motivation, and the shift from node activations to edge functions

The canonical theoretical reference point is the Kolmogorov-Arnold representation theorem, written in the KAN literature as

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

or equivalently with minor index variations across papers. The operative interpretation in KAN research is that high-dimensional continuous functions can be represented through sums and compositions of univariate functions, which motivates learning families of one-dimensional transformations rather than relying exclusively on linear maps plus fixed nodewise nonlinearities (Liu et al., 2024).

This leads to the central architectural inversion relative to MLPs. In an MLP, the edge carries a scalar weight and the node applies a fixed activation. In a spline-based KAN, each edge carries a learnable univariate function, and the node typically performs only summation. The literature repeatedly describes this as “activation on edges, not nodes,” and uses it to explain both the claimed expressivity and the claimed interpretability of the architecture (Vaca-Rubio et al., 2024).

The original KAN formulation also generalizes the shallow theorem-inspired construction to arbitrary depth and width. A theorem-matching two-layer shape such as $[n,2n+1,1]$ appears in the time-series exposition, but practical KANs are stacked compositions of matrices of univariate functions rather than direct literal implementations of the classical representation theorem (Vaca-Rubio et al., 2024).

2. Architectural form and spline parameterization

A standard KAN layer is written as

$x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$

with $\phi_{l,j,i}$ the learnable edge function from input coordinate $i$ in layer $l$ to output coordinate $j$ in layer $l+1$. A full network is then expressed as a composition of such layers,

$$\mathrm{KAN}(\mathbf{x})=(\mathbf{\Phi}_{L-1}\circ\cdots\circ \mathbf{\Phi}_0)(\mathbf{x}),$$

where each $\mathbf{\Phi}_l$ is a matrix of univariate functions (Liu et al., 2024).

The usual spline parameterization is

$[n,2n+1,1]$0

with $[n,2n+1,1]$1 denoting B-spline basis functions and $[n,2n+1,1]$2 trainable coefficients. Several implementations augment the spline branch with a base branch,

$[n,2n+1,1]$3

or an equivalent notation such as $[n,2n+1,1]$4. This hybrid form appears in the original KAN description and in convolutional KAN variants, where it serves as an optimization aid and a fallback outside effective spline support (Liu et al., 2024).

Grid structure is integral to the spline-based formulation. Classical KANs use fixed, bounded grids and a fixed number of B-spline coefficients. This creates well-known range-management issues: if activations move outside the grid, learning can stall or the spline contribution can degenerate. The literature therefore discusses grid extension, batch normalization, and related stabilization mechanisms. UKAN removes the bounded-grid assumption by replacing static spline coefficients with a coefficient generator conditioned on positional encodings of an infinite symmetric grid and feature embeddings; its stated consequence is that evaluation no longer requires data normalization or bounded input domains (Moradzadeh et al., 2024).

The same spline logic has also been transplanted into convolution. In Convolutional KANs, each classical convolutional kernel weight is replaced by a learnable nonlinear function $[n,2n+1,1]$5, yielding

$[n,2n+1,1]$6

so that nonlinearity is integrated into each kernel entry rather than appended after a linear convolution (Bodner et al., 2024).

3. Approximation theory, spline spaces, and relations to MLPs

The theoretical case for spline-based KANs is articulated at several levels. In the original paper, KANs are said to approximate compositional smooth functions with an error bound of the form

$[n,2n+1,1]$7

where $[n,2n+1,1]$8 is grid size and $[n,2n+1,1]$9 is spline order, and to exhibit faster neural scaling laws than MLPs for smooth targets (Liu et al., 2024).

A more formal approximation-theoretic account is given for residual spline-based KANs in Besov spaces. That work states that KANs can optimally approximate any Besov function in $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$0 on a bounded open, or even fractal, domain $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$1 with respect to a weaker Besov norm $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$2, $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$3, and complements this with a dimension-free estimate on sample complexity for learning from noiseless samples. Its construction uses trainable linear combinations of cardinal B-splines inside residual layers, thereby tying KAN approximation guarantees directly to spline and wavelet multiresolution machinery (Kratsios et al., 21 Apr 2025).

A distinct theoretical perspective comes from the change-of-basis analysis of spline KANs. There, KAN layers of the form

$x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$4

are shown to be equivalent, via a linear change of basis, to dense multichannel MLPs with power ReLU activations. The paper’s main point is not merely equivalence of function classes, but the fact that the basis choice alters optimization geometry: in the spline basis, compact support and knot refinement induce a properly nested hierarchy suitable for multilevel training, whereas the ReLU-basis view emphasizes smooth modes and can impede fine-scale correction (Southworth et al., 5 Mar 2026).

Taken together, these results situate spline-based KANs at the intersection of neural approximation theory and classical spline analysis. A plausible implication is that much of the architecture’s distinct behavior is less about universal approximation in the abstract—shared by many neural models—and more about the numerical and statistical consequences of choosing localized spline coordinates for the learned univariate operators.

4. Optimization, initialization, and computational acceleration

The main practical objection to spline-based KANs is computational cost. The original KAN work reports training that is less efficient than MLP training, and later papers repeatedly attribute the bottleneck to B-spline evaluation, grid management, and the lack of mature low-level kernels (Liu et al., 2024).

Several strands of work address this directly. UKAN introduces a GPU-accelerated library for B-spline evaluation whose complexity becomes $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$5, independent of grid size, rather than $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$6 in prior code, and reports 5.5–15x speedup versus previous PyTorch implementations together with feasible grid sizes up to $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$7 rather than $x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$8 (Moradzadeh et al., 2024). LTBs-KAN attacks the same problem algorithmically by replacing recursive B-spline evaluation with a linear-time Bernstein-Bézier coefficient approach and coupling it with product-of-sums matrix factorization, reporting good time complexity and parameter reduction on MNIST, Fashion-MNIST, and CIFAR-10 when used as building blocks (Merin-Martinez et al., 23 Apr 2026).

Training dynamics have also become a topic in their own right. An empirical study of initialization schemes for spline-based KANs proposes LeCun-inspired, Glorot-inspired, and power-law families; it states that the Glorot-inspired scheme significantly outperforms the baseline in parameter-rich models, while the power-law scheme achieves the strongest overall performance across tasks and architecture sizes. The same study analyzes the Neural Tangent Kernel and associates improved initialization with more stable spectra and avoidance of early rank collapse in larger KANs (Rigas et al., 3 Sep 2025).

At the systems level, KAN-specific deployment work has begun to appear. QuantKAN extends QAT and PTQ methods to spline-based layers with branch-specific quantizers for base, spline, and activation components, and concludes that KANs are compatible with low-bit quantization, though with strong method-architecture interactions (Fuad et al., 24 Nov 2025). KAN-SAs analyzes systolic-array inference for B-spline KANs, replacing recursive evaluation with a nonrecursive implementation and exploiting intrinsic KAN sparsity; it reports up to 100% systolic-array utilization and up to 50% clock-cycle reduction versus conventional arrays of equivalent area (Errabii et al., 20 Nov 2025).

5. Variants and applications

Spline-based KANs have been adapted to multiple modeling settings, often by preserving the core idea of learnable spline-parameterized edge functions while changing the surrounding network scaffold.

Variant or setting	Core adaptation	Reported outcome
Convolutional KAN (Bodner et al., 2024)	Replace scalar kernel entries with learnable spline functions	Competitive accuracy on Fashion-MNIST with up to 50% fewer parameters
Time-series KAN (Vaca-Rubio et al., 2024)	Spline-parameterized edge functions for forecasting	Outperforms conventional MLPs on satellite traffic with fewer learnable parameters
Probabilistic KAN (Vaca-Rubio et al., 19 Oct 2025)	KAN heads directly parameterize Gaussian or Student-t outputs	Outperforms MLP baselines in both accuracy and calibration with substantially fewer parameters
Transfer-learning probe (Shen et al., 2024)	Replace final linear probe on frozen ResNet-50 features with a KAN layer	Similar validation accuracy to linear probing on CIFAR-10, but faster convergence
IMU HAR feature extractor (Liu et al., 2024)	Use KAN blocks as feature extractors for sensor streams	Outperforms CNN-based extractors on all studied datasets while being more parameter efficient

In time-series forecasting, the satellite traffic study reports KAN models with markedly smaller parameter counts than its MLP baselines and better error metrics, alongside an ablation over spline grid size and node count. The probabilistic extension P-KAN then moves beyond point prediction by making KAN layers output distributional parameters for Gaussian and Student-$x_{l+1,j}=\sum_{i=1}^{n_l}\phi_{l,j,i}(x_{l,i}),$9 likelihoods, reporting gains in both calibration and efficiency-risk trade-offs for resource allocation (Vaca-Rubio et al., 2024).

In computer vision, Convolutional KANs translate spline edge functions into local receptive-field operators. The reported behavior is nuanced: parameter efficiency improves, especially in small models and few-parameter regimes, but training is significantly slower than for standard convolutions, and large classical CNNs can remain marginally ahead (Bodner et al., 2024).

Transfer-learning results are similarly measured rather than categorical. Replacing the final linear probe on a frozen ImageNet-pretrained ResNet-50 with a KAN layer on CIFAR-10 yields nearly overlapping validation-accuracy curves relative to standard linear probing, with faster convergence but no significant gain in the generalization gap. This is a useful counterexample to blanket claims that spline-based KANs uniformly dominate simpler heads (Shen et al., 2024).

6. Limitations, misconceptions, and neighboring directions

Two misconceptions recur in the literature. The first is that “KAN” names a single fixed architecture. In practice, spline-based KANs define a family organized around learnable univariate edge functions, but the choice of basis, grid treatment, sharing scheme, and surrounding block structure varies substantially. The second is that KANs are uniformly superior to MLPs or CNNs. The empirical record in the cited papers is more conditional: benefits are often strongest in parameter-limited, function-approximation, scientific, or structured forecasting regimes, while gains may narrow or vanish on simpler transfer-learning tasks or at scale (Shen et al., 2024).

The central spline-specific limitations are computational and regularity-related. Convolutional KANs are reported to be up to 6x slower per epoch than standard CNN layers; classical bounded-grid KANs can fail gracefully only if normalization, grid updates, or fallback branches keep activations within usable support; and learned spline shapes do not always yield simple semantic interpretations in practice (Bodner et al., 2024). UKAN can be read as a direct response to the bounded-grid pathology, while initialization, quantization, and hardware papers respond to the efficiency gap (Moradzadeh et al., 2024).

Neighboring directions clarify the domain of validity of spline bases. P1-KAN replaces spline bases with trainable piecewise-linear finite-element bases and is reported to outperform spline-KANs on irregular functions while achieving similar accuracy to the original spline-based KAN on smooth functions (Warin, 2024). FI-KAN augments or replaces the B-spline path with learnable fractal interpolation bases and reports that Hybrid FI-KAN outperforms KAN at every tested Hölder regularity level, while Pure FI-KAN dominates on rough targets but underperforms on smooth ones; the authors explicitly frame this as evidence that basis geometry must match target regularity (N'guessan, 30 Mar 2026).

Other alternatives pursue speed or parameter sharing rather than regularity matching. SineKAN replaces B-spline grids with re-weighted sine functions and reports better or comparable performance together with substantial speed increases, including 4–9x faster execution than B-spline KAN baselines in the reported comparisons (Reinhardt et al., 2024). Sprecher Networks share learnable splines within blocks rather than allocating private splines to every edge, and are described as scaling as $\phi_{l,j,i}$0 rather than the $\phi_{l,j,i}$1 associated with conventional KANs (Hägg et al., 22 Dec 2025).

Viewed in this broader context, spline-based KANs are best understood not as a final endpoint but as a spline-centric point in a growing design space of edge-function networks. Their enduring contribution is the explicit use of trainable univariate function bases on edges; the open question is which basis, hierarchy, and systems substrate best matches a given target class, from smooth PDE solutions to rough or fractal regimes.