Abstract
Artificial intelligence is increasingly deployed in time-critical systems that must convert information into action on sub-microsecond timescales. Integrated photonics offers a route to such low-latency computation, but scalable photonic neural networks remain limited by the lack of compact nonlinear elements. Existing approaches to photonic nonlinearities often rely on optical–electrical–optical conversion that introduces latency overhead, while faster receiverless nonlinear units have primarily been explored in multilayer perceptron architectures requiring large numbers of elements. Here, we experimentally demonstrate a fully CMOS-compatible silicon photonic nonlinear unit based on a photodiode–microring modulator and use it to construct a hardware-grounded model of photonic Kolmogorov–Arnold networks. The programmable photonic nonlinear transfer functions exhibit nanosecond-scale dynamics governed by carrier recombination, with a response time of approximately 8 ns. From static and dynamic measurements, we derive a differentiable physical model and evaluate photonic Kolmogorov–Arnold networks under realistic hardware constraints, including limited photodiode headroom, merge-only routing and finite on-chip integration density. We find that these photonic networks can accurately approximate structured multidimensional functions using compact architectures comprising only a few hundred nonlinear units. These results establish a hardware-grounded route to photonic Kolmogorov–Arnold networks and identify carrier-injection based nonlinearities as a practical building block for ultrafast optical inference.
1 Introduction
As intelligent algorithms become increasingly embedded in control systems and real-time decision-making environments, computational latency poses a fundamental constraint (). In such systems, the time required to convert information into action directly determines performance. Applications including guidance, navigation, and control in autonomous UAVs (), obstacle avoidance in fast autonomous drones (), rapid fault detection in smart electrical grids (), and high-frequency financial trading () require decision times in the microsecond or sub-microsecond regime, where latency directly impacts stability, safety, and competitiveness. Despite the remarkable growth of machine learning accelerators, modern computing platforms remain constrained by the so-called memory wall (), where the disparity between processor throughput and memory bandwidth causes data movement to dominate both latency and energy consumption. Together with the breakdown of Dennard scaling and the slowing of Moore’s law (), this increasingly limits the extent to which improvements in algorithmic efficiency alone can reduce inference latency.
Integrated photonics offer an alternative paradigm well suited for high-speed analog computation (; ). Optical interference naturally performs linear transformations at the speed of light, enabling matrix–vector multiplication without the incremental energy costs associated with digital arithmetic (). Optical signals propagate continuously through photonic circuits rather than in sequential clocked steps, allowing downstream stages to respond while upstream signals are still evolving, enabling sublinear latency scaling with network depth. However, while linear operations can be efficiently realized using optical interference, implementing compact and scalable nonlinearities remains a central challenge in photonic neural networks (). As a result, many implementations rely on optical–electrical–optical (OEO) conversion, which partially negates the latency advantages of optical computation. To address this, a wide range of approaches have been explored to introduce nonlinear behavior in photonic systems, including all-optical mechanisms such as Kerr nonlinearities () and quantum-interference media (), as well as hybrid receiverless optoelectronic schemes employing photodetectors (), electro-optic modulators (), or phase-change materials (). While these approaches enable nonlinear transfer functions, they often require high optical intensities, introduce conversion overhead, or face scalability limitations in deep networks (). Among these approaches, photodiode–microring modulator (PD–MRM) nonlinear units enable compact receiverless operation, where the photodiode directly drives the micro-ring modulator without intermediate electronic amplification or signal conditioning stages, while maintaining nanosecond-scale dynamics. However, prior demonstrations have primarily employed multilayer perceptron architectures, which require large numbers of nonlinear elements to achieve reasonable expressivity, posing a significant challenge for scaling integrated photonic devices.
Kolmogorov–Arnold networks (KANs) present an intriguing alternative architecture capable of representing functions through compositions of univariate nonlinear mappings (). Instead of fixed node activations, KANs place learnable univariate nonlinear functions along network edges according to the Kolmogorov–Arnold representation theorem, enabling expressive function approximation with reduced structural redundancy. The KAN framework shows flexibility across diverse learning settings, including operator learning for mechanical problems (), convolutional KAN architectures for image processing (), and graph-based KAN variants for molecular property prediction (). In addition, the spline-based nonlinear functions of the original formulation have been replaced by alternative functional bases such as sinusoidal (), radial basis () and ReLU () functional expansions, or combinations thereof (), highlighting the adaptability of the architecture to different computational substrates. In light of their parameter-efficiency, implementing photonic KANs poses an exciting trajectory for ultra-low latency neural network inference. While recent work employing ring-assisted Mach–Zehnder interferometer (RAMZI) nonlinearities illustrates the potential of photonic KANs (), practical implementations require compact nonlinear units and network models that explicitly account for realistic hardware constraints (Figure 1).
FIGURE 1
In this work, we experimentally characterize a photodiode–microring modulator nonlinear unit fabricated on a CMOS-compatible silicon photonics platform and derive a compact physical model of its response from static and dynamic measurements, demonstrating nanosecond-scale transient operation governed by carrier recombination dynamics. Using this experimentally validated unit-cell model, we construct photonic Kolmogorov–Arnold network (PKAN) architectures that incorporate realistic hardware constraints, including limited photodiode headroom, merge-only signal routing without optical fan-out, and practical limits on the number of nonlinear units that can be integrated on a chip. With this framework, we reproduce representative benchmark examples from the KAN literature, including standard benchmark problems and a diverse set of structured analytical functions, and show that the Lorentzian-induced nonlinear basis provided by the PD-MRM unit enables accurate approximation of structured functions with compact networks. To our knowledge, this represents the first experimentally validated photonic implementation of a Kolmogorov–Arnold network using compact carrier-injection nonlinear units operating on nanosecond timescales.
2 Materials and methods
2.1 Device design
We fabricated the nonlinear optical unit investigated in this work through the Applied Micro Foundry (AMF) silicon photonics multi-project wafer platform. The device integrates a silicon–germanium PD with a carrier-injection MRM, forming a compact electro–optic structure in which photocurrent generated in the PD is directly injected into the forward-biased PN junction of the MRM. This configuration enables efficient carrier injection in the resonator without intermediate amplification, establishing the basis for the nonlinear optical response described in this work. Surface grating couplers (GCs) provide optical access to the circuit by coupling light from optical fibers into single-mode silicon waveguides connected to the resonator. We routed both the through and drop ports of the MRM to GCs to enable optical characterization of the resonance and nonlinear response.
Figure 2A shows an optical micrograph of the fabricated device. We positioned the PD adjacent to the MRM to enable direct electrical coupling between the two components while maintaining optical separation between the detection and modulation regions. The MRM incorporates electrical terminals for carrier injection together with an integrated resistive microheater positioned above the ring waveguide. The microheater allows thermal tuning of the resonance wavelength, enabling precise alignment of the cavity resonance with respect to the optical signal wavelength. When optical power is incident on the PD, photocurrent injects carriers into the MRM junction. The injected carriers modify the effective refractive index of the ring resonator through the plasma dispersion effect, producing a shift of the resonance wavelength.
FIGURE 2
Importantly, the PD-MRM unit involves two distinct optical signals with different physical roles. Optical power propagating in the resonator bus waveguide defines the available output amplitude of the unit, while a separate optical intensity incident on the photodiode generates the photocurrent that modulates the resonance of the microring. In this way, the optical signal incident on the PD acts as a control signal that shifts the resonance relative to the fixed signal wavelength in the ring. As the resonance position varies with the generated photocurrent, the resonator transmission produces a power-dependent nonlinear response that serves as the fundamental building block of the photonic computing architecture explored in this work. Because both the through and drop ports of the resonator are accessible, we obtain complementary nonlinear responses with opposite slopes. These responses arise from the resonance line shape of the microring and enable flexible realization of nonlinear activation functions when cascading multiple units within photonic computing architectures.
2.2 Electrical packaging and biasing
Figure 2B shows the packaged device used for experimental characterization. We mounted the fabricated photonic chip on a custom carrier printed circuit board (PCB) that provides electrical access to the PD-MRM circuit as well as to the integrated microheater and established electrical connections between the chip and the carrier through wire bonds to the on-chip contact pads. The PCB layout enables independent biasing of the PD and the MRM while maintaining low parasitic impedance to support dynamic measurements. The electrical configuration used to operate the device is illustrated in Figure 2C. We biased the cathode of the PD and connected the cathode of the MRM to ground. Under optical illumination of the PD, the generated photocurrent flows through the MRM junction, thereby establishing a forward-biased carrier-injection condition in the ring resonator. A passive bias-tee network implemented on the carrier PCB provides simultaneous DC biasing and a high-frequency current return path. An independent voltage source drives the integrated microheater of the MRM, enabling controlled thermal tuning of the ring resonance during operation.
2.3 Experimental setup
Multiple GCs distributed across the chip provide optical access to the photonic integrated circuit (PIC). We aligned all optical input and output ports simultaneously using a 127 μm pitch fiber array (SQS) mounted on precision XYZ translation stages (Thorlabs). The experimental configuration consisted of two optical excitation paths coupled to the PIC through the fiber array. In the first optical path, light from a tunable laser (Santec TSL-550) operating at 1,560 nm passes through a polarization controller, corresponding to the peak responsivity of the PD and efficient coupling through the GCs, while a 40 GHz electro-optic modulator (EOM) operating at 1,550 nm modulates the optical signal. An in-house developed FPGA-based arbitrary waveform generator implemented on a Xilinx FPGA drives the EOM, enabling both static optical power control and dynamic modulation up to 500 MS/s. A GC couples the modulated optical signal into the PIC and routes it to an on-chip 1 × 2 multimode interference splitter, dividing the optical power into two branches. We routed one branch to a reference GC used for alignment and optical power monitoring, and directed to the on-chip PD where the optical signal is converted into photocurrent that drives the MRM. In a second optical path, we coupled light from another tunable laser (Santec TSL-550) through a polarization controller into the input waveguide of the MRM, extracting the transmission at the through port via an output GC, while monitoring the drop port using an integrated on-chip PD.
We performed static characterization of the device by recording transmission spectra using a multi-channel optoelectronic data acquisition system (CoreDAQ, Core - Instrumentation) synchronized with the tunable laser. This configuration enabled reconstruction of the MRM transmission spectrum while varying the optical power incident on the PD as well as the heater current applied to the MRM. Using these measurements, we extracted the static nonlinear transfer characteristics of the PD-MRM unit. For dynamic measurements, we detected the transmitted optical signal using high-speed photodetectors (RX10MAF, Thorlabs; 30 kHz–10 GHz bandwidth) connected to a high-bandwidth oscilloscope, enabling direct measurement of the temporal response of the MRM under pulsed optical excitation. A source meter (Keithley 2,450) supplied the forward bias applied to the PD and simultaneously monitored the average photocurrent generated in the circuit.
2.4 Software model of the photonic network architecture
In the present system, the fundamental nonlinear element is the transmission characteristic of the MRM, which exhibits a Lorentzian-induced nonlinear response. This nonlinear response replaces the B-spline basis functions used in the original KAN formulation and features two learnable parameters. The scaling parameter corresponds to the optical power injected into the resonator bus waveguide and determines the maximum amplitude of the transmitted output, whereas the shift parameter corresponds to thermal tuning of the ring resonance, which controls the spectral position of the Lorentzian response. To enable network training and analysis, we implement a differentiable software model of the PD-MRM nonlinear unit in PyTorch (). The model is based on a physical description of the device whose parameters were calibrated and validated using the experimental measurements described in Section 3.2.
In this formulation, we first convert the optical input power into photocurrent at the photodiode according to the measured responsivity. The resulting photocurrent, together with the heater current used for thermal tuning, determines the phase shift of the MRM resonance through carrier-induced refractive index changes in the ring waveguide. We compute the resonator transmission using a standard ring-resonator model,where is the self-coupling coefficient, the round-trip field attenuation, the ring circumference, and the optical wavelength. The effective refractive index, , depends on the photocurrent and heater current through carrier-induced and thermo-optic effects. Drop-port transmission follows as . The final device output equals the selected transmission function multiplied by the optical power propagating in the resonator bus waveguide. This physically grounded formulation produces a differentiable nonlinear transfer function whose parameters correspond directly to experimentally accessible device controls, enabling the PD-MRM unit to be incorporated into network-level simulations and training procedures.
We construct networks by cascading the PD-MRM nonlinear units described above. In the resulting network structure, the MRMs implement the nonlinear functions along network edges, while the PDs act as nodes that aggregate incoming optical signals. Routing the optical output of one resonator to the photodiode of a subsequent unit therefore creates compositions of univariate nonlinear mappings, producing a structure analogous to KANs. The resulting architecture forms a photonic implementation of a constrained KAN in which nonlinear edge functions are realized by the Lorentzian response of the MRM units.
The physical implementation imposes several architectural constraints that influence the resulting network topology. First, the current architecture does not support optical fan-out from a single resonator output to multiple downstream nodes, as implementing such fan-out would require amplification or additional splitter stages that increase loss and device complexity. Consequently, signal propagation after the input stage follows a merge-only connectivity pattern, in which multiple upstream edges may feed into a single photodiode node while each edge connects to only one downstream node. Second, the linear operating range of the photodiodes limits the number of optical signals that can be combined at a single node, since the resulting photocurrent must remain within the diode’s linear regime. Third, the number of nonlinear units that can be integrated on a single chip is constrained by chip area, routing complexity, and the peripheral circuitry required for each MRM, such as heaters, resonance stabilization, and monitoring or feedback control. Without chip–package co-design that integrates these functions efficiently, the number of controllable rings cannot scale arbitrarily, making practical networks likely limited to at most a few hundred MRM units. These constraints shape the resulting network architecture. At the input stage, sensor signals are distributed to multiple nodes using optical splitters, with connections arranged in a round-robin pattern to encourage early interaction between input variables. Subsequent layers follow the merge-only connectivity structure described above, where multiple upstream edges feed into individual photodiode nodes. An example of such a network is shown in Figure 3, illustrating the round-robin input distribution, merge-only connectivity, and the nonlinear responses learned on individual edges for a trained PKAN instance, together with the corresponding network output computed from these learned parameters.
FIGURE 3
To evaluate the function-approximation capabilities of these networks, experiments were performed on benchmark functions spanning multiple input dimensionalities mostly derived from the original KAN study (). All experiments used the differentiable PKAN model described above. For each target function, hyperparameter optimization was performed using Optuna () to determine suitable network architectures and training parameters. The search space included layer count, layer widths, edge parameters, learning rate, and initialization settings. During optimization, architectures were constrained to respect merge-only connectivity and a maximum budget of 500 MRM units, reflecting practical chip-area and routing limits. The optimization objective jointly considered prediction accuracy and architectural complexity, allowing us to analyze the trade-off between approximation error and network size.
3 Results
3.1 Static nonlinear response of the PD-MRM unit
The static nonlinear characteristic of the PD-MRM unit arises from the photocurrent-induced shift of the MRM resonance under optical excitation of the PD. The MRM exhibited an insertion loss of 0.6 dB at 1,550 nm, and the photodiodes showed a responsivity of approximately 0.9 A/W. The PD’s cathode was biased at 0.7 V. The bias voltage appears at the anode of the MRM, forward-biasing the device and enabling current flow through the circuit, while the photodiode remains reverse-biased. Figure 4A shows the measured transmission spectra at the through port of the MRM for increasing photocurrent levels generated in the PD. As the optical power incident on the PD increases, the increasing photocurrent produces a progressive blue shift of the MRM resonance. In addition to the shift, the transmission spectra exhibit a reduction in extinction ratio and a broadening of the resonance linewidth as the photocurrent increases. The decrease in extinction ratio can be attributed to the transition of the resonator away from the critical coupling condition as carrier injection modifies the internal loss and effective coupling balance of the ring. At the same time, increased free-carrier absorption contributes to an effective increase in cavity loss, resulting in a broader resonance linewidth.
FIGURE 4
To quantify this, the measured spectra were fitted using a Lorentzian resonance model. All measured optical powers were normalized to the maximum output power of the GCs, ensuring that variations in coupling loss (∼5 dB per coupler at 1,550 nm) did not affect our model. From these fits, we extracted the resonance wavelength shift as a function of the photocurrent generated in the PD and modelled the relationship between photocurrent and resonance shift using a polynomial fit, which captures the nonlinear dependence observed experimentally. At higher photocurrent levels, the resonance shift exhibits a gradual saturation behavior, resulting from competing thermal effects that red-shift the resonance. In addition to the resonance shift, we also parameterized the extinction ratio as a function of photocurrent using the same polynomial fitting approach. While Figure 4A displays through-port transmission, the corresponding drop-port spectra were also verified to exhibit the typical complementary behavior with inverted transmission characteristics.
3.2 Nonlinear transfer functions and operating point control
Figure 4B shows the extracted nonlinear transfer characteristics obtained by monitoring the MRM transmission at fixed wavelengths corresponding to different offsets from the resonance. These curves represent the optical output power as a function of the photocurrent generated in the PD and illustrate how the Lorentzian resonance response translates into different nonlinear transfer functions depending on the chosen operating point. For operating wavelengths positioned near the steepest slope of the ring resonance, the device exhibits strong nonlinear behavior with high sensitivity to photocurrent variations. In contrast, operating points further from the resonance yield weaker nonlinear responses that approach quasi-linear behavior. This tunability enables a range of nonlinear responses.
We control the operating point of the nonlinear unit through the integrated microheater of the MRM. Thermal tuning shifts the resonance wavelength over approximately one free spectral range (FSR), enabling the resonance position to be adjusted relative to the fixed input wavelength. In the PKAN formulation this resonance shift corresponds directly to the learnable shift parameter of the nonlinear edge function, determining which region of the resonator response is used to implement the nonlinear mapping. In the present device the electrical power required for a π-phase shift of the ring is approximately 20 mW. We parameterized the experimentally extracted nonlinear transfer curves using the Lorentzian resonance model described in Section 3.1 together with the polynomial relationships obtained for resonance shift and extinction ratio. Because photocurrent also modifies the effective resonator loss and linewidth, the resulting power-domain responses deviate slightly from an ideal Lorentzian. These effects are captured by our software model, providing a compact representation of the nonlinear optical response and forming the basis of the unit-cell model used in the PKAN simulations described later.
3.3 Dynamic response of the nonlinear unit
To evaluate the temporal response of the PD-MRM nonlinear unit, we performed dynamic measurements using amplitude-modulated optical pulses applied to the PD. Figure 4C shows the temporal response of the device when square optical pulses are applied to the PD input. The transmitted optical signal follows the temporal envelope of the applied optical pulses. For the measurements shown in Figure 4C, optical pulses corresponding to a photocurrent of approximately 3 mA were applied to the PD. The transmission response of the MRM exhibits a measured rise time of approximately 8 ns. This response time is consistent with the carrier recombination dynamics of forward-biased silicon carrier-injection modulators and represents the dominant timescale governing the nonlinear activation speed of the device.
To further validate the predictive capability of the static nonlinear model under dynamic operation, we applied square optical pulses with varying amplitudes to the PD input. Subsequently, we extracted the peak amplitudes of the transmitted optical pulses and compared them with the nonlinear transfer curve predicted from the static measurements. Figure 4D shows the measured output pulse amplitudes plotted together with the predicted response obtained from the DC model. The strong agreement confirms that the static model accurately predicts the device response under high-speed operation. These results confirm that the PD-MRM structure can operate as a compact all-optical nonlinear unit with nanosecond-scale response time, enabling high-speed optical signal processing suitable for ultrafast network inference.
3.4 Function approximation with PKANs
Table 1 summarizes the results obtained with simulated networks for the set of benchmark functions described in the experimental setup. For each target function, we performed hyperparameter optimization over network architecture and training settings and selected the configuration achieving the lowest root-mean-square error (RMSE). Across these functions, the learned PKAN models achieve RMSEs ranging from slightly above down to nearly . This performance indicates that networks composed of physically realizable photonic nonlinear units can approximate a variety of structured nonlinear mappings using device counts that remain compatible with realistic on-chip PD-MRM integration limits.
TABLE 1
| Function | Definition or scipy.special API | Domain | Units | RMSE |
|---|---|---|---|---|
| Bessel function (1st kind, 1D) | f(x) = J0 (20x) | [0, 1] | 175 | |
| Exp-Sin | f(x, y) = exp(sin(πx) + y2) | [−1, 1]2 | 119 | |
| Product | f(x, y) = xy | [−1, 1]2 | 243 | |
| 2D radial basis function | f(x, y) = exp(−2 (x2 + y2)) | [−1, 1]2 | 186 | |
| Four dimensional | [−0.5, 0.5]4 | 122 | ||
| High dimensional | [−1, 1]100 | 141 | ||
| Jacobi elliptic sine | ellipj (x, y) | [0, 4] × [0, 0.9] | 184 | |
| Incomplete elliptic integral (1st kind) | ellipkinc (x, y) | [0, 2] × [0, 0.8] | 116 | |
| Incomplete elliptic integral (2nd kind) | ellipeinc (x, y) | [0, 2] × [0, 1] | 159 | |
| Bessel function (1st kind) | jv (x, y) | [0, 3] × [0.5, 10] | 312 | |
| Bessel function (2nd kind) | yv (x, y) | [0, 2] × [0.5, 2.5] | 110 | |
| Modified bessel function (2nd kind) | kv (x, y) | [0, 1.5] × [0.5, 1.5] | 243 | |
| Modified bessel function (1st kind) | iv (x, y) | [0, 1] × [0.05, 1] | 216 | |
| Associated legendre function (m = 0) | lpmv (0, x, y) | [0, 5] × [0, 0.99] | 156 | |
| Associated legendre function (m = 1) | lpmv (1, x, y) | [1, 5] × [0, 0.99] | 322 | |
| Associated legendre function (m = 2) | lpmv (2, x, y) | [2, 5] × [0, 0.99] | 322 | |
| Spherical harmonics (m = 0, n = 1) | sph_harm (0, 1, x, y) | [0, 2π] × [0, π] | 322 | |
| Spherical harmonics (m = 1, n = 1) | sph_harm (1, 1, x, y) | [0, 2π] × [0, π] | 200 | |
| Spherical harmonics (m = 0, n = 2) | sph_harm (0, 2, x, y) | [0, 2π] × [0, π] | 166 | |
| Spherical harmonics (m = 1, n = 2) | sph_harm (1, 2, x, y) | [0, 2π] × [0, π] | 322 | |
| Spherical harmonics (m = 2, n = 2) | sph_harm (2, 2, x, y) | [0, 2π] × [0, π] | 313 |
PKAN approximation performance and model complexity across benchmark functions. Each function is defined over the domain listed in the table. “Units” shows the number of MRMs (edges) in the resulting network. Reported errors are root-mean-square error (RMSE) between model prediction and ground truth.
Figure 5 shows representative examples from this set: the Bessel function of the first kind, , the associated Legendre polynomial, , and the real part of the spherical harmonic, . For each example we show the ground-truth function, the prediction produced by the PKAN model, and the relative residual. Even for the spherical harmonic , which represents one of the more challenging functions in this set (as evident from its comparatively high RMSE of ), the relative residual remains below 6% across the entire domain and is considerably smaller over most of the input region. Together, these observations demonstrate that photonic networks constructed from experimentally characterized PD-MRM nonlinear units can accurately approximate complex nonlinear mappings while remaining compatible with realistic photonic hardware constraints.
FIGURE 5
4 Discussion
The results demonstrate that the PD-MRM nonlinear unit provides a compact and physically realizable optical activation mechanism that can be directly integrated into scalable photonic network architectures. Static and dynamic characterization confirms a tunable nonlinear response governed by carrier-induced resonance shifts with nanosecond-scale response time. Using a physically grounded model derived from these measurements, we showed that cascaded networks of such units can approximate structured nonlinear mappings while respecting realistic hardware constraints, including merge-only connectivity, limited photodiode headroom, and practical limits on the number of units that can be integrated on a chip.
A key advantage of the PD–MRM unit is the direct coupling between the photodiode and micro-ring modulator, enabling nonlinear transformation without intermediate electronic amplification or conventional OEO conversion. Many photonic neural network implementations rely on photodetection followed by electronic processing and remodulation (; ), introducing latency, energy overhead, and circuit complexity. In contrast, the directly coupled PD-MRM structure enables compact nonlinear activation with a minimal set of physically controllable parameters: optical MRM input power and heater-controlled resonance position shift. Cascading such units naturally produces a computational structure in which univariate nonlinear functions are composed along directed edges, closely mirroring the formulation of Kolmogorov-Arnold networks (). Compared to multilayer perceptron implementations based on similar nonlinear devices (), this representation enables complex mappings to be constructed using fewer nonlinear elements, which is advantageous in photonic systems where device footprint, loss, and routing complexity limit network scale. Importantly, although grating couplers introduce an insertion loss of approximately 5 dB per coupler, the PD–MRM architecture permits fresh optical carrier injection at each layer. As a result, optical power does not need to propagate through the entire network as in fully passive cascaded photonic systems, mitigating accumulated signal attenuation and removing a fundamental layer-scaling limitation arising from optical loss.
Recent work has explored photonic implementations of KAN-inspired architectures using ring-assisted Mach–Zehnder interferometer (RAMZI) structures (), demonstrating the potential of interferometric nonlinear elements for large-scale accelerator architectures. The unit cell proposed in this work differs substantially from the PD-MRM element studied here: whereas our nonlinear unit consists of a single PD directly coupled to a single MRM and provides two trainable physical controls per edge, the RAMZI-based edge relies on a cascaded interferometric structure with multiple resonators, phase shifters, and intermediate optical amplification, providing nine trainable controls per edge. This reflects a different design trade-off. The RAMZI approach explores a more parameter-rich edge function for fully connected photonic KAN layers, while the present work emphasizes a compact nonlinear element whose static transfer behavior and nanosecond-scale dynamics are directly measured and used to construct the network model.
This distinction is also relevant to connectivity and scalability. Fully connected photonic layers require extensive routing, splitting, and power redistribution between layers, which can become difficult to reconcile with dense, manufacturable integrated layouts. Inter-layer fan-out would require distributing the output of many internal nonlinear units to multiple downstream receivers, increasing routing complexity and reducing the optical power available at each receiver through splitter loss and 1:N power division. For this reason, our architecture confines fan-out to a dedicated input distribution stage, where the externally supplied inputs are mapped once onto the first PKAN layer using a round-robin scheme to promote early mixing of input variables, but avoids inter-layer fan-out by enforcing merge-only routing in subsequent layers. This constraint reduces routing complexity and optical power overhead but also restricts later layers to equal or smaller widths. Our simulations therefore intentionally evaluate PKAN performance under manufacturability-oriented constraints, rather than assuming arbitrary dense connectivity. In this sense, the RAMZI-based and PD-MRM-based approaches are complementary: the former investigates accelerator-scale photonic KANs with complex edge units and dense connectivity, whereas the latter experimentally validates a compact CMOS-compatible nonlinear building block and studies what can be achieved when its measured behavior is embedded into a hardware-constrained network model.
The propagation-based nature of photonic computation further distinguishes this approach from conventional electronic accelerators. Because optical signals propagate continuously through the network, downstream nonlinear units begin responding as soon as the signal reaches them, rather than waiting for sequential execution of preceding layers. As a result, inference latency is primarily determined by the intrinsic response time of the nonlinear unit together with optical propagation delays. The measured rise time of approximately 8 ns therefore establishes the characteristic timescale for network operation and highlights the potential for photonic inference with sublinear latency scaling in network depth.
Despite these promising results, several limitations of the present work should be noted. The hardware constraints inherent to the PD-MRM architecture restrict network design flexibility. In particular, the absence of optical fan-out limits connectivity to merge-only structures, while the finite linear operating range of the photodiodes constrains the number of signals that can be combined at a single node. Because optical intensities and photocurrents are non-negative and are combined additively, the present architecture also does not directly implement signed edge weights or cancellation between different edge contributions. This restricts expressivity relative to KAN implementations with signed coefficients, especially for functions that naturally require subtractive combinations of basis functions or localized positive and negative corrections. Negative-valued target functions can still be represented after appropriate normalization or output-domain shifting, and the results in Table 1 show that a range of structured functions can be approximated accurately despite the positive-weight constraint.
A related limitation is the restricted expressivity of each individual edge function. Each PD-MRM unit provides only two trainable parameters per edge, corresponding to physically realizable scale and shift controls, whereas conventional digital KANs often use spline or basis-function expansions with many effective parameters per edge. As a result, the present architecture may require more edges to reach comparable approximation accuracy. For example, Liu et al. report that the modified Bessel function can be fitted with a (2,1,1) KAN, corresponding to three edges, with an RMSE of , whereas our hardware-constrained PKAN uses 243 edges to reach an RMSE of . This comparison should not be interpreted purely as an edge-count disadvantage, however, because the number of trainable parameters per spline edge in conventional KANs depends on the chosen basis resolution and is not always reported explicitly. The relevant trade-off is therefore between edge-level functional flexibility and physically compact parametrization: PKANs use more hardware edges, but each edge has only two trainable parameters tied directly to experimentally accessible device controls.
The current model also does not explicitly account for noise sources such as optical intensity fluctuations, thermal crosstalk, or fabrication variability, which may influence performance in large-scale implementations. However, previous work has shown that suitable learning algorithms exist that result in parametrizations robust to such physical noise (; ; ). Finally, while the present work demonstrates function approximation across multiple input dimensionalities, high-dimensional problems generally require increased model capacity. In PKANs, this can translate into greater routing complexity and a higher number of required nonlinear units under realistic chip-area constraints. Notably, similar scaling challenges are also observed in conventional KANs, where the number of required basis functions and parameters typically grows rapidly with input dimensionality. A systematic study of noise robustness, thermal drift, fabrication variability, and more challenging benchmark classes, including non-smooth and higher-dimensional functions, is therefore an important direction for future work.
Addressing these limitations provides several directions for further development at both the device and architectural level. Engineering the carrier dynamics through junction design and doping profiles offers a route toward faster nonlinear response and reduced activation energy (; ), while improved photodiode responsivity can enhance power efficiency by increasing photocurrent generation for a given optical input (). Optimizing microring geometry and coupling conditions may further improve the sharpness and tunability of the nonlinear response, enabling more expressive nonlinear transfer characteristics within the same device footprint (). Additionally, we currently use thermal tuning to set the operating point of the microring, which can limit the number of nodes in the network. Looking ahead, this can be replaced with nonvolatile, PCM-based tuning, which is fully CMOS-compatible and requires virtually no electrical energy (), resulting in significant energy savings. At the architectural level, the present implementation relies on intensity-based signal propagation, resulting in purely additive combinations of positive-valued signals. While this already enables accurate function approximation under realistic hardware constraints, it restricts the network to effectively positive weights. Coherent photonic schemes could overcome this limitation by enabling subtraction and signed signal processing through controlled interference (), thereby extending the representational flexibility of the architecture while remaining compatible with integrated photonic platforms.
Taken together, these results illustrate how experimentally characterized nonlinear photonic devices, combined with physically grounded network architectures, provide a pathway from compact optical nonlinearities to scalable photonic computing systems. By linking device-level physics with network-level expressivity under realistic hardware constraints, this work helps bridge the gap between nonlinear photonic components and functional computing architectures. As integrated photonics continues to mature, such approaches offer a promising route toward ultrafast and energy-efficient optical inference engines operating on timescales beyond the reach of conventional electronic hardware.
Statements
Data availability statement
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
Author contributions
RP: Writing – original draft, Writing – review and editing. RJ: Writing – original draft, Writing – review and editing. AV: Writing – review and editing. NG: Conceptualization, Data curation, Formal Analysis, Methodology, Validation, Visualization, Writing – review and editing. FE: Conceptualization, Data curation, Formal Analysis, Methodology, Software, Validation, Visualization, Writing – review and editing. RD: Writing – review and editing. TL: Writing – review and editing. LM: Writing – review and editing. JR: Writing – review and editing. PS: Writing – review and editing. JB: Writing – review and editing. FB-P: Conceptualization, Formal Analysis, Methodology, Writing – review and editing, Writing – original draft. HF: Writing – original draft, Writing – review and editing. WP: Writing – original draft, Writing – review and editing.
Funding
The author(s) declared that financial support was received for this work and/or its publication. The research is funded by the German Research Foundation under Germany´s Excellence Strategy EXC 2181/1—390900948 (the Heidelberg STRUCTURES Excellence Cluster), the Excellence Cluster 3D Matter Made to Order (EXC−2082/1—390761711) and CRC 1459 “Intelligent matter”, the German Research Foundation (grant PE 1832/22-1) and the European Union’s Horizon 2020 research and innovation programme (grant no. 101017237, PHOENICS project) and the European Union’s Innovation Council Pathfinder programme (grant no. 101046878, HYBRAIN project), and the ERC Advanced Grant PICNIC (grant no. 101200429).
Acknowledgments
We thank Jochen Stuhrmann (Illustrato) for his assistance with the illustrations.
Conflict of interest
The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Generative AI statement
The author(s) declared that generative AI was used in the creation of this manuscript. Generative AI tools were used to improve the clarity, grammar, and readability of the manuscript. All technical content, analysis, and conclusions were developed and verified by the authors.
Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.
Publisher’s note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
References
1
AbueiddaD. W.PantidisP.MobasherM. E. (2025). “DeepOKAN: deep operator network based on kolmogorov arnold networks for mechanics problems,” Comput. Methods Appl. Mech. Eng.436, 117699. 10.1016/j.cma.2024.117699
2
AkibaT.SanoS.YanaseT.OhtaT.KoyamaM. (2019). “Optuna: a next-generation hyperparameter optimization framework,” in Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York, NY:Association for Computing Machinery, 2623–2631.
3
AlzahraniS.MitraJ. (2022). “Microgrid fault detection utilizing state observer and multi-agent system,” in 2022 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm) (IEEE).
4
BandyopadhyayS.SluddsA.KrastanovS.HamerlyR.HarrisN.BunandarD.et al (2024). Single-chip photonic deep neural network with forward-only training. Nat. Photonics18, 1335–1343. 10.1038/s41566-024-01567-z
5
BodnerA. D.TepsichA. S.SpolskiJ. N.PourteauS. (2024). “Convolutional Kolmogorov-Arnold Networks”.
6
BorrasH.KleinB.FröningH. (2024). Walking Noise: On Layer-specific Robustness of Neural Architectures Against Noisy Computations and Associated Characteristic Learning Dynamics, 38–55. Cham: Springer.
7
CanoraR.XuX.NiuZ.AlaeianH.DuS. (2025). Engineering Nonlinear Activation Functions for all-optical Neural Networks via Quantum Interference.
8
ChangJ.SitzmannV.DunX.HeidrichW.WetzsteinG. (2018). Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification. Sci. Rep.8, 12324. 10.1038/s41598-018-30619-y
9
ChaoZ.LiuX.WuZ.LiX. (2026). “RBF-KAN: radial basis function-kolmogorov-arnold networks,” in IEEE Internet Things J. Piscataway, NJ:Institute of Electrical and Electronics Engineers.
10
ClaphamB.HaferkornM.ZimmermannK. (2023). The impact of high-frequency trading on modern securities markets: an analysis based on a technical interruption. Bus. Inf. Syst. Eng.65, 7–24. 10.1007/s12599-022-00768-6
11
FalangaD.KleberK.ScaramuzzaD. (2020). Dynamic obstacle avoidance for quadrotors with event cameras. Sci. Robot.5. 10.1126/scirobotics.aaz9712
12
FeldmannJ.YoungbloodN.KarpovM.GehringH.LiX.StappersM.et al (2021). Parallel convolutional processing using an integrated photonic tensor core. Nature589, 52–58. 10.1038/s41586-020-03070-1
13
GholamiA.YaoZ.KimS.HooperC.MahoneyM. W.KeutzerK. (2024). AI and memory wall. IEEE Micro44, 33–39. 10.1109/mm.2024.3373763
14
HennessyJ. L.PattersonD. A. (2019). A new golden age for computer architecture. Commun. ACM62, 48–60. 10.1145/3282307
15
JiangJ.DingB.LiS.ZhangX.WangH.WuJ.et al (2025). All-optical nonlinear activation functions realized on phase-change photonic integrated circuits with microheaters. J. Semicond.46, 022405. 10.1088/1674-4926/24090045
16
JoshiV.Le GalloM.HaefeliS.BoybatI.NandakumarS. R.PiveteauC.et al (2020). Accurate deep neural network inference using computational phase-change memory,” Nat. Commun.11 (1), 2473. 10.1038/s41467-020-16108-9
17
KazanskiyN. L.KhorinP. A.GolovastikovN. V.KhoninaS. N. (2026). Analog optical computing: principles, progress, and prospects. Opt. Laser Technol.193, 114220. 10.1016/j.optlastec.2025.114220
18
LiL.ZhangY.WangG.XiaK. (2025). Kolmogorov–arnold graph neural networks for molecular property prediction,” Nat. Mach. Intell.7 (8), 1346–1354. 10.1038/s42256-025-01087-7
19
LiuZ.TegmarkM.MaP.MatusikW.WangY. (2025a). Kolmogorov-arnold networks meet science. Phys. Rev. X15, 041051. 10.1103/4t7t-v19l
20
LiuZ.WangY.VaidyaS.RuehleF.HalversonJ.SoljacicM.et al (2025b). “KAN: Kolmogorov arnold networks,” in The Thirteenth International Conference on Learning Representations.
21
PaszkeA.GrossS.MassaF.LererA.BradburyJ.ChananG.et al (2019). “PyTorch: an imperative style, high-performance deep learning library,” in Proceedings of the 33rd International Conference on Neural Information Processing Systems (Red Hook, NY, USA: Curran Associates Inc.).
22
PengY.HootenS.YuX.Van VaerenberghT.YuanY.XiaoX.et al (2025). “Photonic KAN: a kolmogorov-arnold network inspired efficient neuromorphic accelerator,” in 2025 Optical Fiber Communications Conference and Exhibition (OFC).
23
QiuQ.ZhuT.GongH.ChenL.NingH. (2025). “ReLU-KAN: new kolmogorov-arnold networks that only need matrix addition, dot multiplication, and ReLU,” in 2025 IEEE Smart World Congress (SWC). IEEE, 1686–1694.
24
ReinhardtE.RamakrishnanD.GleyzerS. (2024). SineKAN: Kolmogorov-arnold networks using sinusoidal activation functions. Front. Artif. Intell.7, 1462952. 10.3389/frai.2024.1462952
25
ShiW.HuangZ.FuT.ChenH. (2025). Review of nonlinear activation functions in optical neural networks. Adv. Photonics7. 10.1117/1.ap.7.6.064004
26
Silva CottaJ. L.AgarD.BertaskaI. R.InnessJ. P.GutierrezH. (2023). Latency reduction and packet synchronization in low-resource devices connected by DDS networks in autonomous UAVs. Sensors23, 9269. 10.3390/s23229269
27
StroevN.BerloffN. G. (2023). Analog photonics computing for information processing, inference, and optimization. Adv. Quantum Technol.6, 2300055. 10.1002/qute.202300055
28
ThubthimthongB.SasakiT.HaneK. (2015). Asymmetrically and vertically coupled hybrid Si/GaN microring resonators for On-Chip optical interconnects. IEEE Photonics J.7, 1–11. 10.1109/jphot.2015.2464721
29
VivienL.RouvièreM.FédéliJ.-M.Marris-MoriniD.DamlencourtJ. F.MangeneyJ.et al (2007). High speed and high responsivity germanium photodetector integrated in a silicon-on-insulator microwaveguide. Opt. Express15, 9843–9848. 10.1364/oe.15.009843
30
WangX.BorrasH.KleinB.FröningH. (2026). “Variance-aware noisy training: hardening DNNs against unstable analog computations,” in Lecture Notes in Computer Science 1B019 LNCS. Cham: Springer, 147–163.
31
WeiM.XuK.TangB.LiJ.YunY.ZhangP.et al (2024). Monolithic back-end-of-line integration of phase change materials into foundry-manufactured silicon photonics. Nat. Commun.15, 2786. 10.1038/s41467-024-47206-7
32
WilliamsonI. A. D.HughesT. W.MinkovM.BartlettB.PaiS.FanS. (2019). Reprogrammable Electro-Optic Nonlinear Activation Functions for Optical Neural Networks.
33
WuR.ChenC.-H.FedeliJ.-M.FournierM.ChengK.-T.BeausoleilR. G. (2015). Compact models for carrier-injection silicon microring modulators. Opt. Express23, 15545–15554. 10.1364/OE.23.015545
34
WuB.HuangC.ZhangJ.ZhouH.WangY.DongJ.et al (2025). Scaling up for end-to-end on-chip photonic neural network inference. Light Sci. Appl.14, 328. 10.1038/s41377-025-02029-z
35
YangZ.HeJ.YanZ.HuY.LiX.DongN.et al (2024). Inverse-designed integrated all-optical nonlinear activators for optical computing. Opt. Express32, 34001–34014. 10.1364/OE.531679
36
YuS.BohmC.ChenH.AllgowerF. (2010). “Robust model predictive control with disturbance invariant sets,” in Proceedings of the 2010 American Control Conference (IEEE).
37
YuanY.PengY.SorinW. V.CheungS.HuangZ.LiangD.et al (2024). A 5 × 200 Gbps microring modulator silicon chip empowered by two-segment Z-shape junctions. Nat. Commun.15, 918. 10.1038/s41467-024-45301-3
38
ZhangL.FanW.LuL. (2026). RSS-KAN: replacing B-splines with ReLU, sigmoid, and sine for accurate and efficient kolmogorov–arnold networks. Neurocomputing672, 132796. 10.1016/j.neucom.2026.132796
Summary
Keywords
all optical nonlinearity, neuromorphic computing, photonic artificial intelligence hardware, photonic neural network, ultra-low latency
Citation
Pradip R, Janssen R, Varri A, Gründel N, Ebert F, Durán RG, Lee T, McRae L, Römer J, Schmidt P, Bankwitz JR, Brückerhoff-Plückelmann F, Fröning H and Pernice W (2026) Enabling photonic Kolmogorov-Arnold networks for ultra-fast inference. Front. Photonics 7:1860010. doi: 10.3389/fphot.2026.1860010
Received
19 April 2026
Revised
20 May 2026
Accepted
05 June 2026
Published
19 June 2026
Volume
7 - 2026
Edited by
Qiming Zhang, University of Shanghai for Science and Technology, China
Reviewed by
Muhammad Shemyal Nisar, University of Shanghai for Science and Technology Sino-British College, China
Jun Wang, Shanghai institute of optics and fine mechanics, China
Updates
Copyright
© 2026 Pradip, Janssen, Varri, Gründel, Ebert, Durán, Lee, McRae, Römer, Schmidt, Bankwitz, Brückerhoff-Plückelmann, Fröning and Pernice.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Wolfram Pernice, wolfram.pernice@kip.uni-heidelberg.de
† These authors have contributed equally to this work and share first authorship
Disclaimer
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.