Wav-KAN: Wavelet Kolmogorov–Arnold Networks

• Wav-KAN is a family of architectures that replaces fixed node activations with learnable wavelet functions, enabling multi-resolution analysis and interpretability.
• The approach employs explicit wavelet parameters such as amplitude, scale, and translation on network edges to achieve parameter efficiency and sparse representations.
• Variants like WavKAN-AE and WaveTuner demonstrate practical benefits through improved performance in fault detection, MNIST classification, and federated learning scenarios.

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 (Bozorgasl et al., 2024, Villagómez et al., 4 Aug 2025, Wang et al., 24 Nov 2025).

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 (Bozorgasl et al., 2024). A second usage is WavKAN-AE, a wavelet-based KAN autoencoder evaluated for unsupervised fault detection on the Tennessee Eastman Process (Villagómez et al., 4 Aug 2025). 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 (Wang et al., 24 Nov 2025).

Source	What “WavKAN” denotes	Functional basis
(Bozorgasl et al., 2024)	Wavelet Kolmogorov–Arnold Networks	Learnable wavelet functions
(Villagómez et al., 4 Aug 2025)	WavKAN-AE autoencoder	Single DoG wavelet per edge
(Wang et al., 24 Nov 2025)	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 (Bozorgasl et al., 2024). 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 $\tau_{q,m}$ (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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 (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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 $\phi_{q,p}$0, 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 (Villagómez et al., 4 Aug 2025). Because each edge carries its own amplitude, shift, and scale, the model can adapt locally at multiple resolutions (Villagómez et al., 4 Aug 2025).

3. Architectural realizations

The original supervised Wav-KAN paper presents a compact classifier in which all models in the MNIST comparison use architecture $\phi_{q,p}$1 (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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. $\phi_{q,p}$2 (Villagómez et al., 4 Aug 2025). 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 (Villagómez et al., 4 Aug 2025).

In federated learning, the FL-Wav-KAN setup uses topology $\phi_{q,p}$3 on MNIST, and explicitly considers wavelet families Mexican hat, Morlet $\phi_{q,p}$4, Derivative-of-Gaussian (DOG), and Shannon (Bozorgasl et al., 2024). The per-neuron scale and translation parameters are initialized as $\phi_{q,p}$5 and $\phi_{q,p}$6 (Bozorgasl et al., 2024).

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 $\phi_{q,p}$7, the default base is $\phi_{q,p}$8, activations inside the branch use tanh candidates, all $\phi_{q,p}$9 parameters are initialized orthogonally, and a residual skip produces $\Phi_q$0 (Wang et al., 24 Nov 2025). 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 $\Phi_q$1 and weight decay $\Phi_q$2 (Bozorgasl et al., 2024). Every Wav-KAN layer is followed by BatchNorm, which is described as essential for rapid training (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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 (Villagómez et al., 4 Aug 2025). 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 $\Phi_q$3 and early stop after 15 validation epochs with no loss improvement (Villagómez et al., 4 Aug 2025). Fault detection thresholds are set by kernel density estimation of the normal SPE with $\Phi_q$4 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 (Villagómez et al., 4 Aug 2025). For WavKAN-AE specifically, hyperparameters were optimized by a Tree-structured Parzen Estimator on a 625-sample split, yielding initial learning rate $\Phi_q$5, weight decay $\Phi_q$6, and scheduler reduction factor $\Phi_q$7 with ReduceLROnPlateau patience $\Phi_q$8 (Villagómez et al., 4 Aug 2025).

The federated-learning variant is described as a FedAvg-style procedure on all parameters $\Phi_q$9, with optional CWT or DWT preprocessing, local optimization by AdamW, $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$0 local epochs, $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$1 global rounds, and 3 independent trials per setting (Bozorgasl et al., 2024). Evaluation includes training accuracy, test accuracy, wall-clock time per round, and robustness to 10% label noise (Bozorgasl et al., 2024). WaveTuner, by contrast, trains end-to-end with elementwise SmoothL1 (Huber) loss, Adam $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$2, initial learning rate $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$3, learning-rate halving on plateau after 5 epochs, and early stopping on validation loss, with weight decay $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$4 on all linear layers (Wang et al., 24 Nov 2025).

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 $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$5, WavKAN-AE reaches 85%, compared with 78% for the OAE baseline, 90% for EfficientKAN, 82% for FastKAN, and 70% for FourierKAN. At $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$6, WavKAN-AE reaches 92%, matching EfficientKAN and exceeding OAE (85%), FastKAN (88%), and FourierKAN (75%). At $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$7, WavKAN-AE reaches 94%, ahead of EfficientKAN (93%), FastKAN (92%), OAE (91%), and FourierKAN (80%) (Villagómez et al., 4 Aug 2025). 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 $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$8 FDR as early as $$\phi_q(u)=\sum_{m=1}^{M_q} w_{q,m}\,\psi\!\Bigl(\frac{u-\tau_{q,m}}{s_{q,m}}\Bigr),$$9 (Villagómez et al., 4 Aug 2025).

The study also identifies a nuanced low-data regime. EfficientKAN-AE is slightly ahead at $w_{q,m}$0, 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 (Villagómez et al., 4 Aug 2025). 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 $w_{q,m}$1 for $w_{q,m}$2, while remaining practically equivalent within a $w_{q,m}$3 FDR region in data-rich regimes and never yielding to the baseline (Villagómez et al., 4 Aug 2025).

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 $w_{q,m}$4 versus Spl-KAN’s approximately $w_{q,m}$5 (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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 (Bozorgasl et al., 2024). The paper reports Morlet $w_{q,m}$6 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 (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024). 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 (Bozorgasl et al., 2024).

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 (Wang et al., 24 Nov 2025). 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 (Wang et al., 24 Nov 2025). 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 (Villagómez et al., 4 Aug 2025). 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 (Villagómez et al., 4 Aug 2025). Because each edge has semantically explicit parameters $w_{q,m}$7, the model also supports post-hoc analysis of which variables, locations, and scales are most indicative of a fault (Villagómez et al., 4 Aug 2025).

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 (Bozorgasl et al., 2024). The same work emphasizes parameter asymptotics: for width $w_{q,m}$8 and depth $w_{q,m}$9, MLPs require $s_{q,m}$0 weights, Spl-KAN requires $s_{q,m}$1 parameters, and Wav-KAN uses exactly three parameters per edge, i.e. $s_{q,m}$2 (Bozorgasl et al., 2024). In the autoencoder setting, this is instantiated concretely by the 6,716-parameter WavKAN-AE model (Villagómez et al., 4 Aug 2025).

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 (Bozorgasl et al., 2024, Villagómez et al., 4 Aug 2025, Wang et al., 24 Nov 2025). 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 (Bozorgasl et al., 2024). 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 $s_{q,m}$3 (Bozorgasl et al., 2024). 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.