Functional Kolmogorov-Arnold Network (FunKAN)

• FunKAN is a neural architecture that processes image feature maps as functions in Hilbert space via Fourier Hermite decomposition.
• It preserves spatial structure in 2D MRI data through dynamic grid adaptation, enhancing tasks like Gibbs ringing suppression.
• Empirical results show a 1 dB PSNR gain over KAN baselines and efficient segmentation performance across BUSI, GlaS, and CVC-ClinicDB datasets.

Functional Kolmogorov-Arnold Network (FunKAN) is a KAN-style neural architecture proposed for medical image enhancement and segmentation that replaces scalar-coordinate processing by functional processing of image feature maps in a Hilbert space, while learning inner functions through Fourier decomposition over basis Hermite functions (Penkin et al., 16 Sep 2025). It is motivated by the claim that standard Kolmogorov-Arnold Networks, although interpretable, rely on flattened feature representations that disrupt the intrinsic spatial structure of imaging data, which is especially problematic for image-to-image tasks such as Gibbs ringing suppression in magnetic resonance imaging and binary segmentation of anatomically structured targets (Penkin et al., 16 Sep 2025). In this formulation, FunKAN sits within the broader KAN family of models in which the basic trainable objects are functions rather than scalar weights, but its distinctive step is to make whole feature maps, rather than scalar coordinates, the objects on which those learned functionals act (Sohail, 2024, Penkin et al., 16 Sep 2025).

1. Historical and conceptual setting

The immediate background to FunKAN is the Kolmogorov-Arnold representation theorem, stated in the FunKAN formulation as

$$f(x_{1}, ..., x_{n}) = \sum_{j=1}^{2n+1} \zeta_{j} \left(\sum_{i=1}^{n} \phi_{ji}(x_{i})\right),$$

with continuous univariate inner functions $\phi_{ji}$ and outer functions $\zeta_j$ (Penkin et al., 16 Sep 2025). Standard KANs relax this theorem into deep architectures whose layerwise form is written as

$$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$

with

$x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$

so that learned scalar-to-scalar functions replace the usual fixed scalar weights on edges (Penkin et al., 16 Sep 2025). In the broader KAN literature, these edge functions are often implemented with B-spline expansions and related basis parameterizations, and the resulting models are understood as architectures whose trainable objects are univariate functions rather than only scalar weights (Sohail, 2024).

FunKAN is proposed against the specific objection that this scalar-coordinate viewpoint is mismatched to image processing. The stated problem is not that KANs lack interpretability, but that their reliance on flattened feature representations fundamentally disrupts spatial organization in 2D medical images (Penkin et al., 16 Sep 2025). That criticism is consistent with a wider theoretical caution in the Kolmogorov-Arnold literature: the theorem itself does not straightforwardly justify a practical shallow architecture, and later analyses argue that its most natural interpretation is closer to a deep interior encoding followed by univariate readout than to a literal two-hidden-layer implementation (Schmidt-Hieber, 2020). FunKAN therefore reuses the theorem-inspired decomposition while changing the underlying objects from scalar coordinates to feature maps.

This places FunKAN in a broader line of KAN generalizations that alter the representation of learned functions rather than abandoning the KAN principle. Other examples include piecewise-linear finite-element parameterizations in P1-KAN (Warin, 2024), adaptive radial basis expansions in Free-RBF-KAN (Chiu et al., 12 Jan 2026), dual-stage function control in DualFlexKAN (Ortiz et al., 9 Mar 2026), and function sharing across symmetry orbits in FS-KAN (Elbaz et al., 29 Sep 2025). FunKAN differs from these by centering the architectural redesign on image feature maps as elements of a Hilbert space (Penkin et al., 16 Sep 2025).

2. Functional formulation

The central formal move in FunKAN is to replace scalar coordinates $x_{l,i}$ by feature maps $\chi_{l,i}$ that are treated as elements of a Hilbert space $H$ (Penkin et al., 16 Sep 2025). The motivating functional analogue of the Kolmogorov-Arnold theorem is stated as

$$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$

where $\chi_i \in H$, $\phi_{ji}$0, and $\phi_{ji}$1 are continuous (Penkin et al., 16 Sep 2025). The formulation is explicitly hypothetical rather than formally proved. Its role is to motivate mappings between collections of feature maps by continuous linear functionals followed by additive recombination and nonlinear composition.

At the layer level, the architecture is written as

$\phi_{ji}$2

with $\phi_{ji}$3 (Penkin et al., 16 Sep 2025). The construction then invokes the Riesz representation theorem to identify continuous linear functionals with inner products against functions in the same Hilbert space. The manuscript subsequently simplifies the notation in a way that is acknowledged to be dimensionally odd; the intended meaning is that each $\phi_{ji}$4 acts on $\phi_{ji}$5 through such an inner-product representation (Penkin et al., 16 Sep 2025).

To obtain a trainable parametric form, each inner functional is expanded in a truncated Hermite basis: $\phi_{ji}$6 The coefficients are defined by

$\phi_{ji}$7

and arranged into an attention matrix

$\phi_{ji}$8

The implementation then factors channel mixing outward into a form resembling a $\phi_{ji}$9 convolution: $\zeta_j$0 This is described as “Fourier decomposition over the basis Hermite functions,” with Hermite functions chosen because they are eigenfunctions of the Fourier transform and have dual localization properties in both space and frequency (Penkin et al., 16 Sep 2025).

The basis truncation parameter is fixed to $\zeta_j$1 in the reported experiments. That choice is attributed to prior work that grid-searched among B-splines, Chebyshev polynomials, and Hermite functions and found six basis functions optimal (Penkin et al., 16 Sep 2025). A plausible implication is that the FunKAN design treats basis selection as part of the inductive bias, in the same broad sense that P-KAN treats training as a search over functional representation spaces such as Fourier, Chebyshev, and Bessel families (Poole et al., 24 Sep 2025).

3. Architectural organization and spatial mechanisms

FunKAN is designed as an image-to-image backbone rather than as a flattened-feature multilayer perceptron (Penkin et al., 16 Sep 2025). The learned basis functions are evaluated directly on a 2D spatial grid, and the model adds a dynamic grid deformation mechanism so that basis evaluation need not remain tied to a rigid lattice.

The deformed grid is

$\zeta_j$2

where $\zeta_j$3 is predicted from the current feature maps. The offset tensors satisfy

$\zeta_j$4

The residual prediction module is given by

$\zeta_j$5

with

$\zeta_j$6

The listed kernel tensors are

$\zeta_j$7

with biases

$\zeta_j$8

This mechanism gives the basis evaluation an explicit content-adaptive spatial component (Penkin et al., 16 Sep 2025).

Two concrete architectures are reported. For MRI enhancement, the input image is embedded by a $\zeta_j$9 convolution into 16 channels, lifted by a $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$0 convolution with ReLU pre-activation to 32 channels, processed by three FunKAN blocks in 32-dimensional feature space with skip connections and $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$1, then projected by a $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$2 convolution back to 16 channels and restored by a $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$3 convolution to the output image (Penkin et al., 16 Sep 2025). For segmentation, U-FunKAN inserts three FunKAN blocks at the bottleneck of a U-shaped encoder-decoder. The encoder uses four U-Net-like residual blocks with channels $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$4, each downsampling by strided $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$5 convolution; the decoder uses four residual blocks with channels $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$6, each with nearest-neighbor upsampling, $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$7 convolution, and skip fusion from the corresponding encoder feature (Penkin et al., 16 Sep 2025).

Interpretability is argued structurally rather than post hoc. The inspectable objects are the Hermite-basis coefficients $$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$8, the basis functions themselves, and the theorem-motivated decomposition into functional components (Penkin et al., 16 Sep 2025). The reported work does not include a dedicated interpretability experiment or coefficient-analysis study, so the interpretability claim remains architectural rather than empirically validated in the same way as symbolic or sparse KAN variants (Penkin et al., 16 Sep 2025, Poole et al., 24 Sep 2025).

4. Optimization protocol and datasets

All reported experiments use Python 3.12, PyTorch 2.5, PyTorch Lightning 2.5.1, CUDA 11.8, cuDNN 9, and an NVIDIA RTX A6000 GPU, with full precision training (Penkin et al., 16 Sep 2025). Optimization uses Adam with

$$KAN(\mathbf{x}) = (\Phi_{L} \circ \Phi_{L-1} \circ ... \circ \Phi_{1})(\mathbf{x}),$$9

and manual learning-rate scheduling across

$x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$0

Models are trained from scratch until convergence. For MRI enhancement, augmentation includes Gaussian noise with $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$1. For segmentation, augmentation includes random vertical and horizontal flips, rotations, and transpositions, each with probability $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$2 (Penkin et al., 16 Sep 2025).

The enhancement benchmark uses the IXI dataset. Approximately 600 high-resolution MRI scans were considered, intersecting modalities to obtain 577 volumes with T1, T2, and PD. The first 400 volumes are used for training, the next 100 for testing, and the remainder for validation. After discarding 25 slices at both ends of each volume and sampling every tenth slice, the final dataset contains 10,427 training pairs, 2,016 validation pairs, and 2,617 test pairs (Penkin et al., 16 Sep 2025). The corruption protocol loads a $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$3 image, applies a Fourier transform, crops the central 25% of frequency space without zero-padding, and then applies the inverse Fourier transform, producing a $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$4 Gibbs-corrupted image (Penkin et al., 16 Sep 2025).

The segmentation benchmarks are BUSI, GlaS, and CVC-ClinicDB. BUSI contains 780 grayscale breast ultrasound images from 600 female patients, of which 647 benign and malignant images are used after resizing to $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$5. GlaS contains 165 RGB histology images, resized to $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$6, with a random 80/20 train-test split using seed 42. CVC-ClinicDB contains 612 colonoscopy RGB frames from 29 videos, resized to $x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$7 (Penkin et al., 16 Sep 2025). Segmentation is evaluated by IoU and F1, averaged over the last 50 epochs and over three runs with seeds 50, 100, and 150 (Penkin et al., 16 Sep 2025).

The enhancement loss is mean squared error,

$x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$8

while segmentation uses

$x_{l+1,j} = \sum_{i=1}^{n} \phi_{l,ji}(x_{l,i}),$9

with batch size $x_{l,i}$0 in both cases (Penkin et al., 16 Sep 2025).

5. Empirical results

On IXI, FunKAN is compared with corrupted input, Kellner deringing, and learned MLP, KAN, ChebyKAN, and HermiteKAN backbones within the same convolutional framework (Penkin et al., 16 Sep 2025). The reported image-enhancement results are:

Method	PSNR	TV
Corrupted input $x_{l,i}$1	31.33	1476.55
Kellner	31.09	1120.05
MLP	37.96	1145.57
KAN	38.10	1161.63
ChebyKAN	38.01	1156.56
HermiteKAN	38.04	1161.31
FunKAN	39.05	1174.86

The reference clean images have $x_{l,i}$2 (Penkin et al., 16 Sep 2025). FunKAN therefore improves PSNR by about 1 dB over the KAN-type learned baselines in this setting, while remaining below the clean-image TV target (Penkin et al., 16 Sep 2025). The reported computational tradeoff is substantial: the MLP backbone uses $x_{l,i}$3 Gflops and $x_{l,i}$4M parameters, standard KAN-type backbones use $x_{l,i}$5 Gflops and $x_{l,i}$6–$x_{l,i}$7M parameters, whereas FunKAN uses $x_{l,i}$8 Gflops and $x_{l,i}$9M parameters (Penkin et al., 16 Sep 2025).

For segmentation, U-FunKAN is compared with U-Net, Attention U-Net, U-Net++, U-NeXt, Rolling-UNet, U-Mamba, UKAGNet, and U-KAN (Penkin et al., 16 Sep 2025). Its reported mean $\chi_{l,i}$0 standard deviation metrics are:

Dataset	U-FunKAN IoU	U-FunKAN F1
BUSI	$\chi_{l,i}$1	$\chi_{l,i}$2
GlaS	$\chi_{l,i}$3	$\chi_{l,i}$4
CVC-ClinicDB	$\chi_{l,i}$5	$\chi_{l,i}$6

The reported claim is that U-FunKAN achieves the best IoU on all three datasets (Penkin et al., 16 Sep 2025). The F1 picture is more qualified: it is best on GlaS, but not on BUSI, where UKAGNet reports $\chi_{l,i}$7, or on CVC-ClinicDB, where U-KAN reports $\chi_{l,i}$8 (Penkin et al., 16 Sep 2025). This makes the segmentation result strong but not metric-wise uniform.

Efficiency results are also reported for the segmentation setting. U-FunKAN uses $\chi_{l,i}$9 Gflops and $H$0M parameters, compared with U-KAN’s $H$1 Gflops and $H$2M parameters (Penkin et al., 16 Sep 2025). Among the listed models, U-FunKAN is the most efficient in Gflops, slightly below U-NeXt’s $H$3 Gflops and far below U-Mamba’s $H$4 Gflops (Penkin et al., 16 Sep 2025).

The ablation study varies channel widths on BUSI. The setting $H$5 yields IoU $H$6, F1 $H$7, $H$8 Gflops, and $H$9M parameters, while larger settings improve performance only modestly at much higher cost, culminating in IoU $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$0, F1 $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$1, $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$2 Gflops, and $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$3M parameters for $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$4 channels (Penkin et al., 16 Sep 2025). The chosen default is therefore the smallest configuration as the preferred efficiency-performance tradeoff (Penkin et al., 16 Sep 2025).

6. Relation to the broader KAN literature, limitations, and open directions

FunKAN belongs to a broader research program in which KANs are treated as architectures whose central trainable objects are functions, but its contribution is specific: it generalizes that principle from scalar-coordinate mappings to feature-map functionals for image processing (Sohail, 2024, Penkin et al., 16 Sep 2025). Other recent directions illuminate what is distinctive about this move. P1-KAN replaces spline edge functions by a $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$5 finite-element basis and shows that basis choice strongly affects behavior on smooth versus irregular targets (Warin, 2024). Free-RBF-KAN shows that adaptive centers and widths in radial-basis edge functions improve multiscale regression, physics-informed learning, and DeepONet trunk design (Chiu et al., 12 Jan 2026). DualFlexKAN generalizes KAN into a dual-stage architecture with independently configurable pre-linear and post-linear function modules, emphasizing the placement and sharing of learned function operators (Ortiz et al., 9 Mar 2026). P-KAN explicitly reframes KAN training as discovery of compact functional representation spaces, using entropy minimization to move spline edges toward Fourier, Chebyshev, or Bessel representations (Poole et al., 24 Sep 2025). FS-KAN makes symmetry act on functions rather than scalar parameters by tying learned univariate functions along group orbits (Elbaz et al., 29 Sep 2025). KKAN reinterprets KAN through a two-block approximate Kolmogorov-Arnold decomposition and demonstrates operator-learning compatibility in DeepONet-like settings (Toscano et al., 2024).

Within this landscape, FunKAN’s main novelty is not a new spline basis, new optimizer, or symmetry construction, but the claim that KAN-like interpretability can be reformulated at the level of feature-map functionals in a Hilbert space (Penkin et al., 16 Sep 2025). That move makes it especially relevant to image-to-image problems, whereas most earlier KAN variants remain fundamentally scalar-coordinate architectures even when used inside more structured pipelines.

The limitations are equally important. The functional Kolmogorov-Arnold theorem used to justify the architecture is explicitly hypothetical and unproven (Penkin et al., 16 Sep 2025). Some expressions after the Riesz-representation step are under-specified or notationally inconsistent, and the method section contains equations whose intended meaning must be inferred from context (Penkin et al., 16 Sep 2025). The work does not provide explicit formulas for Hermite functions, does not deeply analyze the learned coefficient matrices or deformation fields, and does not ablate the number of Hermite basis functions, the deformation module, or alternative basis families within FunKAN itself (Penkin et al., 16 Sep 2025). In enhancement, the method is much heavier than scalar KAN backbones (Penkin et al., 16 Sep 2025). The reported applications are limited to 2D medical enhancement and binary segmentation; there are no experiments on multiclass segmentation, 3D volumetric segmentation, detection, registration, or direct reconstruction from raw $$f(\chi_{1}, ..., \chi_{n}) \leadsto \sum_{j} \zeta_{j} \left(\sum_{i} \varphi_{ji}(\chi_{i})\right),$$6-space (Penkin et al., 16 Sep 2025).

A broader theoretical caution also remains. The Kolmogorov-Arnold theorem has long been interpreted too literally as a shallow constructive template, whereas later analysis argues that practical realizations more naturally require deep interior encoding and careful regularity control (Schmidt-Hieber, 2020). FunKAN should therefore be understood less as a completed function-space theorem and more as a concrete architectural proposal: a KAN-inspired, basis-driven, spatially structured image backbone whose empirical strength lies in preserving spatial organization while keeping a functional decomposition visible at the level of learned Hermite coefficients and grid deformations (Penkin et al., 16 Sep 2025).