KANalogue: KAN Analogues in Diverse Domains

• KANalogue is a family of architectures that reformulate KANs' additive composition via alternative univariate basis functions, enabling applications in quantum, hardware, and operator learning.
• The approach replaces traditional spline functions with alternatives like single-qubit data re-uploading, Fourier series, and analog hardware devices to optimize performance and interpretability.
• KANalogue’s versatile design fosters rigorous interpretability and efficiency in areas such as device modeling, reinforcement learning, and symbolic extraction, paving the way for adaptive, domain-specific strategies.

Searching arXiv for papers on KANalogue and related KAN analogues. KANalogue denotes a family of architectures and implementation paradigms that preserve the Kolmogorov–Arnold Network (KAN) principle—multivariate modeling through compositions and sums of learnable univariate functions—while replacing the original spline-on-edge realization, deployment substrate, or operator-level interpretation with an alternative mechanism. Across recent work, the term spans quantum-inspired edge functions, physics-guided surrogates, reinforcement-learning function approximators, compact transistor models, LUT-native FPGA flows, physical analog hardware, operator-learning frameworks, and interpretability tools. In all of these settings, the defining motif is the retention of KAN’s additive composition of univariate functions together with a change in basis, medium, or modeling role (Sharma et al., 9 Oct 2025).

1. Foundational meaning of a KAN analogue

Classical KANs are built on the Kolmogorov–Arnold representation theorem, stated in one form as

$$f(x_1,x_2, \dots , x_n) = \sum_{q=1}^{2n+1} \Phi_q \left( \sum_{p=1}^{n} \phi_{q,p}(x_p) \right),$$

with univariate inner functions $\phi_{q,p}$ and outer univariate functions $\Phi_q$ (Sharma et al., 9 Oct 2025). In KANs, each edge carries a trainable univariate function rather than a scalar weight, and in the original formulation these functions are represented as B-splines defined over knots with trainable coefficients (Sharma et al., 9 Oct 2025). A KAN layer is correspondingly written as

$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$

or equivalently as a sum of edge-wise univariate transforms (Guo et al., 21 Nov 2025, Kich et al., 2024).

Within this framework, a KANalogue is a system that keeps the KAN compositional pattern while changing the univariate basis, the physical implementation, or the functional role of the architecture. The most direct version is QuIRK, where spline edge functions are replaced by single-qubit data re-uploading circuits while the outer additive structure is preserved exactly (Sharma et al., 9 Oct 2025). In operator learning, KANO extends the same philosophy from finite-dimensional maps to operators between function spaces by parameterizing pseudo-differential symbols and nonlinearities with small KANs (Lee et al., 20 Sep 2025). In hardware-oriented settings, KANELÉ turns spline-defined edge functions into LUT-native FPGA circuits (Hoang et al., 14 Dec 2025), while physical analog KANs use reconfigurable nonlinear-processing units or negative differential resistance devices as in-materia realizations of learned univariate functions (Escudero et al., 7 Feb 2026, Li et al., 24 Oct 2025).

This suggests a useful distinction between a KAN and a KANalogue. A KAN is the original spline-based architecture; a KANalogue is any faithful reformulation that preserves the edge-function paradigm and additive node structure while altering either the basis family, the computational substrate, or the operator domain. That interpretation is explicit in several papers, although the exact scope varies by application (Sharma et al., 9 Oct 2025, Guo et al., 21 Nov 2025, Lee et al., 20 Sep 2025).

2. Replacement of spline edge functions by alternative univariate mechanisms

The clearest KANalogue mechanism is basis substitution at the edge level. QuIRK replaces each spline-based univariate function $\phi(x)$ with a single-qubit data re-uploading model, yielding a layer output

$$h^{(l)}_{u} = \sum_{i=1}^{|I|} DR\big(x_i;\theta^{(l)}_{i,u}\big),$$

which is structurally identical to a KAN layer except that each edge output is the expectation value of a one-qubit circuit (Sharma et al., 9 Oct 2025). The paper gives the resulting univariate function as

$$\phi_\theta(x) := \langle \psi_L(x;\theta) | Z | \psi_L(x;\theta) \rangle,$$

with values in $[-1,1]$ and a rescaling layer to map intermediate sums back into the valid encoding range $[0,\pi]$ (Sharma et al., 9 Oct 2025). The stated motivation is that such DR edge functions are univariate universal approximators, very parameter-efficient, naturally periodic, and classically simulable with simple $2\times 2$ matrix multiplications (Sharma et al., 9 Oct 2025).

Other basis substitutions appear in several domains. Fourier KANs replace B-splines with Fourier series,

$\phi_{q,p}$0

to model transistor current and charge functions (Novkin et al., 19 Mar 2025). InfinityKAN generalizes the edge-function basis viewpoint by treating the number of basis functions as a variational latent variable, starting from an infinite expansion

$\phi_{q,p}$1

and learning an effective truncation order during training (Alesiani et al., 3 Jul 2025). In audio implicit neural representations, KAN edge functions combine a SiLU branch with a spline branch,

$\phi_{q,p}$2

and are used as learned bases for waveform reconstruction (Marszałek et al., 4 Mar 2025).

The same idea also appears in quantum-native proposals. Quantum Kolmogorov–Arnold networks replace learnable univariate edge functions with quantum signal processing polynomials $\phi_{q,p}$3, realized as entries of QSP unitaries and combined through block-diagonal composition and selection gates (Daskin, 2024). Adaptive VQKAN uses spline-parameterized univariate angle functions inside Pauli-string exponentials, so that a classical KAN-like edge function becomes a quantum rotation-angle function (Wakaura et al., 27 Mar 2025).

A plausible implication is that “KANalogue” increasingly denotes not a single architecture but a design pattern: preserve additive composition and move expressive capacity into low-dimensional edge functions whose basis is chosen to match the domain—periodic rotations for periodic functions, Fourier terms for spectral structure, or device I–V characteristics for analog hardware.

3. KANalogue as a physics-guided or domain-specific modeling paradigm

A second major meaning of KANalogue is the use of KANs as interpretable analogues of classical scientific or engineering models. In radiomap estimation, RadioKMoE uses a KAN module as a coarse propagation model and explicitly frames it as a “KANalogue” of analytical propagation: the KAN acts as a data-driven surrogate for large-scale radio propagation patterns, analogous to basis expansion models and path-loss curves (Guo et al., 21 Nov 2025). The KAN stage predicts a coarse coverage map from features such as position, frequency embedding, and side information, and its smooth spline-based inductive bias is reported to align with monotone decay with distance, smooth frequency dependence, and slow spatial variation (Guo et al., 21 Nov 2025).

In transistor compact modeling, KAN and Fourier KAN are used as learned analogues of BSIM-CMG-style compact models for a 7 nm FinFET (Novkin et al., 19 Mar 2025). The modeled quantities are $\phi_{q,p}$4, $\phi_{q,p}$5, and $\phi_{q,p}$6, with training losses that include first and second derivatives such as $\phi_{q,p}$7, $\phi_{q,p}$8, $\phi_{q,p}$9, and $\Phi_q$0 (Novkin et al., 19 Mar 2025). The paper states that KAN and FKAN achieve superior prediction accuracy for critical figures of merit including gate current, drain charge, and source charge, and it emphasizes symbolic extraction as a route toward interpretable compact equations (Novkin et al., 19 Mar 2025). This use of KANalogue is explicitly tied to analog/RF design: a learned, interpretable, data-driven analogue of compact device equations.

A related but more hardware-physical interpretation appears in “Bridging Function Approximation and Device Physics via Negative Differential Resistance Networks,” where KANalogue is the use of negative differential resistance tunnel diodes as physical realizations of KAN basis functions (Li et al., 24 Oct 2025). There, each device I–V curve becomes a univariate basis function $\Phi_q$1, and KAN layers are implemented as linear combinations

$\Phi_q$2

with the basis shapes derived from simulated device physics rather than generic splines (Li et al., 24 Oct 2025). The work positions this as a bridge between function approximation theory and device-level physics.

KANO for image super-resolution uses another domain-specific form of KANalogue: a blind SR operator whose latent degradation fitting process is represented by KAN-style spline operators inside an unrolled optimization framework (Li et al., 28 Dec 2025). The model decomposes the image as $\Phi_q$3, uses a blur kernel $\Phi_q$4, and learns KAN-based proximal operators $\Phi_q$5-Net and $\Phi_q$6-Net for kernel and spectral updates (Li et al., 28 Dec 2025). The paper argues that B-spline activations accurately capture local linear trends and peak-valley structures at nonlinear inflection points in spectral curves, giving the SR process physical interpretability (Li et al., 28 Dec 2025).

These examples show that KANalogue often signifies a learned replacement for an analytical model class: path-loss models, compact transistor equations, degradation operators, or device response functions. The shared claim is not merely approximation accuracy, but interpretability through explicit univariate components.

4. Quantum-inspired, quantum-native, and operator-learning KANalogues

Quantum-related KANalogues span three distinct directions. QuIRK is quantum-inspired but classically tractable: it uses single-qubit data re-uploading circuits per edge, simulated as unentangled one-qubit circuits with tensor-network treewidth $\Phi_q$7, and accelerated with cuTensorNet on GPUs (Sharma et al., 9 Oct 2025). The paper reports that QuIRK often matches or beats classical KAN in RMSE with significantly fewer parameters on Feynman symbolic regression tasks, and that its advantage is “especially apparent in the case of periodic functions” (Sharma et al., 9 Oct 2025).

QKAN is quantum-native. It maps each learnable univariate function to a QSP polynomial $\Phi_q$8, builds a block-diagonal quantum layer

$\Phi_q$9

and forms additive combinations through selection gates (Daskin, 2024). It is explicitly presented as an implementation of KAN on quantum computers by combining quantum signal processing circuits in layers (Daskin, 2024). The paper is conceptual rather than empirical: it proposes the correspondence between KAN edge functions and QSP polynomials, and discusses stacking layers through “qubitization of polynomial terms,” but does not present numerical benchmarks (Daskin, 2024).

Adaptive VQKAN is a variational quantum analogue of KAN that uses spline-parameterized angle functions and adaptive ansatz growth inspired by Adaptive VQE (Wakaura et al., 27 Mar 2025). For a 4D fitting problem, it is reported to obtain a minimum sum of absolute distances of about 12.74 versus 14.98 for the QNN baseline, while using far fewer parametric gates (Wakaura et al., 27 Mar 2025). By contrast, on a 2D classification task and a Fourier heat-equation problem, the adaptive VQKAN is described as less successful, with poorer classification performance and lower PDE accuracy than the QNN in some settings (Wakaura et al., 27 Mar 2025).

Operator learning introduces a different but closely related KANalogue. The operator-learning KANO paper argues that FNO suffers from a “pure-spectral bottleneck” on position-dependent dynamics and that KANO avoids this by parameterizing pseudo-differential symbols $$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$0 and pointwise nonlinearities with small KANs (Lee et al., 20 Sep 2025). A KANO layer is written as

$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$1

with $$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$2 the Kohn–Nirenberg quantization of the symbol (Lee et al., 20 Sep 2025). The paper states that KANO remains expressive over generic position-dependent dynamics, whereas FNO remains practical only for spectrally sparse operators with fast-decaying Fourier tails (Lee et al., 20 Sep 2025). On synthetic operator-learning benchmarks, KANO robustly generalizes to unseen function families while FNO degrades sharply, and in quantum Hamiltonian learning KANO reconstructs symbolic Hamiltonians with coefficients accurate to the fourth decimal place and reaches approximately $$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$3 state infidelity from projective measurement data (Lee et al., 20 Sep 2025).

Across these works, the common structure is the same: retain KAN’s edge-function philosophy, but reinterpret the edge functions as DR expectations, QSP polynomials, adaptive quantum angle functions, or pseudo-differential symbols.

5. Hardware and physical analog realizations

Hardware KANalogues form a large and increasingly specialized subfield. KANELÉ is a LUT-based FPGA deployment framework that exploits the fixed-domain spline structure of KANs (Hoang et al., 14 Dec 2025). It performs quantization-aware training, norm-based structured pruning, and then compiles each surviving edge function into a discrete lookup table. The resulting hardware consists of “LUTs + adders” with no DSPs and no BRAM in the reported designs (Hoang et al., 14 Dec 2025). The paper reports up to a 2700x speedup and over 4000x resource reduction relative to prior KAN-on-FPGA approaches on benchmark tasks, while matching or surpassing other LUT-based architectures particularly on symbolic or physical-formula tasks (Hoang et al., 14 Dec 2025). It also reports an 8-bit KAN HalfCheetah policy implemented at 884 MHz with 4.5 ns latency and 1136 LUT on xczu7ev FPGA (Hoang et al., 14 Dec 2025).

A different hardware path is the large-scale RRAM-ACIM acceleration of KANs (Huang et al., 7 Sep 2025). That work addresses the cost of B-spline evaluation through Alignment-Symmetry and PowerGap hardware-aware quantization, a KAN sparsity-aware mapping strategy, and N:1 Time Modulation Dynamic Voltage input generation for analog-compute-in-memory circuits (Huang et al., 7 Sep 2025). The paper states that for large-scale KAN networks, despite parameter counts increasing by 500Kx to 807Kx compared to tiny-scale tasks, area overhead increases by only 28Kx to 41Kx and power by 51x to 94x, with accuracy degradation only 0.11% to 0.23% (Huang et al., 7 Sep 2025). This is explicitly framed as algorithm–hardware co-design for scalable KAN acceleration.

Physical analog KANs go further by implementing edge nonlinearities directly in hardware. The RNPU-based analog KAN architecture realizes edge functions with reconfigurable nonlinear-processing units—multi-terminal nanoscale silicon devices whose I–V characteristics are tuned by control voltages (Escudero et al., 7 Feb 2026). The paper reports system-level estimates of approximately 250 pJ per inference and approximately 600 ns end-to-end latency for a representative workload, corresponding to a $$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$4–$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$5 reduction in energy and approximately $$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$6 reduction in area compared to a digital fixed-point MLP at similar approximation error (Escudero et al., 7 Feb 2026). It further reports aKAN classification accuracies of 99.5% on noisy Moons and 99.75% on a one-turn spiral task, with smaller parameter counts than corresponding MLPs (Escudero et al., 7 Feb 2026).

The negative-differential-resistance KANalogue work offers a related but distinct device-physics realization (Li et al., 24 Oct 2025). Here, tunnel-diode I–V curves serve as basis functions extracted from NbSi$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$7N$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$8/HfSi$$x^{(\ell+1)}_j \;=\; \sum_{i=1}^{n_\ell} \varphi^{(\ell)}_{j,i}\!\big(x^{(\ell)}_i\big),$$9N$\phi(x)$0 heterostructures, and KANalogue networks are trained on MNIST, Fashion-MNIST, and CIFAR-10 using these device-inspired bases (Li et al., 24 Oct 2025). On MNIST, a 2-basis KANalogue reaches 97.71% accuracy with 26,570 parameters versus 97.78% for a B-spline KAN with 92,288 parameters (Li et al., 24 Oct 2025). The paper uses this to argue that device-level nonlinearities can serve as hardware-native basis functions.

These hardware papers share a specific interpretation of KANalogue: a route to implementing univariate edge functions more efficiently or more natively than dense matrix–vector networks with fixed activations.

6. Interpretability, symbolic extraction, and analytical insight

Interpretability is central to nearly all KANalogue formulations. QuIRK explicitly retains the KAN mechanism of post-hoc closed-form reconstruction by sampling each learned univariate edge function and fitting it to a simple symbolic form (Sharma et al., 9 Oct 2025). In an example approximating $\phi(x)$1, the DR edge functions are fit by 1D polynomials and recombined into a simplified symbolic expression (Sharma et al., 9 Oct 2025).

In transistor compact modeling, iterative symbolic regression is introduced to transform trained KANs into symbolic equations with much better accuracy than naive post-fit extraction (Novkin et al., 19 Mar 2025). For a KAN configuration with 18 edge functions, fixing three activations per iteration and retraining yields fully symbolic models with approximately 1.8% MAPE for $\phi(x)$2 and approximately 0.96% MAPE for $\phi(x)$3, compared with approximately 41% and 7.8% MAPE for direct symbolic regression without iterative retraining (Novkin et al., 19 Mar 2025). The extracted formulas involve trigonometric, polynomial, and hyperbolic components and can, in principle, be embedded into Verilog-A or C (Novkin et al., 19 Mar 2025).

The operator-learning KANO paper pushes interpretability further by recovering symbolic PDE operators and quantum Hamiltonians directly from trained KAN edge functions (Lee et al., 20 Sep 2025). For synthetic operators such as

$\phi(x)$4

the symbolicized KANO reconstructs the operator with tiny coefficient errors such as

$\phi(x)$5

and similarly recovers quartic double-well Hamiltonians with coefficients accurate to the fourth decimal place (Lee et al., 20 Sep 2025).

Interpretability also appears in a matrix-analysis form. KAN-Matrix introduces the Pairwise KAN Matrix (PKAN) and Multivariate KAN Contribution Matrix (MKAN) as visual KAN-based analogues of correlation and feature-importance matrices (Fuente et al., 12 Dec 2025). PKAN fits univariate KAN models $\phi(x)$6 and displays both a strength score and the learned function in each cell; MKAN fits additive multivariate KAN models

$\phi(x)$7

and uses KAN 2.0 attribute scores to quantify contributions (Fuente et al., 12 Dec 2025). The paper reports that PKAN and MKAN yield more robust and informative results than Pearson correlation and Mutual Information in synthetic examples, CAMELS hydrology, and CFD analysis (Fuente et al., 12 Dec 2025). In CAMELS, MKAN-based ranking enables a random forest to reach $\phi(x)$8 skill 0.768 with the top 2 attributes versus 0.620 for Pearson and 0.652 for MI, and it maintains an advantage across top-4 through top-14 selections (Fuente et al., 12 Dec 2025).

A broad pattern emerges across these works: KANalogues are often justified not merely by predictive performance but by their capacity to expose learned 1D functions that can be plotted, ranked, simplified, or converted to closed-form formulas.

7. Performance patterns, limitations, and directions of development

Across applications, several recurring empirical patterns appear. QuIRK often matches or beats B-spline KAN on Feynman equations using factors of 2–10 fewer parameters, though some classical KAN instances still reach lower RMSE at higher parameter counts (Sharma et al., 9 Oct 2025). In online reinforcement learning with PPO, a KAN actor plus MLP critic reaches the highest mean reward, 1540, versus 1482 for the standard MLP baseline, while a full KAN actor–critic reaches 1358 mean reward with only 415 parameters versus 10,490 for the standard MLP (Kich et al., 2024). At the same time, that work reports a roughly $\phi(x)$9 slower runtime per step for KAN than MLP on HalfCheetah, reflecting the cost of spline evaluation in current software implementations (Kich et al., 2024).

Not all KANalogues dominate conventional models uniformly. In transistor compact modeling, KAN achieves the lowest fitting error for current and charge, but higher-order derivatives exhibit oscillatory behavior that degrades multi-stage circuit simulation, especially in a 25-stage ring oscillator (Novkin et al., 19 Mar 2025). In RadioKMoE, the KAN prior alone can be slightly worse than the backbone on BRAT-LabW ablation, and the main gain comes from the combination of KAN prior, MoE refinement, and depth map (Guo et al., 21 Nov 2025). In audio INRs, KAN achieves the best perceptual metrics such as LSD 1.29 and PESQ 3.57 for 1.5 s audio, yet in the FewSound hypernetwork setting a NeRF target slightly outperforms a KAN target on several metrics, though KAN still improves HyperSound when used as its target INR (Marszałek et al., 4 Mar 2025).

Several technical limitations recur. Many KAN-based methods require careful control of basis complexity, spline order, grid size, or equivalent edge-function hyperparameters (Alesiani et al., 3 Jul 2025, Novkin et al., 19 Mar 2025, Marszałek et al., 4 Mar 2025). Hardware implementations often depend on bounded activation domains and quantization-aware training (Hoang et al., 14 Dec 2025, Huang et al., 7 Sep 2025). Quantum-native KANalogues remain mostly small-scale or conceptual, with limited noise analysis and little large-scale empirical validation (Daskin, 2024, Wakaura et al., 27 Mar 2025). Physical analog KANs face mixed-signal overhead, calibration, and variability challenges despite strong energy projections (Escudero et al., 7 Feb 2026). Device-physics KANalogues based on NDR curves remain less competitive on more complex vision tasks such as CIFAR-10, where fixed device I–V libraries may be insufficiently expressive (Li et al., 24 Oct 2025).

The current trajectory nonetheless points toward three converging directions. One is adaptive complexity, exemplified by InfinityKAN’s variational treatment of basis count (Alesiani et al., 3 Jul 2025). A second is operator-level generalization, where KAN-based symbol parameterizations appear to offer a tractable alternative to pure-spectral neural operators on position-dependent dynamics (Lee et al., 20 Sep 2025, Li et al., 28 Dec 2025). A third is hardware-native deployment, where LUT discretization, analog nonlinear devices, or mixed-signal primitives make the edge-function paradigm physically realizable (Hoang et al., 14 Dec 2025, Escudero et al., 7 Feb 2026, Li et al., 24 Oct 2025, Huang et al., 7 Sep 2025).

In that broader sense, KANalogue is not a single model but an organizing concept for KAN-derived systems: preserve learnable univariate functions and additive composition, then tailor the basis, operator calculus, or hardware embodiment to the structure of the problem.