---
title: 'Wav-KAN: Wavelet Kolmogorov–Arnold Networks'
url: https://www.emergentmind.com/topics/wavkan
type: topic
---

# Wav-KAN: Wavelet Kolmogorov–Arnold Networks

WavKAN, usually written **Wav-KAN**, denotes a family of Kolmogorov–Arnold Network constructions in which wavelet functions replace fixed node activations or spline-based univariate maps. The central architectural move is to place learnable wavelet-parameterized functions on edges or channels, typically with learnable amplitude, scale, and translation, so that the network inherits multi-resolution structure from wavelet analysis while retaining the univariate-function decomposition associated with the Kolmogorov–Arnold representation theorem. In the recent literature, the term appears in several related but not identical senses: as **Wavelet Kolmogorov–Arnold Networks** for supervised learning, as **WavKAN-AE** for unsupervised fault detection in the Tennessee Eastman Process, and as a label inside a wavelet-packet forecasting framework for a KAN-based submodule rather than a wavelet-edge network [2405.12832, 2508.02860, 2511.18846].

## 1. Terminology and scope

The most direct usage is the architecture introduced as **“Wav-KAN: Wavelet Kolmogorov-Arnold Networks”**, where wavelet basis functions are used in place of the spline parameterizations associated with Spl-KAN [2405.12832]. A second usage is **WavKAN-AE**, a wavelet-based KAN autoencoder evaluated for unsupervised fault detection on the Tennessee Eastman Process [2508.02860]. A third usage appears in **WaveTuner**, where the authors refer to their Multi-Branch Specialization module as “WavKAN,” even though its branch nonlinearity is a learnable Chebyshev-polynomial expansion applied after wavelet-packet decomposition rather than a wavelet function placed directly on each edge [2511.18846].

| Source | What “WavKAN” denotes | Functional basis |
|---|---|---|
| [2405.12832] | Wavelet Kolmogorov–Arnold Networks | Learnable wavelet functions |
| [2508.02860] | WavKAN-AE autoencoder | Single DoG wavelet per edge |
| [2511.18846] | WaveTuner MBS / “WavKAN” | Chebyshev-polynomial KAN on wavelet subbands |

A separate summary associated with wavelet-based communications also uses the label **“WavKAN”** for modem implementation issues built around the Kaiser electromagnetic wavelet, channel kernels, and fractal modulation rather than a neural-network model [0904.2584]. This suggests that the label is not yet fully stabilized across preprint-style usage, and that the precise meaning must be inferred from context.

## 2. Mathematical basis

The KAN lineage is grounded in the Kolmogorov–Arnold superposition theorem, stated in the Wav-KAN literature as
$$
f(x_1,\dots,x_n)=\sum_{q=1}^{2n+1}\Phi_q\Bigl(\sum_{p=1}^n \phi_{q,p}(x_p)\Bigr),
$$
where the inner maps $\phi_{q,p}$ and outer maps $\Phi_q$ are continuous univariate functions [2405.12832]. Wav-KAN preserves this decomposition but replaces spline parameterizations with wavelet parameterizations. One formulation writes each hidden-channel function as
$$
\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),
$$
with learnable coefficient $w_{q,m}$, scale $s_{q,m}$, and translation $\tau_{q,m}$ [2405.17836]. In the original Wav-KAN presentation, each scalar weight in an MLP is replaced by a learnable univariate function, and the next layer is obtained by a row-sum operator over the resulting matrix of edge evaluations [2405.12832].

Wavelet theory supplies the multi-resolution component. The same literature discusses both the Continuous Wavelet Transform and the Discrete Wavelet Transform; in the dyadic DWT setting, high-pass and low-pass recursion yields a multiresolution analysis in which finer details can be added without recomputing coarse-scale coefficients [2405.12832]. In federated variants, clients may preprocess raw signals through either CWT or DWT, then feed the resulting multi-scale coefficients directly to the KAN input layer or use them to initialize scale and translation parameters [2405.17836].

The autoencoder instantiation used for process monitoring specializes this general construction to a **single wavelet per edge**. For the connection from neuron $j$ in layer $\ell-1$ to neuron $i$ in layer $\ell$, the edge function is parameterized by amplitude, translation, and scale, and the layer output is the sum over all incoming edge transforms. In the reported experiments, the mother wavelet was a **single-difference-of-Gaussians (DoG) wavelet** [2508.02860]. Because each edge carries its own amplitude, shift, and scale, the model can adapt locally at multiple resolutions [2508.02860].

## 3. Architectural realizations

The original supervised Wav-KAN paper presents a compact classifier in which all models in the MNIST comparison use architecture **$[28\times 28,\,32,\,10]$** [2405.12832]. In this setting, Wav-KAN is positioned against Spl-KAN and standard MLP baselines, with wavelet families such as Derivative-of-Gaussian and Mexican hat reported as the best-performing options [2405.12832].

For unsupervised industrial monitoring, the WavKAN-AE architecture adopts a **3-layer bottleneck** with **33 inputs, a 25-dimensional latent code, and 33 outputs**, i.e. **$[33 \to 25 \to 33]$** [2508.02860]. The edge parameterization uses a **single DoG wavelet, one wavelet per edge**, with **no B-spline or RBF expansion**. The model has **6,716 learnable parameters**, fewer than the other KAN variants in that study, and uses **standard weight-decay only**, with **no extra L1/entropy or orthogonality terms** [2508.02860].

In federated learning, the FL-Wav-KAN setup uses topology **$[28\times 28 \to 64 \text{ hidden units} \to 10 \text{ outputs}]$** on MNIST, and explicitly considers wavelet families **Mexican hat, Morlet $(\omega_0=5)$, Derivative-of-Gaussian (DOG), and Shannon** [2405.17836]. The per-neuron scale and translation parameters are initialized as **$s \sim \mathrm{Uniform}(0.5,1.5)$** and **$\tau \sim \mathrm{Uniform}(-1,1)$** [2405.17836].

WaveTuner introduces another architectural usage of the term. After wavelet-packet decomposition and adaptive refinement, each subband embedding is fed into its own KAN branch, whose complexity grows with branch index: the branch order is **$n_i=b+i$**, the default base is **$b=2$**, activations inside the branch use **tanh candidates**, all $\Theta$ parameters are **initialized orthogonally**, and a **residual skip** produces $\hat f_i=KAN_i(f_i)+f_i$ [2511.18846]. Here, “WavKAN” denotes a subband-specialized KAN module operating on a wavelet-domain representation rather than a wavelet-edge network in the narrow sense.

## 4. Training protocols and optimization

In the original supervised formulation, Wav-KAN is trained with **cross-entropy loss** and **AdamW** with learning rate **$1\times 10^{-3}$** and weight decay **$10^{-4}$** [2405.12832]. Every Wav-KAN layer is followed by **BatchNorm**, which is described as essential for rapid training [2405.12832]. The same work also notes that, in DWT mode, hidden-layer maps can be implemented via a discrete wavelet filter bank, and that the nested multiresolution structure avoids recomputation of coarse-scale coefficients when finer-scale details are added [2405.12832].

The Tennessee Eastman Process study follows a unified autoencoder protocol across all variants. Each time step contains **33 process variables (11 manipulated, 22 measurements)**, the data are **z-score normalized**, and **13 training set sizes** are used: **625, 1 250, 1 875, 3 125, 5 000, 8 125, 13 750, 23 125, 38 125, 64 375, 107 500, 180 000, 250 000** [2508.02860]. Each subset is split **80/20 into training/validation at the simulation-run level**. Optimization uses **AdamW**, **batch size 256**, and **mixed-precision (PyTorch AMP)**, with **maximum epochs $E_{\max}=600$** and **early stop after 15 validation epochs with no loss improvement** [2508.02860]. Fault detection thresholds are set by **kernel density estimation of the normal SPE** with **$\alpha=0.05$ false alarm rate**, and the test protocol uses **500 runs per fault (960 samples each)** for **faults 1–21 from the Russell et al. TEP benchmark** [2508.02860]. For WavKAN-AE specifically, hyperparameters were optimized by a **Tree-structured Parzen Estimator on a 625-sample split**, yielding **initial learning rate $\eta_0=4.99\times 10^{-3}$**, **weight decay $\lambda_{\mathrm{wd}}=7.60\times 10^{-3}$**, and **scheduler reduction factor $0.95$ with ReduceLROnPlateau patience $=5$** [2508.02860].

The federated-learning variant is described as a **FedAvg-style** procedure on all parameters **$\{\alpha,\beta;w,s,\tau\}$**, with optional CWT or DWT preprocessing, local optimization by **AdamW**, **$K=5$ local epochs**, **$T=50$ global rounds**, and **3 independent trials per setting** [2405.17836]. Evaluation includes **training accuracy, test accuracy, wall-clock time per round, and robustness to 10% label noise** [2405.17836]. WaveTuner, by contrast, trains end-to-end with **elementwise SmoothL1 (Huber) loss**, **Adam $(\beta_1=0.9,\beta_2=0.999)$**, **initial learning rate $10^{-3}$**, learning-rate halving on plateau after **5 epochs**, and **early stopping on validation loss**, with **weight decay $10^{-5}$** on all linear layers [2511.18846].

## 5. Empirical performance

The clearest quantitative case study for WavKAN as a standalone model is the Tennessee Eastman Process benchmark. Average Fault Detection Rate across **21 faults**, averaged over **30 seeds**, is reported at three representative training sizes. At **$n=500$**, WavKAN-AE reaches **85%**, compared with **78%** for the OAE baseline, **90%** for EfficientKAN, **82%** for FastKAN, and **70%** for FourierKAN. At **$n=4\,000$**, WavKAN-AE reaches **92%**, matching EfficientKAN and exceeding OAE (**85%**), FastKAN (**88%**), and FourierKAN (**75%**). At **$n=50\,000$**, WavKAN-AE reaches **94%**, ahead of EfficientKAN (**93%**), FastKAN (**92%**), OAE (**91%**), and FourierKAN (**80%**) [2508.02860]. The same study states that WavKAN-AE **surpasses 92% overall FDR with only 4,000 training samples**, whereas OAE requires **approximately 30,500 samples** to reach that level, and that on **uncontrollable faults** WavKAN-AE achieves **$>92\%$ FDR as early as $n=2,500$** [2508.02860].

The study also identifies a nuanced low-data regime. **EfficientKAN-AE** is slightly ahead at **$n=500$**, reaching **approximately 90%**, but is eventually overtaken by WavKAN-AE; **FastKAN** requires **approximately 50,000 samples** to match WavKAN’s **92% FDR**; **FourierKAN** remains below the other variants at all scales [2508.02860]. Bayesian signed-rank tests are reported to show that WavKAN-AE is **decisively superior to OAE and other KAN-AEs in data-scarce regimes** with **$P_{\mathrm{right}}>0.94$ for $n\leq 6,500$**, while remaining **practically equivalent within a $\pm1\%$ FDR region** in data-rich regimes and **never yielding to the baseline** [2508.02860].

On supervised MNIST, the original Wav-KAN paper reports that Wav-KAN **converges in fewer epochs and achieves higher test accuracy than Spl-KAN and standard MLP baselines**, with **Derivative-of-Gaussian (DOG)** and **Mexican hat** performing best and achieving **test accuracy $>98\%$** versus **Spl-KAN’s approximately $97\%$** [2405.12832]. Spl-KAN is described as tending to overfit, with training accuracy near **100%** but lower test-time generalization, while Wav-KAN’s orthogonal wavelet basis is said to enforce spectral sparsity and improve robustness [2405.12832].

In federated MNIST, the best reported Wav-KAN configuration is **Mexican hat**, with **99.4 (±0.1)** train accuracy, **98.1 (±0.2)** test accuracy, **0.82 s** per round, and **1.3** percentage-point drop under **10% label noise** [2405.17836]. The paper reports **Morlet $(\omega_0=5)$** at **97.9 (±0.3)** test accuracy and **0.85 s**, **DOG** at **97.8 (±0.2)** and **0.80 s**, and **Shannon** at **96.4 (±0.4)** and **0.78 s** [2405.17836]. The comparators are **Spl-KAN** with **95.2 (±0.5)** test accuracy and **1.10 s**, and **FedAvg-CNN** with **96.8 (±0.3)** and **0.95 s** [2405.17836]. The same study states that Wav-KAN with Mexican hat gives **+2.9% test accuracy vs. Spl-KAN** and is **25% faster per-round**, while **DWT-KAN** runs **approximately 20% faster** than **CWT-KAN** at a **slight cost of approximately 0.3% accuracy** [2405.17836].

Within WaveTuner, the WavKAN-labeled KAN branches contribute to a forecasting system that achieves the **lowest MSE/MAE on 28/32 settings**, including a reduction on **ETTh1@96→96** from **0.374** to **0.368** in MSE [2511.18846]. The ablation table reports **+0.009 MSE** when all KANs are replaced by simple MLPs, and **+0.005–0.006** when fixed low-order or high-order KANs are used instead of increasing-order branches [2511.18846]. These results belong to a broader wavelet-packet framework rather than a pure wavelet-edge Wav-KAN, but they demonstrate that the label is being associated with subband-specialized KAN computation in later work.

## 6. Interpretability, efficiency, and current research directions

Across the literature, the principal rationale for WavKAN is the conjunction of **locality**, **multi-resolution structure**, and **explicit parameterization**. The fault-detection study attributes WavKAN-AE’s performance to three factors: a **localized, multiresolution basis**, **parameter efficiency and rapid saturation**, and **enhanced interpretability** [2508.02860]. In that account, a single-wavelet-per-edge parameterization allows each connection to tune its own center and scale of sensitivity, so that small-scale wavelets can capture sharp transient deviations and large-scale wavelets can capture broad drift-like trends [2508.02860]. Because each edge has semantically explicit parameters $(w,t,s)$, the model also supports post-hoc analysis of which variables, locations, and scales are most indicative of a fault [2508.02860].

The original Wav-KAN paper makes a closely related interpretability claim in signal-processing terms. Each univariate wavelet function can be read as a localized feature extractor, analogous to a matched filter at a learned scale and location, and orthogonal or semi-orthogonal wavelet bases are said to reduce superposition interference and yield sparser, more disentangled feature spaces than dense MLPs [2405.12832]. The same work emphasizes parameter asymptotics: for width $N$ and depth $L$, **MLPs require $O(N^2L)$ weights**, **Spl-KAN requires $O(N^2L(G+k+1))$ parameters**, and **Wav-KAN uses exactly three parameters per edge**, i.e. **$O(3N^2L)$** [2405.12832]. In the autoencoder setting, this is instantiated concretely by the **6,716-parameter** WavKAN-AE model [2508.02860].

A common misconception is that WavKAN denotes a single fixed implementation. The available literature suggests otherwise. The name currently covers at least a wavelet-edge KAN for supervised learning, a single-DoG-per-edge autoencoder for industrial process monitoring, and a wavelet-packet forecasting module whose KAN branches are Chebyshev rather than wavelet-edge expansions [2405.12832, 2508.02860, 2511.18846]. The shared theme is not a unique formula but the use of KAN-style univariate decomposition together with wavelet-domain structure, whether at the edge-function level, the preprocessing level, or the subband-specialization level.

The research directions explicitly proposed for Wav-KAN include extension to **time-series, genomics, and audio**, **hybrid wavelet-spline architectures**, integration into **PyTorch and TensorFlow**, and **automatic selection or learning of mother-wavelet families** [2405.12832]. In federated settings, the literature additionally points to heterogeneous client configurations in which **CWT and DWT clients can be mixed** while the server aggregates only the shared parameter set **$\{w,s,\tau\}$** [2405.17836]. Taken together, these lines of work position WavKAN as a compact and interpretable KAN family whose most mature empirical demonstrations presently lie in MNIST-scale supervised learning, federated learning on MNIST, and data-constrained fault detection on the Tennessee Eastman Process.

Source: https://www.emergentmind.com/topics/wavkan