---
title: 'KAN-FIF: Kolmogorov-Arnold Feature Interaction'
url: https://www.emergentmind.com/topics/kolmogorov-arnold-network-based-feature-interaction-framework-kan-fif
type: topic
---

# KAN-FIF: Kolmogorov-Arnold Feature Interaction

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" [2602.12117], 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 [2406.13597], [2504.03923], [2504.04669], [2408.08713].

## 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 [2602.12117]. The same theorem is used in GraphKAN and AFBR-KAN as the theoretical basis for replacing standard dense transformations with learnable univariate operators [2406.13597], [2504.03923].

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 [2602.12117]. 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 [2602.12117], 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 [2406.13597]. 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 [2504.03923]. 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 [2408.08713].

## 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 [2602.12117]. 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
$$
y_q=\sum_{p=1}^m \phi_{q,p}(x_p), \quad q=1,\dots,n.
$$
In practice, the tropical-cyclone study fixes $G=5$ and $k=3$ and implements SiLU plus the spline as a single fused operator [2602.12117].

GraphKAN uses a closely related spline parameterization, representing each univariate function as a cubic B-spline with $k$ knots, with the experiments using $k=3$ [2406.13597]. In that system, the node-update map after local aggregation is expressed as
$$
h_v^{t+1}
=
\sum_{q=1}^{m}
\Phi_{t,q}\left(
\sum_{p=1}^{d_t}\phi_{t,p,q}(m_{v,p}^{t+1})
\right),
$$
and a LayerNorm is optionally applied after each KAN block [2406.13597].

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 [2504.03923]. 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 [2603.28288]. 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 [2602.12117]. The architecture has four major modules.

The **shared feature extraction** module contains a temporal path and a spatial path. The temporal path takes $X_{\mathrm{seq}}\in\mathbb{R}^{B\times T\times 5}$, 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
$$
F_{\mathrm{seq}}=KAN_{\mathrm{Linear}}^{64\to 32}(F_{\mathrm{LSTM}})\in\mathbb{R}^{B\times 32}.
$$
The spatial path uses a multi-scale ConvBlock with kernels $5\times 5$, $3\times 3 +$ maxpool, $1\times 1$ residual, and dilated convolution, producing a 256-dimensional flattened feature that is projected by
$$
F_{\mathrm{img}}=KAN_{\mathrm{Linear}}^{256\to 32}(\mathrm{Flatten}(F_{\mathrm{multiscale}}))\in\mathbb{R}^{B\times 32}.
$$
These are fused as
$$
F_{\mathrm{shared}}=\mathrm{concat}[F_{\mathrm{seq}},F_{\mathrm{img}}]\in\mathbb{R}^{B\times 64}.
$$
All of these elements are stated in the architectural description of Fig. 2 [2602.12117].

The **task-specific feature extraction via KAN-Attention** module operates on annular pooling features from IR channel 7, yielding $P\in\mathbb{R}^{B\times (39\cdot 4)}$. A distance encoder uses $G=\mathrm{linspace}(0,1,39)$ and computes
$$
K=KAN_{\mathrm{Linear}}(G),
$$
while the content encoder computes
$$
[Q,V]=\mathrm{Split}(KAN_{\mathrm{Linear}}(P),2).
$$
After multi-head attention and temporal fusion, the task feature is formed as
$$
A_{\mathrm{task}}=KAN_{\mathrm{Linear}}(\mathrm{concat}[C_{\mathrm{avg}},F_{\mathrm{seq}}])\in\mathbb{R}^{B\times 32}.
$$
This places KAN not only in projection roles but also in the content and distance encoders of an attention-style mechanism [2602.12117].

The **physics-guided constraint module** imposes bidirectional residual KAN mappings between Maximum Sustained Wind (MSW) and Radius of Maximum Wind (RMW) task features:
$$
\Gamma_{\mathrm{msw}\to \mathrm{rmw}}(A_{\mathrm{msw}})=A_{\mathrm{rmw}}+KAN_{\mathrm{msw2rmw}}(A_{\mathrm{msw}})
$$
and
$$
\Gamma_{\mathrm{rmw}\to \mathrm{msw}}(A_{\mathrm{rmw}})=A_{\mathrm{msw}}+KAN_{\mathrm{rmw2msw}}(A_{\mathrm{rmw}}).
$$
This is the paper’s explicit mechanism for injecting physics-guided cross-task coupling [2602.12117].

The **multimodal fusion and decoding** stage concatenates task feature, cross-task constrained feature, and shared feature, then decodes with $KAN_{\mathrm{Linear}}^{128\to 1}$:
$$
\bar{Y}_{\mathrm{msw}}=D_{\mathrm{msw}}(\mathrm{concat}[A_{\mathrm{msw}},\Gamma_{\mathrm{rmw}\to \mathrm{msw}},F_{\mathrm{shared}}]),
$$
$$
\bar{Y}_{\mathrm{rmw}}=D_{\mathrm{rmw}}(\mathrm{concat}[A_{\mathrm{rmw}},\Gamma_{\mathrm{msw}\to \mathrm{rmw}},F_{\mathrm{shared}}]).
$$
In this realization, KAN-FIF is therefore a full-stack interaction design, not an isolated layer replacement [2602.12117].

## 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:
$$
m_v^{t+1}=
\sum_{w\in N(v)}
(\deg(v)\deg(w))^{-1/2}
A_{v,w}h_w^t.
$$
The subsequent KAN-based feature extraction replaces the conventional post-aggregation MLP [2406.13597]. 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 [2504.03923]. The data-flow summary is
fMRI-patch embeddings $\to$ positional-encoding + tokenization $\to$ stacked (self-attention + KAN) layers $\to$ [CLS] token $\to$ KAN head $\to$ SoftMax. The model is described as explicitly realizing a 2-stage high-order feature interaction rather than relying on node-wise activations [2504.03923].

In **ridge-function approximation and scientific computing**, asKAN alternates KAN layers with active-subspace projections:
$$
\mathrm{asKAN}(x)=KAN_L\circ W_L^T\circ KAN_{L-1}\circ W_{L-1}^T\circ \dots \circ KAN_0(x),
$$
with each level computing a gradient covariance
$$
C=E_x[\nabla f(x)\nabla f(x)^T]
$$
and projecting onto dominant eigendirections [2504.04669]. 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 [2504.04669]. 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 [2408.08713]. 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 [2603.28288]. 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 [2602.12117].

| 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 $94.8\%$ reduction in parameters, $68.7\%$ faster inference per sample, $32.5\%$ lower MAE, and $31.9\%$ lower RMSE for MSW prediction [2602.12117]. 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 $<10$ W, with a deployable model size of 0.92 MB and per-sample latency of 14.41 ms [2602.12117].

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 [2406.13597]. AFBR-KAN reports that the best configuration on ABIDE-NYU attained ACC $=0.743\pm 0.088$, AUC $=0.734\pm 0.131$, SEN $=0.897\pm 0.065$, and SPE $=0.716\pm 0.105$, and that ViT KAN-KAN achieved AUC $=0.687$ versus $0.628$ for MLP-MLP in cross-site generalization from NYU to UM [2504.03923]. asKAN reports, with the same per-block neuron counts as vanilla KAN, losses of approximately $1.8\times 10^{-1}$ versus $2.7\times 10^{-2}$ on ridge-function fitting after 3 levels, approximately $1.2\times 10^{-2}$ versus $4.5\times 10^{-4}$ on the Poisson PDE, and approximately $3.6\times 10^{-9}$ versus $1.2\times 10^{-10}$ on sound-field reconstruction [2504.04669]. 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 [2408.08713]. 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 [2603.28288].

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 [2602.12117], stronger graph feature extraction in GraphKAN [2406.13597], higher-order functional representation in transformers [2504.03923], better ridge approximation in asKAN [2504.04669], sparse efficient multiplicative interaction modeling in CTR prediction [2408.08713], and regularity-matched basis adaptation in non-smooth approximation [2603.28288].

## 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 [2602.12117]. 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 [2408.08713]. asKAN likewise identifies an inflexibility of KAN in modeling ridge functions and remedies it by embedding active-subspace projections between KAN blocks [2504.04669]. FI-KAN shows that fixed-grid B-spline bases provide no intrinsic multi-scale decomposition for non-smooth function approximation, motivating fractal interpolation bases [2603.28288]. 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 $\Lambda_i$ to decide how many directions to keep [2504.04669]. GraphKAN reports 2×–3× more compute than a dense MLP layer because of spline evaluation [2406.13597]. AFBR-KAN reports that fully KAN integration obtains the best results at the cost of approximately 2× the MLP parameter count [2504.03923]. FI-KAN documents complementary behavior between Pure and Hybrid variants, including underperformance of Pure FI-KAN on smooth targets [2603.28288].

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 [2504.04669]. GraphKAN suggests inserting KAN blocks into tabular and sequential models and recommends feature normalization to $[0,1]$, moderate spline knot counts, and regularization via $L_2$ penalty or total variation when overfitting is a concern [2406.13597]. FI-KAN establishes “regularity-matched” basis design as a principled strategy for neural function approximation [2603.28288]. 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.

Source: https://www.emergentmind.com/topics/kolmogorov-arnold-network-based-feature-interaction-framework-kan-fif