Small-scale photonic Kolmogorov-Arnold networks using standard telecom nonlinear modules

Luca Nogueira Calçado; Sergei K. Turitsyn; Egor Manuylovich

physics.opticsApr 9, 2026

Small-scale photonic Kolmogorov-Arnold networks using standard telecom nonlinear modules

Authors: Luca Nogueira Calçado, Sergei K. Turitsyn, Egor Manuylovich

Organizations: Aston Institute of Photonic Technologies (AiPT), Aston University, Birmingham, B4 7ET, United Kingdom.

Abstract

Photonic neural networks promise ultrafast inference, yet most architectures rely on linear optical meshes with electronic nonlinearities, reintroducing optical-electrical-optical bottlenecks. Here we introduce small-scale photonic Kolmogorov-Arnold networks (SSP-KANs) implemented entirely with standard telecommunications components. Each network edge employs a trainable nonlinear module composed of a Mach-Zehnder interferometer, semiconductor optical amplifier, and variable optical attenuators, providing a four-parameter transfer function derived from gain saturation and interferometric mixing. Despite the constrained functional form of these optical nonlinearities, SSP-KANs comprising only a few optical modules achieve strong nonlinear inference performance across classification, regression, and image recognition tasks, approaching software baselines with significantly fewer parameters. A four-module network achieves $94.3$ % (IQR: $90.3$ -- $97.4$ %, 10~seeds) accuracy on nonlinear classification benchmarks; a seven-module network attains $R^2 = 0.986 \pm 0.015$ on six-input regression. Performance remains robust under realistic hardware impairments, maintaining high accuracy down to 6-bit input resolution and 14 dB signal-to-noise ratio. By using a fully differentiable physics model for end-to-end optimisation of optical parameters, this work establishes a practical pathway from simulation to experimental demonstration of photonic KANs using commodity telecom hardware.

Figures & tables

Figure 1: SSP-KAN [2,2] network. Two input features are encoded as optical power and split to four MZI-VOA-SOA-VOA modules on a fibre-coupled breadboard. Each module (detail, lower left) has four trainable parameters: input attenuation $\alpha_{1}$ , injection current $I$ , output attenuation $\alpha_{2}$ , and interferometric phase $\phi$ . Insets show the learned transfer function $P_{\text{out}}(P_{\text{in}})$ of each module. The per-edge outputs are summed on two photodetectors (PD 1 , PD 2 ), producing the nonlinear decision boundary on the concentric rings task (right).

Figure 2: Two Moons classification with SSP-KAN [2,2]. (a) Learned decision boundary ( $94.3$ % (IQR: $90.3$ – $97.4$ %), 10 seeds; test accuracy, 16 parameters). The network separates the two crescents using a smooth nonlinear boundary generated by four MZI-VOA-SOA-VOA modules. (b) Test accuracy and parameter count across models. The linear baseline (89.2%, 6 parameters) confirms the task is intrinsically nonlinear. pyKAN ( $G{=}1$ , 40 parameters) reaches 99.9%; SSP-KAN achieves $94.3$ % with 16 parameters using physically realisable optical activation functions. Inset: [2,2] network schematic showing 2 inputs, 4 modules, and 2 outputs.

Figure 3: Hardware robustness of SSP-KAN [2,2] on Two Moons. (a) Test accuracy as a function of input DAC resolution (12-bit to 4-bit) and optical noise (SNR 30 dB to 6 dB). The smooth colour gradient confirms that quantization has negligible impact, while performance degrades vertically with decreasing SNR. The 90% and 95% contour lines indicate the practical operating boundary. Inset: [2,2] network schematic (16 parameters). (b) Decision boundaries at four corner conditions. Top row: 30 dB noise at 12-bit (94.4%) and 4-bit (94.7%) resolution. Bottom row: 6 dB noise at 12-bit (86.8%) and 4-bit (86.8%). The boundary shape is preserved across quantization levels but simplifies under severe noise.

Figure 4: Regression performance on yacht hydrodynamics. Test

R^{2}

scores (mean

\pm

std, 10 seeds) and parameter counts for linear regression, MLP, software KAN (pyKAN,

G{=}1

), and SSP-KAN architectures [6,1] and [6,1,1]. SSP-KAN [6,1,1] exceeds pyKAN (

R^{2}=0.977

) with 28 parameters versus 60, demonstrating that compositional depth compensates for the constrained per-edge activation family.

Figure 5: Hardware robustness of SSP-KAN on yacht hydrodynamics regression. Test $R^{2}$ as a function of input DAC resolution (12-bit to 4-bit) and optical noise (SNR 30 dB to 6 dB) for (a) the single-layer [6,1] architecture (6 modules, 24 parameters) and (b) the two-layer [6,1,1] architecture (7 modules, 28 parameters). Contour lines mark $R^{2}$ thresholds. The [6,1] panel shows a compressed performance range ( $R^{2}\approx 0.86$ – $0.92$ ), while [6,1,1] spans a wider range ( $R^{2}\approx 0.82$ – $0.99$ ), reflecting both higher peak performance and greater noise sensitivity.

Figure 6: Image classification results. Training curves showing accuracy and loss for (a) MNIST and (b) Fashion-MNIST using the [784,20,10] architecture. Insets show the final training epochs at expanded scale. Confusion matrices for the [784,20,10] model on (c) MNIST and (d) Fashion-MNIST, showing that classification errors concentrate among visually similar classes.

Figure 7: Distinct roles of the two VOAs in the MZI-VOA-SOA-VOA module transfer function. (a) Sweeping VOA 1 changes the SOA operating point and saturation strength, producing pronounced changes in the nonlinearity shape. (b) Sweeping VOA 2 primarily rescales the SOA-arm contribution at recombination, mainly adjusting interference contrast and overall scaling. Fixed parameters: $I=1200$ mA, $\phi=\pi/2$ , $P_{\mathrm{sat}}=18$ dBm, $\alpha_{H}=5$ , with $h_{0}(I)$ calibrated from $G_{\max}=35$ dB at $I_{\max}=1700$ mA and $I_{\mathrm{tr}}=600$ mA. In (a) VOA 2 is fixed at $A_{2}=10$ dB; in (b) VOA 1 is fixed at $A_{1}=10$ dB.

Dataset	Samples	Input dim.	Task	Split	Power range (mW)
Two Moons	2,000	2	Classification	50/50	10–80
Yacht Hydro.	308	6	Regression	70/15/15	10–80
MNIST	70,000	784	Classification	72/8/20	1–10
Fashion-MNIST	70,000	784	Classification	72/8/20	1–10

Table 1: Benchmark datasets and preprocessing details.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Tasks requiring feature interaction. Left column: single-layer [2,2]; right column: two-layer [2,2,2]. Top row: XOR. The [2,2] boundary is at chance (51.8%) while [2,2,2] recovers 89.4%. Bottom row: Two Moons rotated 45°. The [2,2] network (89.0%) falls to the linear baseline because the additive structure cannot represent diagonal boundaries, while [2,2,2] reaches 99.9%.

Figure 9: Coherent versus incoherent operation on Two Moons classification (single seed). Top row: single-layer [2,2]. The incoherent regime (99.4%, 16 parameters) outperforms the coherent regime (91.8%, 20 parameters), indicating that interference at the output couplers complicates optimization without providing a net benefit at this depth. Bottom row: two-layer [2,2,2]. The coherent regime (99.9%, 40 parameters) slightly exceeds the incoherent regime (99.5%, 32 parameters), as the interference cross-terms at intermediate nodes introduce input-dependent effective weights that increase network expressivity. Multi-seed statistics for the incoherent regime are reported in the main text.

Module	$I$ (mA)	$\alpha_{1}$ (dB)	$\alpha_{2}$ (dB)	$\phi$ (rad)
$\varphi_{0,0}$	1045.26	0.54	0.28	1.4926
$\varphi_{0,1}$	1363.19	0.70	3.69	3.2287
$\varphi_{1,0}$	1136.59	3.09	0.07	1.4761
$\varphi_{1,1}$	1219.03	0.28	7.91	1.8061

Appendix

Table 2: Trained parameters for the [2,2] SSP-KAN on Two Moons (4 modules, 16 parameters; mean test accuracy

94.3

% (IQR:

90.3

–

97.4

%) across 10 seeds; values shown are for a representative seed).

Layer	Module	$I$ (mA)	$\alpha_{1}$ (dB)	$\alpha_{2}$ (dB)	$\phi$ (rad)
1	$\varphi_{0,0}$	1035.54	2.17	2.15	4.6215
1	$\varphi_{0,1}$	1358.99	0.02	0.22	3.3808
1	$\varphi_{1,0}$	1117.07	3.25	1.67	2.1629
1	$\varphi_{1,1}$	1222.74	6.14	4.19	4.6445
2	$\varphi_{0,0}$	931.23	2.53	0.25	4.5404
2	$\varphi_{0,1}$	951.79	1.97	4.62	2.5132

Appendix

Table 3: Trained parameters for the [2,2,2] SSP-KAN on Two Moons (8 modules, 32 parameters; mean test accuracy

99.1

% (IQR:

99.0

–

99.1

%) across 10 seeds; values shown are for a representative seed).

Module	$I$ (mA)	$\alpha_{1}$ (dB)	$\alpha_{2}$ (dB)	$\phi$ (rad)
$\varphi_{0,0}$	955.69	10.50	0.09	0.4920
$\varphi_{1,0}$	1169.22	15.78	1.13	3.2527
$\varphi_{2,0}$	950.79	10.64	0.11	0.7227
$\varphi_{3,0}$	1056.65	13.26	0.56	5.0791
$\varphi_{4,0}$	971.82	11.24	0.07	0.2775
$\varphi_{5,0}$	1308.35	3.07	7.73	3.2592

Appendix

Table 4: Trained parameters for the [6,1] SSP-KAN on yacht hydrodynamics (6 modules, 24 parameters; mean test

R^{2}=0.921\pm 0.017

across 10 seeds; values shown are for a representative seed).

Layer	Module	$I$ (mA)	$\alpha_{1}$ (dB)	$\alpha_{2}$ (dB)	$\phi$ (rad)
1	$\varphi_{0,0}$	960.90	6.73	3.18	0.8875
1	$\varphi_{1,0}$	1176.99	7.65	7.22	4.7838
1	$\varphi_{2,0}$	956.79	3.84	8.26	2.2538
1	$\varphi_{3,0}$	1060.58	8.77	5.24	5.9785
1	$\varphi_{4,0}$	977.85	6.16	6.10	1.1401
1	$\varphi_{5,0}$	1300.58	2.62	2.18	3.6593

Appendix

Table 5: Trained parameters for the [6,1,1] SSP-KAN on yacht hydrodynamics (7 modules, 28 parameters; mean test

R^{2}=0.986\pm 0.015

across 10 seeds; values shown are for a representative seed).

Noise model	Moons [2,2] (%)	Moons [2,2,2] (%)	Yacht [6,1,1] $R^{2}$
Baseline (30 dB)	$95.2\pm 3.6$	$98.0\pm 3.3$	$0.977\pm 0.011$
Power 14 dB	$90.6\pm 2.5$	$91.5\pm 4.0$	$0.945\pm 0.014$
Power 20 dB	$91.5\pm 3.0$	$94.8\pm 3.4$	$0.967\pm 0.012$
Power 30 dB	$92.2\pm 3.4$	$97.3\pm 3.3$	$0.975\pm 0.012$
Amplitude 14 dB	$73.9\pm 0.3$	$64.0\pm 0.5$	$0.121\pm 0.173$
Amplitude 20 dB	$84.0\pm 0.5$	$75.9\pm 0.3$	$0.462\pm 0.127$

Appendix

Table 6: Noise model comparison across architectures and tasks (10 seeds, mean

\pm

std). Both physics-based models produce performance statistically indistinguishable from the 30 dB baseline.

Figure 10: Noise residual distributions (noisy output

-

baseline output) at the output of the [2,2,2] network for four noise models (rows) and two layers (columns), pooled across three representative inputs. The physics-based models (datasheet NF and current-dependent) produce residuals with

\sigma<0.1

mW, approximately

10^{4}\times

smaller than the phenomenological models. Dashed lines show Gaussian fits.

Figure 11: SNR at each layer output through a 7-layer [2,2,…,2] cascade under current-dependent ASE noise (10 seeds, mean

\pm

std). The dashed line marks the 14 dB phenomenological threshold from the main text. SNR decreases from

{\sim}

67 dB to

{\sim}

49 dB across seven layers but remains above the threshold by more than 35 dB.

Noise model	Min var. [mW 2 ]	Max var. [mW 2 ]	Ratio	Signal dependence
Power 20 dB	$4.16\times 10^{-1}$	$5.46\times 10^{-1}$	1.3	Weak
Amplitude 20 dB	$1.70\times 10^{1}$	$2.52\times 10^{1}$	1.5	Moderate
Datasheet NF 8 dB	$1.13\times 10^{-3}$	$1.78\times 10^{-3}$	1.6	$\sigma^{2}\propto P_{\text{signal}}$
Current-dependent	$8.89\times 10^{-4}$	$1.15\times 10^{-3}$	1.3	$\sigma^{2}\propto P_{\text{signal}}$

Appendix

Table 7: Output variance range across input power for each noise model. The physics-based models produce variance approximately

10^{4}\times

smaller than the phenomenological power-domain model.

Figure 12: Output variance versus input power for four noise models. The physics-based models (bottom curves) show variance proportional to signal power, consistent with signal–ASE beat noise. Log-scale

y

-axis.

Task	Optimiser	Mean metric	Std	Best	Worst
Moons [2,2]	Adam (500 epochs)	94.2%	1.0%	95.8%	92.4%
Moons [2,2]	CMA-ES (4,000 evals)	93.8%	0.0%	93.8%	93.8%
Yacht [6,1,1]	Adam (1,000 epochs)	$R^{2}=0.881$	0.198	0.983	0.447
Yacht [6,1,1]	CMA-ES (7,200 evals)	$R^{2}=0.955$	0.048	0.986	0.879

Appendix

Table 8: CMA-ES versus Adam+backpropagation. CMA-ES achieves comparable mean performance with substantially lower variance across seeds.

Parameter	Nominal	Sampling range	Units
$G_{\text{max}}$	35.0	$\mathcal{U}[32.0,\,35.0]$	dB
$P_{\text{sat}}$	18.0	$\mathcal{U}[17.0,\,19.0]$	dBm
$I_{\text{tr}}$	600	$\mathcal{U}[\text{nom}\!\cdot\!(1{-}f),\;\text{nom}\!\cdot\!(1{+}f)]$	mA
$\alpha_{H}$	5.0	$\mathcal{U}[3.0,\,7.0]$	—

Appendix

Table 9: Device parameter sampling ranges. Nominal values correspond to the Thorlabs BOA1554P parameters used throughout the main text.

Condition	Yacht [6,1,1] $R^{2}$
Nominal	$0.965\pm 0.041$
$G_{\text{max}}$ ablation	$0.976\pm 0.019$
$P_{\text{sat}}$ ablation	$0.974\pm 0.024$
$I_{\text{tr}}\pm 5$ %	$0.986\pm 0.003$
$I_{\text{tr}}\pm 10$ %	$0.972\pm 0.028$
$\alpha_{H}$ ablation	$0.961\pm 0.047$

Appendix

Table 10: Yacht [6,1,1]

R^{2}

when training with device variability (5 seeds per condition, sigmoid reparameterisation). Device parameters are sampled per module at model initialisation; the optimiser compensates during training.

Figure 13: Sensitivity of yacht [6,1,1]

R^{2}

to single-parameter and combined device variation when deploying a nominally trained model without recalibration (250 evaluations per condition).

\alpha_{H}

, sampled over

\pm 40

% of its nominal value, dominates all other parameter axes by two orders of magnitude. Sampling ranges for the remaining parameters are listed in Supplementary Table 9 .

Figure 14: Distribution of yacht [6,1,1]

R^{2}

across 250 Monte Carlo device draws per condition, for a nominally trained model deployed without recalibration. Solid vertical lines mark the median; dashed red lines mark the 5th and 95th percentiles. The

\alpha_{H}

, combined (

\pm 5

I_{\text{tr}}

), and combined (

\pm 10

I_{\text{tr}}

) conditions produce uniformly negative

R^{2}

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Efficient Neural Network Inference Kolmogorov-Arnold Networks Neuromorphic Computing

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Kolmogorov-Arnold Networks (KANs) shift neural computation from linear layers to learnable nonlinear edge functions, but implementing these nonlinearities efficiently in hardware remains an open challenge. Here we introduce a physical analogue KAN architecture in which edge functions are realized in materia using reconfigurable nonlinear-processing units (RNPUs): multi-terminal nanoscale silicon devices whose input-output characteristics are tuned via control voltages. By combining multiple RNPUs into an edge processor and assembling these blocks into a reconfigurable analogue KAN (aKAN) architecture with integrated mixed-signal interfacing, we establish a realistic system-level hardware implementation that enables compact KAN-style regression and classification with programmable nonlinear transformations. Using experimentally calibrated RNPU models and hardware measurements, we demonstrate accurate function approximation across increasing task complexity while requiring fewer or comparable trainable parameters than multilayer perceptrons (MLPs). System-level estimates indicate an energy per inference of roughly 200 pJ and an end-to-end inference latency of roughly 0.6

μ

s for a representative workload, corresponding to over 100

\times

reduction in energy accompanied by

>

\times

reduction in area compared to a digital fixed-point MLP at similar approximation error. These results establish RNPUs as scalable, hardware-native nonlinear computing primitives and identify analogue KAN architectures as a realistic silicon-based pathway toward energy-, latency-, and footprint-efficient analogue neural-network hardware, particularly for edge inference.

Abstract

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