LuKAN: Versatile KAN Frameworks Overview

• LuKAN is a family label for KAN adaptations that use learnable univariate functions with distinct polynomial and spline bases for domain-specific tasks.
• In 3D motion prediction, Lucas polynomial-based LuKAN employs wavelet transforms and residual blocks to achieve efficient pose forecasting with competitive error metrics.
• Other variants integrate Lyapunov-based adaptive control and LUT-compiled splines for IoT DoS detection, balancing accuracy, stability, and low-latency performance.

LuKAN is a non-unique acronym used in recent arXiv literature for several distinct Kolmogorov–Arnold Network (KAN)–derived methods. In current usage, it denotes a Lucas-polynomial KAN framework for 3D human motion prediction, a Lyapunov-based KAN adaptive controller for uncertain nonlinear systems, and a LUT-compiled KAN pipeline for lightweight denial-of-service detection on IoT edge devices (Hasan et al., 6 Aug 2025, Shen et al., 24 Dec 2025, Kuznetsov, 12 Jan 2026). The common substrate is the KAN design pattern in which learnable univariate functions are placed on edges, but the surrounding temporal encoding, control-theoretic guarantees, basis-function choices, and deployment constraints differ substantially.

1. Terminological scope and variant definitions

The name “LuKAN” identifies multiple domain-specific constructions rather than a single canonical architecture. One usage expands “Lu” as Lucas polynomials in a 3D motion predictor; another uses “LuKAN” for a Lyapunov-based adaptive control method; a third abbreviates a LUT-compiled KAN for CPU-oriented IoT intrusion detection (Hasan et al., 6 Aug 2025, Shen et al., 24 Dec 2025, Kuznetsov, 12 Jan 2026).

Variant	Domain	Distinguishing mechanism
LuKAN	3D human motion prediction	DWT/IDWT, spatial projection, Lucas-polynomial KAN blocks
LuKAN	Adaptive control	KAN approximator inside a Lyapunov-based control and update law
LuKAN	IoT DoS detection	Cubic B-spline KAN compiled into quantized lookup tables

This naming overlap is technically significant because the three systems share only the broad KAN paradigm. Their basis functions, optimization criteria, and system-level objectives are not interchangeable. A plausible implication is that the acronym is best interpreted as a family label for KAN adaptations rather than as the name of a single model class.

2. Lucas-polynomial LuKAN for 3D human motion prediction

In "LuKAN: A Kolmogorov-Arnold Network Framework for 3D Human Motion Prediction" (Hasan et al., 6 Aug 2025), the task is to forecast future 3D poses from historical motion data. The model takes a history of $L$ poses $X_{1:L}\in\mathbb R^{L\times K}$, where $K=3\times \#\text{joints}$, and predicts $T$ future poses $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$. Its pipeline consists of temporal encoding by one-dimensional discrete wavelet transform (DWT) on each joint trajectory, an initial spatial projection into a $D$-dimensional embedding, $B$ repeated Temporal Dependency Learner blocks, a second spatial projection back to coefficient-space joint dimension, and inverse DWT (IDWT) for time-domain reconstruction. The repeated block has a single KAN layer with Lucas-polynomial activations, LayerNorm, and a residual skip connection: $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$

The KAN layer is formulated as a learnable matrix $\Phi$ of scalar functions $\phi_{q,p}:\mathbb R\to\mathbb R$, with

$X_{1:L}\in\mathbb R^{L\times K}$0

Each scalar function is a learned linear combination of Lucas polynomials: $X_{1:L}\in\mathbb R^{L\times K}$1 where the Lucas polynomial basis satisfies

$X_{1:L}\in\mathbb R^{L\times K}$2

The recurrence is central to the model’s efficiency claims. After temporal learning, the model reconstructs coefficient space via $X_{1:L}\in\mathbb R^{L\times K}$3, applies IDWT to obtain $X_{1:L}\in\mathbb R^{L\times K}$4, then takes the first $X_{1:L}\in\mathbb R^{L\times K}$5 rows and adds the last input pose: $X_{1:L}\in\mathbb R^{L\times K}$6

Training supervises both poses and velocities over future frames. With $X_{1:L}\in\mathbb R^{L\times K}$7 and analogous predicted velocities, the loss is

$X_{1:L}\in\mathbb R^{L\times K}$8

The paper attributes efficiency to several factors: Lucas polynomials admit a two-term linear recurrence; each scalar channel learns only $X_{1:L}\in\mathbb R^{L\times K}$9 coefficients; and DWT/IDWT are $K=3\times \#\text{joints}$0 per joint channel with small constant factors. The reported empirical configuration uses $K=3\times \#\text{joints}$1 KAN blocks, embedding $K=3\times \#\text{joints}$2, and $K=3\times \#\text{joints}$3–$K=3\times \#\text{joints}$4, yielding $K=3\times \#\text{joints}$5 parameters.

The experimental evaluation covers Human3.6M, AMASS-BMLrub, and 3DPW. On Human3.6M, average MPJPE at horizons $K=3\times \#\text{joints}$6 ms is reported as $K=3\times \#\text{joints}$7 mm for LuKAN, compared with $K=3\times \#\text{joints}$8 for SiMLPe. The model-size versus accuracy plot reports error $K=3\times \#\text{joints}$9 mm at $T$0 ms with $T$1M parameters, whereas other GCN, Transformer, and MLP methods use $T$2–$T$3M+ parameters. Ablations report that DWT yields uniform improvements of $T$4–$T$5 relative over DCT, Lucas polynomials outperform B-Splines, Chebyshev, Legendre, and Hermite at $T$6 ms with $T$7 mm versus $T$8 mm respectively, and $T$9 is optimal among $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$0.

The architectural significance lies in the coupling of multi-resolution temporal encoding with a single-function-approximation layer per block. The paper argues that Lucas polynomials capture both oscillatory and slow-varying components, which matches the multi-scale character of human joint trajectories. This suggests that the method’s empirical advantage is not only parameter economy but also basis selection aligned with the signal structure of articulated motion.

3. Lyapunov-based LuKAN adaptive control

In "Lyapunov-Based Kolmogorov-Arnold Network (KAN) Adaptive Control" (Shen et al., 24 Dec 2025), LuKAN denotes a controller that integrates a KAN function approximator into a Lyapunov-based adaptive control framework for uncertain nonlinear systems. The control setting is a fully actuated system

$\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$1

with bounded disturbance $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$2, tracking error $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$3, and approximation

$\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$4

where $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$5 on a compact operating domain $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$6.

The KAN construction is motivated by the Kolmogorov–Arnold representation theorem, written in the paper as

$\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$7

A general $\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$8-layer KAN is then realized as

$\hat X_{L+1:L+T}\in\mathbb R^{T\times K}$9

with layerwise components

$D$0

Here $D$1 is a fixed sigmoid-like basis and $D$2 is a B-spline with grid size $D$3 and order $D$4. The full parameter vector is assembled from all edge-wise parameters.

The control law embeds the KAN approximation directly: $D$5 where $D$6 and $D$7. Online adaptation is driven by the Jacobian

$D$8

and the projected update law

$D$9

with $B$0. The paper further states an approximation theorem with bound

$B$1

and employs the composite Lyapunov candidate

$B$2

where $B$3. Using a Filippov derivative, the almost-everywhere bound becomes

$B$4

If $B$5 and the initial condition is sufficiently small, the paper states that $B$6 as $B$7, $B$8 remains bounded, and $B$9.

The simulation study uses a four-state nonlinear drift system with disturbance $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$0, a $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$1 s horizon, and $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$2 s step size. The comparative controllers are Lb-DNN, Lb-LSTM, and Lb-KAN, with Lb-KAN instantiated as network shape $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$3, spline grid $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$4, and order $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$5. Over 20 randomized runs, the KAN controller achieves RMS tracking error $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$6, versus $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$7 for LSTM and $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$8 for DNN. RMS function approximation error is $Z_{b+1}=\mathrm{LN}(\mathrm{KAN}(Z_b))+Z_b.$9, versus $\Phi$0 and $\Phi$1, corresponding to a $\Phi$2 and $\Phi$3 reduction. The paper also emphasizes interpretability: because $\Phi$4 is decomposed into sums of univariate edge-wise functions $\Phi$5, plotting these functions provides a direct visual representation of learned components.

In this usage, LuKAN is not primarily an efficiency mechanism but a control architecture in which approximation, stability certification, and online adaptation are co-designed. The significance is therefore analytical as much as empirical: the method couples KAN-based functional decomposition to explicit Lyapunov arguments rather than relying only on offline predictive performance.

4. LUT-compiled LuKAN for lightweight DoS detection on IoT edge devices

In "LUT-Compiled Kolmogorov-Arnold Networks for Lightweight DoS Detection on IoT Edge Devices" (Kuznetsov, 12 Jan 2026), LuKAN refers to a KAN model whose spline branches are compiled into lookup tables for latency-critical intrusion detection on CPU-only hardware. The baseline network has input dimension $\Phi$6 flow-level features, two KAN layers with $\Phi$7 and $\Phi$8 neurons, and a single-unit output with logistic activation. Each layer computes

$\Phi$9

with

$\phi_{q,p}:\mathbb R\to\mathbb R$0

The spline branch is cubic, with degree $\phi_{q,p}:\mathbb R\to\mathbb R$1, grid $\phi_{q,p}:\mathbb R\to\mathbb R$2 equal-length intervals, and $\phi_{q,p}:\mathbb R\to\mathbb R$3 control points per spline. Training minimizes binary cross-entropy.

The central contribution is the LUT compilation pipeline. Offline, each spline branch $\phi_{q,p}:\mathbb R\to\mathbb R$4 is discretized over its domain, sampled at $\phi_{q,p}:\mathbb R\to\mathbb R$5 equally spaced points per segment, evaluated in float32, and quantized into symmetric int8 tables with a per-segment scale. At runtime, inference locates the segment, computes a normalized position $\phi_{q,p}:\mathbb R\to\mathbb R$6, and reconstructs the spline value by linear interpolation: $\phi_{q,p}:\mathbb R\to\mathbb R$7 The stated out-of-bounds policy is half_open + zero_spline, meaning that if $\phi_{q,p}:\mathbb R\to\mathbb R$8, the spline branch returns $\phi_{q,p}:\mathbb R\to\mathbb R$9. Total LUT memory is modeled as approximately $X_{1:L}\in\mathbb R^{L\times K}$00 bytes, where $X_{1:L}\in\mathbb R^{L\times K}$01.

The experimental setting uses the CICIDS2017 Wednesday DoS subset. Preprocessing removes constant and duplicate features, applies outlier removal by the $X_{1:L}\in\mathbb R^{L\times K}$02 rule, median imputation, standardization, and stratified sampling to balance classes at $X_{1:L}\in\mathbb R^{L\times K}$03 samples each. The final split is $X_{1:L}\in\mathbb R^{L\times K}$04 train and $X_{1:L}\in\mathbb R^{L\times K}$05 test, with AdamW, learning rate $X_{1:L}\in\mathbb R^{L\times K}$06, weight decay $X_{1:L}\in\mathbb R^{L\times K}$07, $X_{1:L}\in\mathbb R^{L\times K}$08 epochs, and batch size $X_{1:L}\in\mathbb R^{L\times K}$09. The float32 baseline reports Accuracy $X_{1:L}\in\mathbb R^{L\times K}$10, Precision $X_{1:L}\in\mathbb R^{L\times K}$11, Recall $X_{1:L}\in\mathbb R^{L\times K}$12, F1 $X_{1:L}\in\mathbb R^{L\times K}$13, ROC-AUC $X_{1:L}\in\mathbb R^{L\times K}$14, PR-AUC $X_{1:L}\in\mathbb R^{L\times K}$15, Parameters $X_{1:L}\in\mathbb R^{L\times K}$16, and Size $X_{1:L}\in\mathbb R^{L\times K}$17 MB.

For LUT compilation, the paper reports a baseline Accuracy $X_{1:L}\in\mathbb R^{L\times K}$18, F1 $X_{1:L}\in\mathbb R^{L\times K}$19, and ROC-AUC $X_{1:L}\in\mathbb R^{L\times K}$20. At $X_{1:L}\in\mathbb R^{L\times K}$21 with symmetric int8, the compiled model attains Accuracy $X_{1:L}\in\mathbb R^{L\times K}$22, F1 $X_{1:L}\in\mathbb R^{L\times K}$23, and ROC-AUC $X_{1:L}\in\mathbb R^{L\times K}$24, with $X_{1:L}\in\mathbb R^{L\times K}$25F1 $X_{1:L}\in\mathbb R^{L\times K}$26. Latency comparisons against a Numba B-spline baseline give, at batch size $X_{1:L}\in\mathbb R^{L\times K}$27, $X_{1:L}\in\mathbb R^{L\times K}$28 ms/sample and $X_{1:L}\in\mathbb R^{L\times K}$29 speedup for Numba LUT at $X_{1:L}\in\mathbb R^{L\times K}$30, versus $X_{1:L}\in\mathbb R^{L\times K}$31 ms for Numba B-spline; at batch size $X_{1:L}\in\mathbb R^{L\times K}$32, the same configuration yields $X_{1:L}\in\mathbb R^{L\times K}$33 ms/sample and $X_{1:L}\in\mathbb R^{L\times K}$34 speedup versus $X_{1:L}\in\mathbb R^{L\times K}$35 ms/sample. The abstract summarizes the compiled system as achieving $X_{1:L}\in\mathbb R^{L\times K}$36 accuracy at $X_{1:L}\in\mathbb R^{L\times K}$37, with $X_{1:L}\in\mathbb R^{L\times K}$38 speedup at batch size $X_{1:L}\in\mathbb R^{L\times K}$39, over $X_{1:L}\in\mathbb R^{L\times K}$40 speedup at batch size $X_{1:L}\in\mathbb R^{L\times K}$41, and only $X_{1:L}\in\mathbb R^{L\times K}$42 memory overhead.

This LuKAN variant is deployment-centered. Its primary innovation is not a new approximation theorem or a new temporal representation, but a systems transformation that replaces recursive spline evaluation with table lookup and interpolation. The result is deterministic inference latency under tight CPU-only constraints.

5. Shared KAN substrate and basis-function heterogeneity

Across the three usages, LuKAN inherits the KAN principle that outputs are formed by summing learnable univariate functions attached to edges rather than by applying a fixed node activation to affine combinations (Hasan et al., 6 Aug 2025, Shen et al., 24 Dec 2025, Kuznetsov, 12 Jan 2026). In the motion-prediction variant, the KAN output is

$X_{1:L}\in\mathbb R^{L\times K}$43

with Lucas-polynomial expansions. In the control variant, each layer uses edge functions of the form $X_{1:L}\in\mathbb R^{L\times K}$44, combining a fixed sigmoid-like basis and B-splines. In the IoT detection variant, each edge similarly combines a base nonlinearity and a cubic B-spline branch, but the spline branch is subsequently compiled into quantized LUTs.

The basis functions differ materially. Lucas polynomials are chosen in the motion paper for their two-term recurrence and their ability to handle oscillatory behaviors. B-splines appear in both the control and edge-detection papers, but for different reasons: in control they support approximation analysis and visualizable decomposition, whereas in edge deployment they are the target of a compilation pipeline that eliminates runtime spline overhead. These are not minor implementation details; they determine approximation properties, memory behavior, and what counts as interpretability in each setting.

A plausible synthesis is that “LuKAN” does not identify a particular polynomial family, spline family, or training regime. It identifies a pattern of domain specialization around the KAN formalism. The commonality is structural; the technical content resides in how each paper modifies the univariate edge functions and surrounding system.

6. Scholarly significance and recurrent points of confusion

A recurrent source of confusion is the assumption that LuKAN names one architecture. The cited literature shows otherwise: the same acronym spans motion forecasting, adaptive control, and IoT intrusion detection, with different expansions, mathematical assumptions, and evaluation protocols (Hasan et al., 6 Aug 2025, Shen et al., 24 Dec 2025, Kuznetsov, 12 Jan 2026). The motion paper evaluates MPJPE on Human3.6M, AMASS-BMLrub, and 3DPW; the control paper evaluates RMS tracking and approximation error under Lyapunov-based online adaptation; the edge paper evaluates Accuracy, F1, ROC-AUC, latency, and memory on CICIDS2017. These are incomparable objective functions and deployment regimes.

Another point of divergence concerns interpretability. The control paper explicitly frames KANs as providing visualizable functional decomposition through univariate edge-wise functions. The motion paper concludes that LuKAN is compact, interpretable, and efficient, but its core empirical emphasis is the balance between prediction accuracy and computational efficiency through wavelet encoding and Lucas-polynomial recurrence. The edge paper focuses less on interpretability than on deterministic inference latency, CPU-only execution, and Pareto trade-offs among speedup, memory, and detection quality. Thus, the interpretability claim has different operational meanings in different LuKAN variants.

Taken together, the three works situate LuKAN at the intersection of function decomposition and domain-specific systems design. One variant targets multi-scale articulated dynamics, one targets stability-certified online control, and one targets quantized low-latency deployment. This suggests that the enduring scholarly value of the label lies less in a fixed architecture than in a reusable KAN-centered design template that can be specialized by polynomial choice, control law, or compilation strategy.