KAN-FIF: Kolmogorov-Arnold Feature Interaction

• The paper demonstrates that KAN-FIF, using spline-parameterized KAN layers, accurately models high-order polynomial feature interactions while reducing model size and inference time.
• KAN-FIF is a framework that replaces standard affine transforms with learnable spline operators to introduce explicit, high-order feature couplings across various domains.
• Empirical results reveal up to 94.8% parameter reduction and 31.9% lower error in tropical cyclone estimation, highlighting the efficiency and accuracy of the approach.

Searching arXiv for KAN-FIF and closely related KAN-based feature-interaction papers. Kolmogorov-Arnold Network-based Feature Interaction Framework (KAN-FIF) denotes a spline-parameterized architectural framework that uses Kolmogorov-Arnold Network (KAN) layers to model feature interactions within larger multimodal or task-specific pipelines. In the usage established by the tropical-cyclone estimation work "KAN-FIF: Spline-Parameterized Lightweight Physics-based Tropical Cyclone Estimation on Meteorological Satellite" (Shen et al., 12 Feb 2026), the framework integrates MLP and CNN layers with spline-parameterized KAN layers, with the stated objective of overcoming linear feature interactions that fail to capture high-order polynomial relationships between domain attributes. More broadly, related KAN-based systems in graph learning, transformer-based neuroimaging, active-subspace modeling, and CTR prediction illustrate a common design pattern: replacing or augmenting conventional affine or MLP interaction modules with univariate spline operators arranged according to the Kolmogorov-Arnold representation, thereby constructing explicit, high-order feature couplings in a compact form (Zhang et al., 2024, Ward et al., 4 Apr 2025, Zhou et al., 7 Apr 2025, Shi et al., 2024).

1. Conceptual basis and formal definition

KAN-FIF is grounded in the Kolmogorov-Arnold representation theorem, which is stated in multiple related works as a decomposition of a continuous multivariate function into sums of univariate inner and outer functions. In the KAN-FIF tropical-cyclone paper, this is written as

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

where $\phi_{q,p}$ and $\phi_q$ are continuous univariate functions (Shen et al., 12 Feb 2026). The same theorem is used in GraphKAN and AFBR-KAN as the theoretical basis for replacing standard dense transformations with learnable univariate operators (Zhang et al., 2024, Ward et al., 4 Apr 2025).

Within KAN-FIF, the decisive architectural move is not merely the adoption of KAN as a standalone network, but the placement of KAN layers at feature-combination bottlenecks. In the tropical-cyclone formulation, KAN layers replace or follow linear and convolutional transforms so that interaction structure is introduced at shared-feature extraction, task-specific attention-style processing, physics-guided cross-task mapping, and final decoding stages (Shen et al., 12 Feb 2026). This suggests a general interpretation of KAN-FIF as a feature-interaction framework rather than a single canonical network topology.

The framework’s distinctiveness, as described in (Shen et al., 12 Feb 2026), lies in the claim that KAN layers can exactly model high-order polynomial interactions in one shot, drastically reducing depth/width requirements. Related papers formulate analogous motivations in different domains. GraphKAN presents KAN blocks as substitutes for MLPs and scalar activations in GNN node-update layers (Zhang et al., 2024). AFBR-KAN replaces transformer MLP sublayers and optionally the classifier head with KAN blocks to induce higher-order feature interactions in fMRI-based ASD diagnosis (Ward et al., 4 Apr 2025). KarSein adopts the KAN mechanism as inspiration for adaptive CTR interaction modeling while explicitly addressing KAN’s stated inability to spontaneously capture multiplicative relationships among features (Shi et al., 2024).

2. Spline-parameterized KAN layers as interaction operators

In the KAN-FIF paper, a KAN layer is defined by parameterizing each edge function with a learnable spline plus a self-gating activation:

$$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$

with $\mathrm{silu}(x)=x/(1+e^{-x})$ and $\mathrm{spline}_{q,p}(x)$ given as a B-spline of order $k$ whose coefficients are learned (Shen et al., 12 Feb 2026). The spline term is constructed from a uniform knot sequence and the Cox-de Boor recursion:

$$B_{i,0}(x)= \begin{cases} 1, & \tau_i\le x<\tau_{i+1}\ 0, & \text{otherwise} \end{cases}$$

and

$$B_{i,k}(x)= \frac{x-\tau_i}{\tau_{i+k}-\tau_i}B_{i,k-1}(x) + \frac{\tau_{i+k+1}-x}{\tau_{i+k+1}-\tau_{i+1}}B_{i+1,k-1}(x),$$

followed by

$$\mathrm{spline}_{q,p}(x)=\sum_{i=0}^{G-1}\alpha_{q,p,i}\,B_{i,k}(x).$$

The resulting layer output is

$\phi_{q,p}$0

In practice, the tropical-cyclone study fixes $\phi_{q,p}$1 and $\phi_{q,p}$2 and implements SiLU plus the spline as a single fused operator (Shen et al., 12 Feb 2026).

GraphKAN uses a closely related spline parameterization, representing each univariate function as a cubic B-spline with $\phi_{q,p}$3 knots, with the experiments using $\phi_{q,p}$4 (Zhang et al., 2024). In that system, the node-update map after local aggregation is expressed as

$\phi_{q,p}$5

and a LayerNorm is optionally applied after each KAN block (Zhang et al., 2024).

AFBR-KAN offers a different implementation choice, using learnable spline functions described as reflectional-switch functions for the inner and outer univariate maps in transformer feed-forward replacements (Ward et al., 4 Apr 2025). FI-KAN, by contrast, replaces or augments the B-spline path with learnable fractal interpolation function bases, yielding Pure FI-KAN and Hybrid FI-KAN variants (N'guessan, 30 Mar 2026). A plausible implication is that KAN-FIF, as a framework concept, is compatible with multiple basis designs so long as the central Kolmogorov-Arnold decomposition is preserved.

3. Architectural realization in tropical-cyclone estimation

The formulation explicitly named KAN-FIF appears in a multimodal, physics-guided tropical-cyclone estimation model for meteorological satellite data (Shen et al., 12 Feb 2026). The architecture has four major modules.

The shared feature extraction module contains a temporal path and a spatial path. The temporal path takes $\phi_{q,p}$6, corresponding to 3 timestamps of 5 evolution features, processes it with an LSTM to a hidden size of 64, and then applies a KAN projection

$\phi_{q,p}$7

The spatial path uses a multi-scale ConvBlock with kernels $\phi_{q,p}$8, $\phi_{q,p}$9 maxpool, $\phi_q$0 residual, and dilated convolution, producing a 256-dimensional flattened feature that is projected by

$\phi_q$1

These are fused as

$\phi_q$2

All of these elements are stated in the architectural description of Fig. 2 (Shen et al., 12 Feb 2026).

The task-specific feature extraction via KAN-Attention module operates on annular pooling features from IR channel 7, yielding $\phi_q$3. A distance encoder uses $\phi_q$4 and computes

$\phi_q$5

while the content encoder computes

$\phi_q$6

After multi-head attention and temporal fusion, the task feature is formed as

$\phi_q$7

This places KAN not only in projection roles but also in the content and distance encoders of an attention-style mechanism (Shen et al., 12 Feb 2026).

The physics-guided constraint module imposes bidirectional residual KAN mappings between Maximum Sustained Wind (MSW) and Radius of Maximum Wind (RMW) task features:

$\phi_q$8

and

$\phi_q$9

This is the paper’s explicit mechanism for injecting physics-guided cross-task coupling (Shen et al., 12 Feb 2026).

The multimodal fusion and decoding stage concatenates task feature, cross-task constrained feature, and shared feature, then decodes with $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$0:

$$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$1

$$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$2

In this realization, KAN-FIF is therefore a full-stack interaction design, not an isolated layer replacement (Shen et al., 12 Feb 2026).

4. Generalizations across domains

Several papers provide concrete realizations of a broader KAN-based feature-interaction framework, even when they do not use the exact term KAN-FIF as the model name.

In graph learning, GraphKAN embeds a KAN block into each GNN node-update step in place of the usual MLP+ReLU, while leaving the local aggregation unchanged from GCN:

$$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$3

The subsequent KAN-based feature extraction replaces the conventional post-aggregation MLP (Zhang et al., 2024). The paper states that there are no fixed affine transforms or scalar activations anywhere in the GraphKAN block. This establishes a graph-specific KAN feature-interaction pipeline in which neighborhood aggregation is conventional but feature mixing is spline-based.

In transformer-based neuroimaging, AFBR-KAN replaces every MLP sublayer in the Transformer encoder, and optionally the final classification head, by a two-layer KAN block (Ward et al., 4 Apr 2025). The data-flow summary is fMRI-patch embeddings $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$4 positional-encoding + tokenization $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$5 stacked (self-attention + KAN) layers $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$6 [CLS] token $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$7 KAN head $$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$8 SoftMax. The model is described as explicitly realizing a 2-stage high-order feature interaction rather than relying on node-wise activations (Ward et al., 4 Apr 2025).

In ridge-function approximation and scientific computing, asKAN alternates KAN layers with active-subspace projections:

$$\phi_{q,p}(x)=w_{q,p}\cdot [\mathrm{silu}(x)+\mathrm{spline}_{q,p}(x)],$$9

with each level computing a gradient covariance

$\mathrm{silu}(x)=x/(1+e^{-x})$0

and projecting onto dominant eigendirections (Zhou et al., 7 Apr 2025). The paper explicitly frames this as a route toward a general KAN-based feature-interaction framework in which linear projections such as PCA, CCA, supervised linear discriminants, or even small learned MLPs could replace the active-subspace step (Zhou et al., 7 Apr 2025). This suggests that KAN-FIF can be understood abstractly as alternating KAN nonlinearities with structure-revealing interaction operators.

In CTR prediction, KarSein introduces a Kolmogorov-Arnold Represented Sparse Efficient Interaction Network that adapts KAN-inspired univariate spline transformations to vector-wise and bit-wise streams, while adding pairwise multiplication in early layers (Shi et al., 2024). The paper states that it extends the capabilities of KAN by supporting two-dimensional embedding vectors and by overcoming KAN’s inability to spontaneously capture multiplicative relationships among features. This is a notable point of divergence within the KAN-interaction literature.

In non-smooth function approximation, FI-KAN introduces fractal interpolation KANs, with a Hybrid FI-KAN that combines a spline path and a learnable fractal correction (N'guessan, 30 Mar 2026). Although its principal focus is basis geometry rather than multimodal feature fusion, it is directly relevant to KAN-FIF because it shows that the univariate basis used on KAN edges can be redesigned to match target regularity.

5. Empirical results and reported performance

The tropical-cyclone KAN-FIF paper reports substantial gains in model size, inference speed, and MSW prediction accuracy relative to the baseline model Phy-CoCo (Shen et al., 12 Feb 2026).

Metric	Phy-CoCo	KAN-FIF
Size	19 MB	0.99 MB
Infer Time	7.35 ms	2.30 ms
MSW MAE	4.76 kt	3.21 kt
MSW RMSE	6.33 kt	4.31 kt

These correspond to a $\mathrm{silu}(x)=x/(1+e^{-x})$1 reduction in parameters, $\mathrm{silu}(x)=x/(1+e^{-x})$2 faster inference per sample, $\mathrm{silu}(x)=x/(1+e^{-x})$3 lower MAE, and $\mathrm{silu}(x)=x/(1+e^{-x})$4 lower RMSE for MSW prediction (Shen et al., 12 Feb 2026). The paper further reports an offline deployment experiment on the FY-4 series meteorological satellite processor using a Qingyun-1000 board with Atlas 200I A2, Huawei Ascend 310 NPU, 8 GB LPDDR4, 22 TOPS (INT8)/11 TOPS (FP16), and $\mathrm{silu}(x)=x/(1+e^{-x})$5 W, with a deployable model size of 0.92 MB and per-sample latency of 14.41 ms (Shen et al., 12 Feb 2026).

Other KAN-based interaction systems report domain-specific performance gains. GraphKAN states that it consistently outperforms GCN by 2–5 points in test accuracy on four basic-graph node-classification datasets, with the gap widening when fewer labeled nodes are available, while runtime is 2–3× that of GCN but remains under 60 seconds per full training run (Zhang et al., 2024). AFBR-KAN reports that the best configuration on ABIDE-NYU attained ACC $\mathrm{silu}(x)=x/(1+e^{-x})$6, AUC $\mathrm{silu}(x)=x/(1+e^{-x})$7, SEN $\mathrm{silu}(x)=x/(1+e^{-x})$8, and SPE $\mathrm{silu}(x)=x/(1+e^{-x})$9, and that ViT KAN-KAN achieved AUC $\mathrm{spline}_{q,p}(x)$0 versus $\mathrm{spline}_{q,p}(x)$1 for MLP-MLP in cross-site generalization from NYU to UM (Ward et al., 4 Apr 2025). asKAN reports, with the same per-block neuron counts as vanilla KAN, losses of approximately $\mathrm{spline}_{q,p}(x)$2 versus $\mathrm{spline}_{q,p}(x)$3 on ridge-function fitting after 3 levels, approximately $\mathrm{spline}_{q,p}(x)$4 versus $\mathrm{spline}_{q,p}(x)$5 on the Poisson PDE, and approximately $\mathrm{spline}_{q,p}(x)$6 versus $\mathrm{spline}_{q,p}(x)$7 on sound-field reconstruction (Zhou et al., 7 Apr 2025). KarSein reports, on MovieLens-1M excluding the embedding table, AUC values of 0.8273 for KAN, 0.8403 for DNN, 0.8531 for EulerNet, and 0.8555 for KarSein, with 0.018 M parameters for KarSein (Shi et al., 2024). FI-KAN reports that Hybrid FI-KAN outperforms KAN at every regularity level on a Hölder benchmark and achieves up to 79× improvement on rough-coefficient diffusion and 3.5× on L-shaped domain singularities (N'guessan, 30 Mar 2026).

Taken together, these results indicate that KAN-centered interaction modules have been used to pursue several distinct goals: parameter efficiency and deployability in KAN-FIF (Shen et al., 12 Feb 2026), stronger graph feature extraction in GraphKAN (Zhang et al., 2024), higher-order functional representation in transformers (Ward et al., 4 Apr 2025), better ridge approximation in asKAN (Zhou et al., 7 Apr 2025), sparse efficient multiplicative interaction modeling in CTR prediction (Shi et al., 2024), and regularity-matched basis adaptation in non-smooth approximation (N'guessan, 30 Mar 2026).

6. Limitations, misconceptions, and research directions

A common misconception is that KAN-FIF refers to any KAN-based model. In the narrow sense, KAN-FIF is the named framework for spline-parameterized lightweight physics-based tropical-cyclone estimation on meteorological satellite data (Shen et al., 12 Feb 2026). In a broader editorial sense, the term can also denote the family of architectures that use KAN blocks as explicit feature-interaction modules across domains, but that broader usage is an interpretive generalization grounded in related papers rather than a uniform author-defined standard.

Another misconception is that KAN mechanisms alone universally solve feature-interaction problems. KarSein explicitly argues that vanilla KAN has an inability to spontaneously capture multiplicative relationships among features and introduces pairwise multiplication to address this limitation (Shi et al., 2024). asKAN likewise identifies an inflexibility of KAN in modeling ridge functions and remedies it by embedding active-subspace projections between KAN blocks (Zhou et al., 7 Apr 2025). FI-KAN shows that fixed-grid B-spline bases provide no intrinsic multi-scale decomposition for non-smooth function approximation, motivating fractal interpolation bases (N'guessan, 30 Mar 2026). These critiques are not contradictions of the KAN-FIF idea; rather, they delimit the conditions under which vanilla spline-parameterized KAN layers are sufficient.

Methodological constraints are also explicit. asKAN notes that active subspace relies on accurate gradient estimates, that an under-trained early KAN layer may identify spurious directions, and that one should monitor eigenvalue decay in $\mathrm{spline}_{q,p}(x)$8 to decide how many directions to keep (Zhou et al., 7 Apr 2025). GraphKAN reports 2×–3× more compute than a dense MLP layer because of spline evaluation (Zhang et al., 2024). AFBR-KAN reports that fully KAN integration obtains the best results at the cost of approximately 2× the MLP parameter count (Ward et al., 4 Apr 2025). FI-KAN documents complementary behavior between Pure and Hybrid variants, including underperformance of Pure FI-KAN on smooth targets (N'guessan, 30 Mar 2026).

The literature also points toward several research directions already articulated in source papers. asKAN proposes replacing the active-subspace linear projection by any differentiable projection, including PCA, CCA, supervised linear discriminants, or small learned MLPs, and notes compatibility with physics-informed residuals, multi-task losses, or attention-style gating (Zhou et al., 7 Apr 2025). GraphKAN suggests inserting KAN blocks into tabular and sequential models and recommends feature normalization to $\mathrm{spline}_{q,p}(x)$9, moderate spline knot counts, and regularization via $k$0 penalty or total variation when overfitting is a concern (Zhang et al., 2024). FI-KAN establishes “regularity-matched” basis design as a principled strategy for neural function approximation (N'guessan, 30 Mar 2026). A plausible implication is that future KAN-FIF systems may vary not only in where KAN blocks are inserted, but also in how projection, basis geometry, and domain constraints are co-designed around them.