FourierKAN: A Fourier-Parameterized KAN Variant

• FourierKAN is a variant of Kolmogorov–Arnold Networks that replaces spline coefficients with truncated Fourier series for univariate functions, addressing grid-bound issues.
• It is applied across various domains—from NLP classification heads and Vision Transformer feed-forward layers to graph collaborative filtering—demonstrating versatile efficiency and trainability.
• While boosting computational speed and spectral expressiveness, FourierKAN may underperform in tasks requiring local adaptivity due to its global trigonometric basis.

FourierKAN denotes a family of Kolmogorov–Arnold Network variants in which the learnable univariate functions are parameterized by Fourier-style basis expansions rather than spline bases. In the recent KAN literature, it is commonly described as a model that constructs KAN using 1D Fourier coefficients instead of spline coefficients, thereby replacing spline parameterization with sinusoidal representations built from $\sin$ and $\cos$ terms (Chen et al., 2024). Across applications, the label covers closely related but not fully identical instantiations: a Fourier-series edge-function replacement in KAN heads for frozen-backbone text classification, a feed-forward replacement inside Vision Transformers, an interaction transform in graph collaborative filtering, a KAN autoencoder basis for fault detection, and a broader Fourier-reparameterized KAN architecture built from trainable Random Fourier Features and a hybrid GELU–Fourier mechanism (Imran et al., 2024, S et al., 3 Mar 2025, Xu et al., 2024, Villagómez et al., 4 Aug 2025, Zhang et al., 9 Feb 2025). A consistent theme is the substitution of spline-based local basis functions with global trigonometric ones, usually motivated by computational efficiency, trainability, and spectral expressiveness.

1. Terminology, lineage, and position within the KAN ecosystem

Within the KAN ecosystem, FourierKAN is treated as an important alternative to spline-based KAN variants. One paper explicitly characterizes it as a model “which constructs KAN using 1D Fourier coefficients instead of spline coefficients,” and credits it with two advantages over spline-based KANs: it “circumvents the issue of grids running out of bounds, a common problem in Spl-KAN, while also offering faster computation” (Chen et al., 2024). In that framing, FourierKAN is a periodic or trigonometric KAN variant whose defining move is the replacement of spline coefficients by Fourier-style basis representations.

The bibliographic lineage is somewhat diffuse. In one source, FourierKAN is cited through a GitHub repository attributed to Gist Noesis rather than through an archival paper, which indicates that the term circulated as an implementation and reference point before or alongside formal comparative studies (Chen et al., 2024). Later papers adopt the name in domain-specific settings and sometimes rename it locally. In text classification, for example, the variant is denoted FR-KAN, but the paper explicitly equates FR-KAN with FourierKAN and presents it as a drop-in replacement for an MLP classification head (Imran et al., 2024).

The literature also uses the term at different levels of generality. Some papers use “FourierKAN” narrowly to denote a truncated Fourier-series parameterization of KAN edge functions (Imran et al., 2024, Villagómez et al., 4 Aug 2025). Others embed the idea into a larger architecture, such as FourierKAN-GCF for graph collaborative filtering (Xu et al., 2024). The paper “Kolmogorov-Arnold Fourier Networks” introduces KAF, described as a reworking of KAN that replaces spline parameterization with trainable Random Fourier Features and a hybrid GELU-Fourier activation; this is a broader Fourierized KAN formulation rather than merely a direct substitution of spline coefficients by fixed Fourier modes (Zhang et al., 9 Feb 2025). This suggests that “FourierKAN” is best understood as a design family rather than a single canonical architecture.

2. Mathematical formulations and basis parameterizations

A recurring mathematical pattern is the parameterization of each univariate function by a truncated Fourier series. In the text-classification formulation, the KAN layer is written as

$$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$

with $\phi_{ij}$ as univariate continuous functions and $\Phi_j$ as learnable activation functions. The original spline-based residual form is

$\phi_b(x) = w(b(x) + \text{spline}(x)),$

where

$b(x) = \text{silu}(x) = \frac{x}{1 + e^{-x}}$

and

$\text{spline}(x) = \sum_{i=1}^{G} c_i B_i(x).$

For FR-KAN, the spline term is replaced by

$$\phi_f(x) = \sum_{k=0}^{G} \left( a_{k} \cdot \cos(kx) + b_{k} \cdot \sin(kx) \right),$$

with $a_k$ and $\cos$0 as trainable Fourier coefficients and $\cos$1 as the grid size (Imran et al., 2024). The same paper states a convergence theorem asserting that the truncated Fourier series converges uniformly to the target continuous univariate function on a finite interval as $\cos$2, with truncation error $\cos$3 (Imran et al., 2024).

The same structural idea appears in other domains with small notational variations. In the Burmese news-classification study, the Kolmogorov–Arnold inner functions are written as

$\cos$4

and the layer-level computation is given by

$\cos$5

(Aung et al., 26 Nov 2025). In the autoencoder setting, FourierKAN edge functions are parameterized as

$\cos$6

where $\cos$7 is the number of Fourier modes (Villagómez et al., 4 Aug 2025). In graph collaborative filtering, the FourierKAN transformation is

$\cos$8

with $\cos$9 the feature dimension, $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$0 the gridsize, and $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$1 trainable coefficients (Xu et al., 2024).

The ViKANformer paper describes FourierKAN less formally but in the same spirit. Its generic KAN feed-forward form is

$$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$2

and FourierKAN is specified as using Fourier basis functions $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$3 and $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$4, with $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$5 basis terms per dimension, frequencies and phases initialized uniformly in $$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$6, and coefficients learned by backpropagation (S et al., 3 Mar 2025).

KAF extends this line of development. Instead of explicit spline recursion or a conventional truncated Fourier series per edge, it uses trainable Random Fourier Features

$$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$7

together with a hybrid activation

$$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$8

or, at layer level,

$$\text{KAN}(\mathbf{H}) = f(\mathbf{H}) = \sum_{j=1}^{2n+1} \Phi_j \left( \sum_{i=1}^{n} \phi_{ij}(h_i) \right),$$9

(Zhang et al., 9 Feb 2025). This formulation preserves the Fourier parameterization but changes the computational structure substantially.

3. Architectural roles across application domains

FourierKAN is not tied to a single architectural locus. In frozen-backbone NLP classification, it serves as the classification head while the transformer encoder remains fixed. The text-classification paper evaluates BART, BERT, DeBERTa, DistilBERT, ELECTRA, RoBERTa, and XLNet under a head-only fine-tuning protocol, treating the head as the only trainable component (Imran et al., 2024). In this setting, FourierKAN is presented as a drop-in replacement for 1-layer and 2-layer MLP heads.

In ViKANformer, FourierKAN replaces the standard MLP/feed-forward sub-layer inside each Vision Transformer block while attention, residual connections, and layer normalization remain intact. The architecture is described as image $\phi_{ij}$0 patchify $\phi_{ij}$1 patch embeddings plus positional embeddings $\phi_{ij}$2 Transformer blocks, where the feed-forward MLP is replaced by one of Vanilla KAN, SineKAN, FourierKAN, Fast-KAN, or Efficient-KAN (S et al., 3 Mar 2025). The replacement is explicitly characterized as plug-and-play.

In graph collaborative filtering, FourierKAN is inserted more selectively. FourierKAN-GCF removes $\phi_{ij}$3 from NGCF-style message passing and replaces the interaction transform $\phi_{ij}$4 with $\phi_{ij}$5. The modified propagation applies FourierKAN only to the element-wise user–item interaction term: $\phi_{ij}$6 on the user side and symmetrically on the item side (Xu et al., 2024). This is therefore not a whole-network replacement but an interaction-specific transformation.

In unsupervised fault detection, FourierKAN is instantiated as a fully KAN-based autoencoder. FourierKAN-AE uses an encoder–decoder structure with layer sizes $\phi_{ij}$7, trained on normal operating data only and optimized with reconstruction MSE (Villagómez et al., 4 Aug 2025). Unlike hybrid designs, the paper emphasizes that the KAN autoencoders are implemented using only KAN layers so that the effect of function parameterization is isolated.

The following table summarizes these roles.

Setting	Architectural role of FourierKAN	Reported formulation
Frozen-backbone text classification	Classification head replacing MLP	Truncated Fourier series with trainable $\phi_{ij}$8 (Imran et al., 2024)
Vision Transformer	Feed-forward sub-layer replacing MLP block	Dimension-wise $\phi_{ij}$9, $\Phi_j$0 expansion with $\Phi_j$1 (S et al., 3 Mar 2025)
Graph collaborative filtering	Interaction transform in message passing	$\Phi_j$2 (Xu et al., 2024)
Fault-detection autoencoder	Encoder and decoder KAN layers	Truncated Fourier edge functions with $\Phi_j$3 (Villagómez et al., 4 Aug 2025)
High-dimensional KAF model	Fourier-reparameterized KAN layer	Trainable RFF plus hybrid GELU-Fourier activation (Zhang et al., 9 Feb 2025)

This distribution of uses indicates that FourierKAN is less a monolithic model than a reusable basis-function strategy for replacing spline-based or MLP-style transformations.

4. Empirical performance in supervised learning and representation learning

In text classification, FourierKAN is reported to outperform MLP heads decisively in the studied frozen-backbone regime. Across seven pre-trained transformer backbones and seven datasets spanning sentiment analysis, topic classification, question classification, and language identification, FR-KAN shows an average improvement of 10% in accuracy and 11% in F1-score over MLP heads (Imran et al., 2024). The aggregated averages are reported as 0.582 accuracy and 0.559 macro-F1 for the MLP baseline versus 0.672 accuracy and 0.669 macro-F1 for FR-KAN (Imran et al., 2024). The paper also states that FR-KAN outperforms the original KAN head on average, while noting XLNet as the main exception where FR-KAN is often slightly worse or similar to MLP (Imran et al., 2024). Detailed DistilBERT results include AgNews 0.877 vs 0.835 accuracy, DBpedia 0.970 vs 0.892, Papluca 0.986 vs 0.816, and TREC-50 0.351 vs 0.179 for FR-KAN relative to MLP (Imran et al., 2024).

In Vision Transformers, the picture is more mixed. Under a common MNIST setup with image size $\Phi_j$4, patch size $\Phi_j$5, 16 patches, embedding dimension 8, 2 Transformer blocks, 2 attention heads, batch size 128, 10 epochs, Adam, and cross-entropy loss, FourierKAN reaches 96.6% accuracy, 0.9662 F1, and 0.9991 ROC AUC, at about 8 min/epoch on an A100 GPU (S et al., 3 Mar 2025). These are strong results, but the same paper places FourierKAN below Vanilla KAN (98.0%), SineKAN (97.8%), Fast-KAN (97.8%), and Efficient-KAN (97.4%) in accuracy on MNIST (S et al., 3 Mar 2025). The significance of the result is therefore primarily architectural feasibility and moderate cost rather than best-in-class accuracy in that benchmark.

FourierKAN is also used as a baseline in comparisons among KAN variants. In the MNIST study introducing LSin-SKAN, LCos-SKAN, and LArctan-SKAN, FourierKAN is part of the explicit benchmark set alongside LSS-SKAN, MLP+rKAN, MLP+fKAN, Spl-KAN, FastKAN, and WavKAN (Chen et al., 2024). There, LArctan-SKAN is reported to improve test accuracy by 0.93% over FourierKAN and training speed by 14.89% over FourierKAN under the reported 30-epoch setting (Chen et al., 2024). These comparisons do not establish a universal ordering across KAN variants, but they show that FourierKAN has quickly become a standard comparison point for trigonometric or alternative-basis KAN proposals.

In the broader KAF study, Fourier-reparameterized KANs are reported to be effective across vision, NLP, audio, function approximation, and PDE solving. Representative results include ResNet-18 on CIFAR-10: KAF 91.72% vs MLP 91.19%, MLP-Mixer/S on ImageNet-1K: KAF 64.7% vs MLP 63.5%, ViT-T/16 on ImageNet-1K: KAF 73.2% vs MLP 72.3%, and MLP-KAN on CIFAR-100: KAF 53.8% vs MLP 49.0% (Zhang et al., 9 Feb 2025). In GPT-2 experiments, replacing the FFN MLP with KAF improves PLL and reduces training time: WikiText: PLL from 184.53 to 180.85, training time from 20h37m to 19h20m; Open WebText: PLL from 151.27 to 145.64, training time from 60h57m to 52h45m (Zhang et al., 9 Feb 2025). Because KAF is not identical to the truncated-series FourierKAN used in other papers, these results are best read as evidence for the broader viability of Fourier-parameterized KAN designs.

5. Efficiency, trainability, and parameterization trade-offs

Efficiency is one of the most persistent motivations for FourierKAN. The LArctan-SKAN paper attributes to FourierKAN faster computation than spline-based KANs and the avoidance of out-of-bounds grid issues in Spl-KAN (Chen et al., 2024). The graph collaborative filtering paper makes a related claim in stronger terms: “The Fourier Coefficients has a significant advantage in computational efficiency and solves the training difficulty caused by the spline function” (Xu et al., 2024). In that work, FourierKAN is specifically introduced because standard KAN is described as difficult to train due to spline functions.

In NLP head fine-tuning, FR-KAN is reported to be more computationally efficient, to train faster, and to use fewer or comparable trainable parameters than the MLP head (Imran et al., 2024). The detailed DistilBERT table shows AgNews with MLP-40 at 30.9k parameters versus FR-KAN-5 at 30.7k, and DBpedia with MLP-138 at 108.1k versus FR-KAN-5 at 107.5k (Imran et al., 2024). The paper does not give a single wall-clock speedup ratio, but repeatedly states that FR-KAN “trains faster” and “requires similar time to fine-tune” (Imran et al., 2024).

The Burmese news-classification results show that FourierKAN’s efficiency profile depends strongly on the embedding regime. For TF-IDF, FourierKAN has 2.99M params, 28.1 s train, 0.56 ms forward, 1.23 ms backward, versus 0.13M params, 10.7 s train, 0.26 ms forward, 0.64 ms backward for MLP (Aung et al., 26 Nov 2025). For random embeddings, FourierKAN has 0.27M params, 8.0 s, 0.56 ms, 1.23 ms versus 2.00M params, 13.5 s, 0.44 ms, 0.97 ms for MLP (Aung et al., 26 Nov 2025). For mBERT, FourierKAN has 180M params, 1481.2 s, 115.33 ms, 215.38 ms, while MLP has 178M params, 1284.8 s, 203.03 ms, 418.72 ms (Aung et al., 26 Nov 2025). The paper’s interpretation is correspondingly nuanced: FourierKAN is usually slower to train than MLP on TF-IDF and transformer setups, but its inference latency is generally better than the transformer-baseline MLP heads (Aung et al., 26 Nov 2025).

ViKANformer again shows a middle-ground profile. Its time-per-epoch table reports FourierKAN: about 8 min/epoch, compared with Vanilla KAN: 7 min/epoch, SineKAN: 9 min/epoch, Fast-KAN: 20 min/epoch, and Efficient-KAN: 47 min/epoch (S et al., 3 Mar 2025). FourierKAN is therefore neither the cheapest nor the most expensive KAN option in that setting.

KAF turns efficiency into a primary architectural objective. The paper states that standard KAN suffers from parameter explosion and gives the KAN parameter complexity as

$\Phi_j$6

whereas KAF reduces this to

$\Phi_j$7

by merging the dual-matrix structure and replacing spline recursion with trainable RFF (Zhang et al., 9 Feb 2025). This is the most radical efficiency claim in the surveyed literature, but it applies specifically to KAF’s reparameterized design rather than to all FourierKAN variants.

6. Limitations, failure modes, and contested advantages

FourierKAN is not uniformly dominant across tasks, and several papers identify clear limitations. In head-only NLP fine-tuning, the gains are “not universal,” with XLNet explicitly described as a consistent exception where FR-KAN is often slightly worse or similar to MLP (Imran et al., 2024). The same paper also notes that FR-KAN’s improved performance comes at some cost to interpretability compared with spline KAN (Imran et al., 2024).

The Burmese news-classification study is more critical. Weighted F1-scores for FourierKAN are reported as 0.538 with TF-IDF, 0.699 with random embeddings, 0.829 with fastText, 0.788 with Distil-mBERT, and 0.877 with mBERT (Aung et al., 26 Nov 2025). In the same benchmark, the best scores in those families are 0.791, 0.917, 0.928, 0.873, and 0.917, respectively (Aung et al., 26 Nov 2025). The authors describe FourierKAN as structurally elegant and efficient at inference but less adaptive than spline-based KANs, and state that it struggles with complex localized decision boundaries (Aung et al., 26 Nov 2025). Their practical conclusion is that EfficientKAN is usually best for accuracy and FasterKAN offers the best speed–accuracy trade-off in that task (Aung et al., 26 Nov 2025).

The strongest negative result comes from unsupervised fault detection on the Tennessee Eastman Process. FourierKAN-AE is reported to consistently underperform across all training sizes and fault categories (Villagómez et al., 4 Aug 2025). The paper attributes this to “limitations of its low-order global basis functions” and to the “limitations of its global basis representation in capturing fault-specific deviations” (Villagómez et al., 4 Aug 2025). For the 500-sample regime, example FDRs include Fault 5: 30.41 ± 1.20, Fault 10: 47.48 ± 1.20, Fault 16: 36.00 ± 1.35, Fault 19: 14.71 ± 2.57, Fault 20: 50.85 ± 1.76, and Fault 21: 43.85 ± 1.35, all substantially below EfficientKAN-AE and often below other baselines (Villagómez et al., 4 Aug 2025). The authors explicitly conclude that FourierKAN-AE is not a good fit for low-data unsupervised fault detection on TEP, because its global periodic basis lacks the locality and adaptivity needed for localized, non-periodic, abrupt, or transient process deviations (Villagómez et al., 4 Aug 2025).

These results qualify a common misconception that replacing splines by Fourier bases automatically improves KANs. The evidence instead suggests that global trigonometric bases are advantageous when smooth spectral structure, trainability, or compact function modeling is central, but can be a poor inductive bias when the target phenomenon is dominated by local, piecewise, or fault-specific deviations.

7. Conceptual interpretation and relation to adjacent KAN variants

The comparative literature makes clear that FourierKAN occupies a specific conceptual position among KAN variants. Relative to spline-based KANs, it replaces local spline expansions with global sinusoidal bases (Chen et al., 2024, Imran et al., 2024). Relative to single-parameter trigonometric KANs such as LSin-SKAN, LCos-SKAN, and LArctan-SKAN, it remains a coefficient-expansion model rather than an edge-wise activation with a single learnable parameter (Chen et al., 2024). Relative to SineKAN in ViKANformer, it uses both $\Phi_j$8 and $\Phi_j$9 basis functions rather than only sinusoidal activations of the form

$\phi_b(x) = w(b(x) + \text{spline}(x)),$0

(S et al., 3 Mar 2025). Relative to EfficientKAN, it is generally described as less locally adaptive but computationally lighter or easier to train in at least some regimes (Xu et al., 2024, Aung et al., 26 Nov 2025).

The KAF paper broadens this interpretation by arguing that conventional KANs have two practical shortcomings: parameter explosion and high-frequency feature capture challenges in high-dimensional tasks (Zhang et al., 9 Feb 2025). Its answer is not merely to truncate a Fourier series on each edge, but to introduce trainable RFF and an adaptive hybrid activation that begins near GELU-dominated behavior with

$\phi_b(x) = w(b(x) + \text{spline}(x)),$1

and gradually increases the Fourier branch’s contribution during training (Zhang et al., 9 Feb 2025). This suggests a second strand in FourierKAN research: not only replacing splines with Fourier bases, but reorganizing KAN computation to make spectral modeling a first-class design principle.

Taken together, the literature supports a layered interpretation. At its narrowest, FourierKAN is a KAN whose univariate functions are expanded in sine and cosine modes. At a broader level, it names a line of research that uses Fourier-domain parameterization to address known KAN issues involving spline complexity, trainability, and frequency representation. Its empirical record is correspondingly heterogeneous: strong as a frozen-backbone classification head, competitive as a transformer feed-forward replacement, effective in graph collaborative filtering, useful as a benchmark baseline for newer trigonometric KANs, but weak in fault-detection settings where global periodic bases provide the wrong inductive bias (Imran et al., 2024, S et al., 3 Mar 2025, Xu et al., 2024, Chen et al., 2024, Villagómez et al., 4 Aug 2025).