Photonic information processing beyond Turing: an optoelectronic implementation of reservoir computing

Experimental control of mode-competition dynamics in a chaotic multimode semiconductor laser for decision making

Photonic computing is widely used to accelerate the computational performance in machine learning. Photonic decision making is a promising approach utilizing photonic computing technologies to solve the multi-armed bandit problems based on reinforcement learning. Photonic decision making using chaotic mode-competition dynamics has been proposed. However, the experimental conditions for achieving a superior decision-making performance have not yet been established. Herein, we experimentally investigate mode-competition dynamics in a chaotic multimode semiconductor laser in the presence of optical feedback and injection. We control the chaotic mode-competition dynamics via optical injection and observe that positive wavelength detuning results in an efficient mode concentration to one of the longitudinal modes with a small optical injection power. We experimentally investigate two-dimensional bifurcation diagram of the total intensity of the laser dynamics. Complex mixed dynamics are observed in the presence of optical feedback and injection. We experimentally conduct decision making to solve the bandit problem using chaotic mode-competition dynamics. A fast mode-concentration property is observed at positive wavelength detunings, resulting in fast convergence of the correct decision rate. Our findings could be useful in accelerating the decision-making performance in adaptive optical networks using reinforcement learning.

Hybrid integrated optical chaos circuits with optoelectronic feedback

A chip-scale chaotic laser system with optoelectronic delayed feedback is proposed and analyzed by numerical simulation. This chip eliminates the need for bulky delay components such as long optical fibers, free propagation and external cavities, relying solely on internal devices and waveguides to achieve feedback delay. This approach simplifies integration, maintaining a compact chip size. According to the results, the chip-scale system exhibits rich dynamics, including periodicity, quasi-periodicity, and chaotic states. Chaos resembling Gaussian white noise is achieved with picosecond-level delay time, highlighting the complexity of chip-scale signals. Furthermore, time delay signature (TDS) concealment is enhanced with a short delay comparable to the inverse bandwidth τ, albeit at a cost of sacrificing chaotic signal complexity. Applying the photonic integrated circuits to practical applications, 1 Gbps back-to-back communication transmission is feasible. Results demonstrate low bit error rates (BERs) for authorizers (<10-6) and high BERs for eavesdroppers (>10-2), ensuring communication confidentiality and chaotic synchronization. Lastly, preliminary experiments validate the feasibility. Our theoretical work has demonstrated the feasibility of hybrid integrated optical chaos circuits with optoelectronic feedback based on photonic wire bonding, which can provide a stable and flexible integrated chaos source.

Embedding Bifurcations into Pneumatic Artificial Muscle

Harnessing complex body dynamics has long been a challenge in robotics, particularly when dealing with soft dynamics, which exhibit high complexity in interacting with the environment. Recent studies indicate that these dynamics can be used as a computational resource, exemplified by the McKibben pneumatic artificial muscle, a common soft actuator. This study demonstrates that bifurcations, including periodic and chaotic dynamics, can be embedded into the pneumatic artificial muscle, with the entire bifurcation structure using the framework of physical reservoir computing. These results suggest that dynamics not present in training data can be embedded through bifurcation embedment, implying the capability to incorporate various qualitatively different patterns into pneumatic artificial muscle without the need to design and learn all required patterns explicitly. Thus, this study introduces a novel approach to simplify robotic devices and control training by reducing reliance on external pattern generators and the amount and types of training data needed for control.

Time delay reservoir computing with a silicon microring resonator and a fiber-based optical feedback loop

Silicon microring resonators serve as critical components in integrated photonic neural network implementations, owing to their compact footprint, compatibility with CMOS technology, and passive nonlinear dynamics. Recent advancements have leveraged their filtering properties as weighting functions, and their nonlinear dynamics as activation functions with spiking capabilities. In this work, we investigate experimentally the linear and nonlinear dynamics of microring resonators for time delay reservoir computing, by introducing an external optical feedback loop. After effectively mitigating the impact of environmental noise on the fiber-based feedback phase dependencies, we evaluate the computational capacity of this system by assessing its performance across various benchmark tasks at a bit rate of few Mbps. We show that the additional memory provided by the optical feedback is necessary to achieve error-free operation in delayed-boolean tasks that require up to 3 bits of memory. In this case the microring was operated in the linear regime and the photodetection was the nonlinear activation function. We also show that the Santa Fe and Mackey Glass prediction tasks are solved when the microring nonlinearities are activated. Notably, our study reveals competitive outcomes even when employing only 7 virtual nodes within our photonic reservoir. Our findings illustrate the silicon microring's versatile performance in the presence of optical feedback, highlighting its ability to be tailored for various computing applications.

Asymmetric leader-laggard cluster synchronization for collective decision-making with laser network

Photonic accelerators have recently attracted soaring interest, harnessing the ultimate nature of light for information processing. Collective decision-making with a laser network, employing the chaotic and synchronous dynamics of optically interconnected lasers to address the competitive multi-armed bandit (CMAB) problem, is a highly compelling approach due to its scalability and experimental feasibility. We investigated essential network structures for collective decision-making through quantitative stability analysis. Moreover, we demonstrated the asymmetric preferences of players in the CMAB problem, extending its functionality to more practical applications. Our study highlights the capability and significance of machine learning built upon chaotic lasers and photonic devices.

Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction

Reservoir computing is a machine learning framework that has been shown to be able to replicate the chaotic attractor, including the fractal dimension and the entire Lyapunov spectrum, of the dynamical system on which it is trained. We quantitatively relate the generalized synchronization dynamics of a driven reservoir during the training stage to the performance of the trained reservoir computer at the attractor reconstruction task. We show that, in order to obtain successful attractor reconstruction and Lyapunov spectrum estimation, the maximal conditional Lyapunov exponent of the driven reservoir must be significantly more negative than the most negative Lyapunov exponent of the target system. We also find that the maximal conditional Lyapunov exponent of the reservoir depends strongly on the spectral radius of the reservoir adjacency matrix; therefore, for attractor reconstruction and Lyapunov spectrum estimation, small spectral radius reservoir computers perform better in general. Our arguments are supported by numerical examples on well-known chaotic systems.

Emerging opportunities and challenges for the future of reservoir computing

Reservoir computing originates in the early 2000s, the core idea being to utilize dynamical systems as reservoirs (nonlinear generalizations of standard bases) to adaptively learn spatiotemporal features and hidden patterns in complex time series. Shown to have the potential of achieving higher-precision prediction in chaotic systems, those pioneering works led to a great amount of interest and follow-ups in the community of nonlinear dynamics and complex systems. To unlock the full capabilities of reservoir computing towards a fast, lightweight, and significantly more interpretable learning framework for temporal dynamical systems, substantially more research is needed. This Perspective intends to elucidate the parallel progress of mathematical theory, algorithm design and experimental realizations of reservoir computing, and identify emerging opportunities as well as existing challenges for large-scale industrial adoption of reservoir computing, together with a few ideas and viewpoints on how some of those challenges might be resolved with joint efforts by academic and industrial researchers across multiple disciplines.

Conflict-free joint decision by lag and zero-lag synchronization in laser network

With the end of Moore's Law and the increasing demand for computing, photonic accelerators are garnering considerable attention. This is due to the physical characteristics of light, such as high bandwidth and multiplicity, and the various synchronization phenomena that emerge in the realm of laser physics. These factors come into play as computer performance approaches its limits. In this study, we explore the application of a laser network, acting as a photonic accelerator, to the competitive multi-armed bandit problem. In this context, conflict avoidance is key to maximizing environmental rewards. We experimentally demonstrate cooperative decision-making using zero-lag and lag synchronization within a network of four semiconductor lasers. Lag synchronization of chaos realizes effective decision-making and zero-lag synchronization is responsible for the realization of the collision avoidance function. We experimentally verified a low collision rate and high reward in a fundamental 2-player, 2-slot scenario, and showed the scalability of this system. This system architecture opens up new possibilities for intelligent functionalities in laser dynamics.

Harnessing synthetic active particles for physical reservoir computing

The processing of information is an indispensable property of living systems realized by networks of active processes with enormous complexity. They have inspired many variants of modern machine learning, one of them being reservoir computing, in which stimulating a network of nodes with fading memory enables computations and complex predictions. Reservoirs are implemented on computer hardware, but also on unconventional physical substrates such as mechanical oscillators, spins, or bacteria often summarized as physical reservoir computing. Here we demonstrate physical reservoir computing with a synthetic active microparticle system that self-organizes from an active and passive component into inherently noisy nonlinear dynamical units. The self-organization and dynamical response of the unit are the results of a delayed propulsion of the microswimmer to a passive target. A reservoir of such units with a self-coupling via the delayed response can perform predictive tasks despite the strong noise resulting from the Brownian motion of the microswimmers. To achieve efficient noise suppression, we introduce a special architecture that uses historical reservoir states for output. Our results pave the way for the study of information processing in synthetic self-organized active particle systems.

Effects of cavity nonlinearities and linear losses on silicon microring-based reservoir computing

Microring resonators (MRRs) are promising devices for time-delay photonic reservoir computing, but the impact of the different physical effects taking place in the MRRs on the reservoir computing performance is yet to be fully understood. We numerically analyze the impact of linear losses as well as thermo-optic and free-carrier effects relaxation times on the prediction error of the time-series task NARMA-10. We demonstrate the existence of three regions, defined by the input power and the frequency detuning between the optical source and the microring resonance, that reveal the cavity transition from linear to nonlinear regimes. One of these regions offers very low error in time-series prediction under relatively low input power and number of nodes while the other regions either lack nonlinearity or become unstable. This study provides insight into the design of the MRR and the optimization of its physical properties for improving the prediction performance of time-delay reservoir computing.

High-speed parallel processing with photonic feedforward reservoir computing

High-speed photonic reservoir computing (RC) has garnered significant interest in neuromorphic computing. However, existing reservoir layer (RL) architectures mostly rely on time-delayed feedback loops and use analog-to-digital converters for offline digital processing in the implementation of the readout layer, posing inherent limitations on their speed and capabilities. In this paper, we propose a non-feedback method that utilizes the pulse broadening effect induced by optical dispersion to implement a RL. By combining the multiplication of the modulator with the summation of the pulse temporal integration of the distributed feedback-laser diode, we successfully achieve the linear regression operation of the optoelectronic analog readout layer. Our proposed fully-analog feed-forward photonic RC (FF-PhRC) system is experimentally demonstrated to be effective in chaotic signal prediction, spoken digit recognition, and MNIST classification. Additionally, using wavelength-division multiplexing, our system manages to complete parallel tasks and improve processing capability up to 10 GHz per wavelength. The present work highlights the potential of FF-PhRC as a high-performance, high-speed computing tool for real-time neuromorphic computing.

On-chip phonon-magnon reservoir for neuromorphic computing

Reservoir computing is a concept involving mapping signals onto a high-dimensional phase space of a dynamical system called "reservoir" for subsequent recognition by an artificial neural network. We implement this concept in a nanodevice consisting of a sandwich of a semiconductor phonon waveguide and a patterned ferromagnetic layer. A pulsed write-laser encodes input signals into propagating phonon wavepackets, interacting with ferromagnetic magnons. The second laser reads the output signal reflecting a phase-sensitive mix of phonon and magnon modes, whose content is highly sensitive to the write- and read-laser positions. The reservoir efficiently separates the visual shapes drawn by the write-laser beam on the nanodevice surface in an area with a size comparable to a single pixel of a modern digital camera. Our finding suggests the phonon-magnon interaction as a promising hardware basis for realizing on-chip reservoir computing in future neuromorphic architectures.

Experimental results on nonlinear distortion compensation using photonic reservoir computing with a single set of weights for different wavelengths

Photonics-based computing approaches in combination with wavelength division multiplexing offer a potential solution to modern data and bandwidth needs. This paper experimentally takes an important step towards wavelength division multiplexing in an integrated waveguide-based photonic reservoir computing platform by using a single set of readout weights for up to at least 3 ITU-T channels to efficiently scale the data bandwidth when processing a nonlinear signal equalization task on a 28 Gbps modulated on-off keying signal. Using multiple-wavelength training, we obtain bit error rates well below that of the [Formula: see text] forward error correction limit at high fiber input powers of 18 dBm, which result in high nonlinear distortion. The results of the reservoir chip are compared to a tapped delay line filter and clearly show that the system performs nonlinear equalization. This was achieved using only limited post processing which in future work can be implemented in optical hardware as well.

Resonant properties of the memory capacity of a laser-based reservoir computer with filtered optoelectronic feedback

We provide a comprehensive analysis of the resonant properties of the memory capacity of a reservoir computer based on a semiconductor laser subjected to time-delayed filtered optoelectronic feedback. Our analysis reveals first how the memory capacity decreases sharply when the input-data clock cycle is slightly time-shifted from the time delay or its multiples. We attribute this effect to the inertial properties of the laser. We also report on the damping of the memory-capacity drop at resonance with a decrease of the virtual-node density and its broadening with the filtering properties of the optoelectronic feedback. These results are interpretated using the eigenspectrum of the reservoir obtained from a linear stability analysis. Then, we unveil an invariance in the minimum value of the memory capacity at resonance with respect to a variation of the number of nodes if the number is big enough and quantify how the filtering properties impact the system memory in and out of resonance.

Synchronizing chaos using reservoir computing

We attempt to achieve complete synchronization between a drive system unidirectionally coupled with a response system, under the assumption that limited knowledge on the states of the drive is available at the response. Machine-learning techniques have been previously implemented to estimate the states of a dynamical system from limited measurements. We consider situations in which knowledge of the non-measurable states of the drive system is needed in order for the response system to synchronize with the drive. We use a reservoir computer to estimate the non-measurable states of the drive system from its measured states and then employ these measured states to achieve complete synchronization of the response system with the drive.

Exploiting oscillatory dynamics of delay systems for reservoir computing

Nonlinear dynamical systems exhibiting inherent memory can process temporal information by exploiting their responses to input drives. Reservoir computing is a prominent approach to leverage this ability for time-series forecasting. The computational capabilities of analog computing systems often depend on both the dynamical regime of the system and the input drive. Most studies have focused on systems exhibiting a stable fixed-point solution in the absence of input. Here, we go beyond that limitation, investigating the computational capabilities of a paradigmatic delay system in three different dynamical regimes. The system we chose has an Ikeda-type nonlinearity and exhibits fixed point, bistable, and limit-cycle dynamics in the absence of input. When driving the system, new input-driven dynamics emerge from the autonomous ones featuring characteristic properties. Here, we show that it is feasible to attain consistent responses across all three regimes, which is an essential prerequisite for the successful execution of the tasks. Furthermore, we demonstrate that we can exploit all three regimes in two time-series forecasting tasks, showcasing the versatility of this paradigmatic delay system in an analog computing context. In all tasks, the lowest prediction errors were obtained in the regime that exhibits limit-cycle dynamics in the undriven reservoir. To gain further insights, we analyzed the diverse time-distributed node responses generated in the three regimes of the undriven system. An increase in the effective dimensionality of the reservoir response is shown to affect the prediction error, as also fine-tuning of the distribution of nonlinear responses. Finally, we demonstrate that a trade-off between prediction accuracy and computational speed is possible in our continuous delay systems. Our results not only provide valuable insights into the computational capabilities of complex dynamical systems but also open a new perspective on enhancing the potential of analog computing systems implemented on various hardware platforms.

Performance-enhanced time-delayed photonic reservoir computing system using a reflective semiconductor optical amplifier

We propose a time-delayed photonic reservoir computing (RC) architecture utilizing a reflective semiconductor optical amplifier (RSOA) as an active mirror. The performance of the proposed RC structure is investigated by two benchmark tasks, namely the Santa Fe time-series prediction task and the nonlinear channel equalization task. The simulation results show that both the prediction and equalization performance of the proposed system are significantly improved with the contribution of RSOA, with respect to the traditional RC system using a mirror. By increasing the drive current of the RSOA, the greater nonlinearity of the RSOA gain saturation is achieved, as such the prediction and equalization performance are enhanced. It is also shown that the proposed RC architecture shows a wider consistency interval and superior robustness than the traditional RC structure for most of the measured parameters such as coupling strength, injection strength, and frequency detuning. This work provides a performance-enhanced time-delayed RC structure by making use of the nonlinear transformation of the RSOA feedback.

High speed human action recognition using a photonic reservoir computer

The recognition of human actions in videos is one of the most active research fields in computer vision. The canonical approach consists in a more or less complex preprocessing stages of the raw video data, followed by a relatively simple classification algorithm. Here we address recognition of human actions using the reservoir computing algorithm, which allows us to focus on the classifier stage. We introduce a new training method for the reservoir computer, based on "Timesteps Of Interest", which combines in a simple way short and long time scales. We study the performance of this algorithm using both numerical simulations and a photonic implementation based on a single non-linear node and a delay line on the well known KTH dataset. We solve the task with high accuracy and speed, to the point of allowing for processing multiple video streams in real time. The present work is thus an important step towards developing efficient dedicated hardware for video processing.

Excellent predictive-performances of photonic reservoir computers for chaotic time-series using the fusion-prediction approach

In this work, based on two parallel reservoir computers realized by the two polarization components of the optically pumped spin-VCSEL with double optical feedbacks, we propose the fusion-prediction scheme for the Mackey-Glass (MG) and Lorenz (LZ) chaotic time series. Here, the direct prediction and iterative prediction results are fused in a weighted average way. Compared with the direct-prediction errors, the fusion-prediction errors appear great decrease. Their values are far less than the values of the direct-prediction errors when the iteration step-size are no more than 15. By the optimization of the temporal interval and the sampling period, under the iteration step-size of 3, the fusion-prediction errors for the MG and LZ chaotic time-series can be reduced to 0.00178 and 0.004627, which become 8.1% of the corresponding direct-prediction error and 28.68% of one, respectively. Even though the iteration step-size reaches to 15, the fusion-prediction errors for the MG and LZ chaotic time-series can be reduced to 55.61% of the corresponding direct-prediction error and 77.28% of one, respectively. In addition, the fusion-prediction errors have strong robustness on the perturbations of the system parameters. Our studied results can potentially apply in the improvement of prediction accuracy for some complex nonlinear time series.

A refined information processing capacity metric allows an in-depth analysis of memory and nonlinearity trade-offs in neurocomputational systems

Since dynamical systems are an integral part of many scientific domains and can be inherently computational, analyses that reveal in detail the functions they compute can provide the basis for far-reaching advances in various disciplines. One metric that enables such analysis is the information processing capacity. This method not only provides us with information about the complexity of a system's computations in an interpretable form, but also indicates its different processing modes with different requirements on memory and nonlinearity. In this paper, we provide a guideline for adapting the application of this metric to continuous-time systems in general and spiking neural networks in particular. We investigate ways to operate the networks deterministically to prevent the negative effects of randomness on their capacity. Finally, we present a method to remove the restriction to linearly encoded input signals. This allows the separate analysis of components within complex systems, such as areas within large brain models, without the need to adapt their naturally occurring inputs.

Photonic reservoir computing enabled by stimulated Brillouin scattering

Artificial intelligence (AI) drives the creation of future technologies that disrupt the way humans live and work, creating new solutions that change the way we approach tasks and activities, but it requires a lot of data processing, large amounts of data transfer, and computing speed. It has led to a growing interest of research in developing a new type of computing platform which is inspired by the architecture of the brain specifically those that exploit the benefits offered by photonic technologies, fast, low-power, and larger bandwidth. Here, a new computing platform based on the photonic reservoir computing architecture exploiting the non-linear wave-optical dynamics of the stimulated Brillouin scattering is reported. The kernel of the new photonic reservoir computing system is constructed of an entirely passive optical system. Moreover, it is readily suited for use in conjunction with high performance optical multiplexing techniques to enable real-time artificial intelligence. Here, a methodology to optimise the operational condition of the new photonic reservoir computing is described which is found to be strongly dependent on the dynamics of the stimulated Brillouin scattering system. The new architecture described here offers a new way of realising AI-hardware which highlight the application of photonics for AI.

Semiconductor Optical Amplifier (SOA)-Driven Reservoir Computing for Dense Wavelength-Division Multiplexing (DWDM) Signal Compensation

Optical signal processing (OSP) technology is a crucial part of the optical switching node in the modern optical-fiber communication system, especially when advanced modulation formats, e.g., quadrature amplitude modulation (QAM), are applied. However, the conventional on-off keying (OOK) signal is still widely used in access or metro transmission systems, which leads to the compatibility requirement of OSP for incoherent and coherent signals. In this paper, we propose a reservoir computing (RC)-OSP scheme based on nonlinear mapping behavior through a semiconductor optical amplifier (SOA) to deal with the non-return-to-zero (NRZ) signals and the differential quadrature phase-shift keying (DQPSK) signals in the nonlinear dense wavelength-division multiplexing (DWDM) channel. We optimized the key parameters of SOA-based RC to improve compensation performance. Based on the simulation investigation, we observed a significant improvement in signal quality over 10 dB compared to the distorted signals on each DWDM channel for both the NRZ and DQPSK transmission cases. The compatible OSP achieved by the proposed SOA-based RC could be a potential application of the optical switching node in the complex optical fiber communication system, where incoherent and coherent signals meet.

Effect of temporal resolution on the reproduction of chaotic dynamics via reservoir computing

Reservoir computing is a machine learning paradigm that uses a structure called a reservoir, which has nonlinearities and short-term memory. In recent years, reservoir computing has expanded to new functions such as the autonomous generation of chaotic time series, as well as time series prediction and classification. Furthermore, novel possibilities have been demonstrated, such as inferring the existence of previously unseen attractors. Sampling, in contrast, has a strong influence on such functions. Sampling is indispensable in a physical reservoir computer that uses an existing physical system as a reservoir because the use of an external digital system for the data input is usually inevitable. This study analyzes the effect of sampling on the ability of reservoir computing to autonomously regenerate chaotic time series. We found, as expected, that excessively coarse sampling degrades the system performance, but also that excessively dense sampling is unsuitable. Based on quantitative indicators that capture the local and global characteristics of attractors, we identify a suitable window of the sampling frequency and discuss its underlying mechanisms.

Photonic elementary cellular automata for simulation of complex phenomena

Cellular automata are a class of computational models based on simple rules and algorithms that can simulate a wide range of complex phenomena. However, when using conventional computers, these 'simple' rules are only encapsulated at the level of software. This can be taken one step further by simplifying the underlying physical hardware. Here, we propose and implement a simple photonic hardware platform for simulating complex phenomena based on cellular automata. Using this special-purpose computer, we experimentally demonstrate complex phenomena, including fractals, chaos, and solitons, which are typically associated with much more complex physical systems. The flexibility and programmability of our photonic computer present new opportunities to simulate and harness complexity for efficient, robust, and decentralized information processing using light.

High-resolution dynamic consistency analysis of photonic time-delay reservoir computer

We numerically investigate a time-delayed reservoir computer architecture based on a single-mode laser diode with optical injection and optical feedback. Through a high-resolution parametric analysis, we reveal unforeseen regions of high dynamic consistency. We demonstrate furthermore that the best computing performance is not achieved at the edge of consistency, as previously suggested in a coarser parametric analysis. This region of high consistency and optimal reservoir performances is highly sensitive to the data input modulation format.

Experimental demonstration of bandwidth enhancement in photonic time delay reservoir computing

Time delay reservoir computing (TDRC) using semiconductor lasers (SLs) has proven to be a promising photonic analog approach for information processing. One appealing property is that SLs subject to delayed optical feedback and external optical injection, allow for tuning the response bandwidth by changing the level of optical injection. Here we use strong optical injection, thereby expanding the SL's modulation response up to tens of gigahertz. Performing a nonlinear time series prediction task, we demonstrate experimentally that for appropriate operating conditions, our TDRC system can operate with sampling times as small as 11.72 ps, without sacrificing computational performance.

Numerical and experimental investigation of a dispersive optoelectronic oscillator for chaotic time-delay signature suppression

Chaos generation from a novel single-loop dispersive optoelectronic oscillator (OEO) with a broadband chirped fiber Bragg grating (CFBG) is numerically and experimentally investigated. The CFBG has much broader bandwidth than the chaotic dynamics such that its dispersion effect rather than filtering effect dominates the reflection. The proposed dispersive OEO exhibits chaotic dynamics when sufficient feedback strength is guaranteed. Suppression of chaotic time-delay signature (TDS) is observed as the feedback strength increases. The TDS can be further suppressed as the amount of grating dispersion increases. Without compromising bandwidth performance, our proposed system extends the parameter space of chaos, enhances the robustness to modulator bias variation, and improves TDS suppression by at least five times comparing to the classical OEO. Experimental results qualitatively agree well with numerical simulations. In addition, the advantage of dispersive OEO is further verified by experimentally demonstrating random bit generation with tunable rate up to 160 Gbps.

Real-time respiratory motion prediction using photonic reservoir computing

Respiration induced motion is a well-recognized challenge in many clinical practices including upper body imaging, lung tumor motion tracking and radiation therapy. In this work, we present a recurrent neural network algorithm that was implemented in a photonic delay-line reservoir computer (RC) for real-time respiratory motion prediction. The respiratory motion signals are quasi-periodic waveforms subject to a variety of non-linear distortions. In this work, we demonstrated for the first time that RC can be effective in predicting short to medium range of respiratory motions within practical timescales. A double-sliding window technology is explored to enable the real-time establishment of an individually trained model for each patient and the real-time processing of live-streamed respiratory motion data. A breathing dataset from a total of 76 patients with breathing speeds ranging from 3 to 20 breaths per minute (BPM) is studied. Motion prediction of look-ahead times of 66.6, 166.6, and 333 ms are investigated. With a 333 ms look-ahead time, the real-time RC model achieves an average normalized mean square error (NMSE) of 0.025, an average mean absolute error (MAE) of 0.34 mm, an average root mean square error (RMSE) of 0.45 mm, an average therapeutic beam efficiency (TBE) of 94.14% for an absolute error (AE) < 1 mm, and 99.89% for AE < 3 mm. This study demonstrates that real-time RC is an efficient computing framework for high precision respiratory motion prediction.

Reservoir computing with noise

This paper investigates in detail the effects of measurement noise on the performance of reservoir computing. We focus on an application in which reservoir computers are used to learn the relationship between different state variables of a chaotic system. We recognize that noise can affect the training and testing phases differently. We find that the best performance of the reservoir is achieved when the strength of the noise that affects the input signal in the training phase equals the strength of the noise that affects the input signal in the testing phase. For all the cases we examined, we found that a good remedy to noise is to low-pass filter the input and the training/testing signals; this typically preserves the performance of the reservoir, while reducing the undesired effects of noise.

Information-theoretical analysis of statistical measures for multiscale dynamics

Multiscale entropy (MSE) has been widely used to examine nonlinear systems involving multiple time scales, such as biological and economic systems. Conversely, Allan variance has been used to evaluate the stability of oscillators, such as clocks and lasers, ranging from short to long time scales. Although these two statistical measures were developed independently for different purposes in different fields, their interest lies in examining the multiscale temporal structures of physical phenomena under study. We demonstrate that from an information-theoretical perspective, they share some foundations and exhibit similar tendencies. We experimentally confirmed that similar properties of the MSE and Allan variance can be observed in low-frequency fluctuations (LFF) in chaotic lasers and physiological heartbeat data. Furthermore, we calculated the condition under which this consistency between the MSE and Allan variance exists, which is related to certain conditional probabilities. Heuristically, natural physical systems including the aforementioned LFF and heartbeat data mostly satisfy this condition, and hence, the MSE and Allan variance demonstrate similar properties. As a counterexample, we demonstrate an artificially constructed random sequence, for which the MSE and Allan variance exhibit different trends.

Editorial for the Special Issue on Advances in Optoelectronic Devices

Optoelectronic devices are fabricated based on an optoelectronic conversion effect, which is a developing research field of modern optoelectronic technology and microelectronics technology [...].

Optical neural ordinary differential equations

Increasing the layer number of on-chip photonic neural networks (PNNs) is essential to improve its model performance. However, the successive cascading of network hidden layers results in larger integrated photonic chip areas. To address this issue, we propose the optical neural ordinary differential equations (ON-ODEs) architecture that parameterizes the continuous dynamics of hidden layers with optical ODE solvers. The ON-ODE comprises the PNNs followed by the photonic integrator and optical feedback loop, which can be configured to represent residual neural networks (ResNets) and implement the function of recurrent neural networks with effectively reduced chip area occupancy. For the interference-based optoelectronic nonlinear hidden layer, the numerical experiments demonstrate that the single hidden layer ON-ODE can achieve approximately the same accuracy as the two-layer optical ResNets in image classification tasks. In addition, the ON-ODE improves the model classification accuracy for the diffraction-based all-optical linear hidden layer. The time-dependent dynamics property of ON-ODE is further applied for trajectory prediction with high accuracy.

Enhanced optoelectronic reservoir computation using semiconductor laser with double delay feedbacks

We numerically explored the enhanced performance and physical mechanism of semiconductor laser (SL) based reservoir computation (RC) with double optoelectronic feedback (DOEF). One-step and multistep Santa Fe time series predictions were used as standard test benchmarks in this work. We found that in the optimized parameter region the normalized mean square error (NMSE) of an SL-based RC under DOEF is smaller than an SL-based RC with single optoelectronic feedback (SOEF). In addition, the performance improvement is more obvious for multistep prediction, which is particularly suitable for more complex tasks that requires a higher memory capability (MC). The enriched node states (optical intensity of the virtual nodes for each sample) and the enhanced MC of the proposed DOEF were verified by a comparison to SOEF under the optimized feedback strength. The influence of the feedback strength and the delay difference on the NMSE and the MC was also investigated. Our study should be helpful in the design of a high-performance optoelectronic RC based on an SL.

Asynchronous photonic time-delay reservoir computing

Time-delay reservoir computing uses a nonlinear node associated with a feedback loop to construct a large number of virtual neurons in the neural network. The clock cycle of the computing network is usually synchronous with the delay time of the feedback loop, which substantially constrains the flexibility of hardware implementations. This work shows an asynchronous reservoir computing network based on a semiconductor laser with an optical feedback loop, where the clock cycle (20 ns) is considerably different to the delay time (77 ns). The performance of this asynchronous network is experimentally investigated under various operation conditions. It is proved that the asynchronous reservoir computing shows highly competitive performance on the prediction task of Santa Fe chaotic time series, in comparison with the synchronous counterparts.

Impact of filtering on photonic time-delay reservoir computing

We analyze the modification of the computational properties of a time-delay photonic reservoir computer with a change in its feedback bandwidth. For a reservoir computing configuration based on a semiconductor laser subject to filtered optoelectronic feedback, we demonstrate that bandwidth selection can lead to a flat-topped eigenvalue spectrum for which a large number of system frequencies are weakly damped as a result of the attenuation of modulational instability by feedback filtering. This spectral configuration allows for the optimization of the reservoir in terms of its memory capacity, while its computational ability appears to be only weakly affected by the characteristics of the filter.

Pseudolaminar chaos from on-off intermittency

In finite-dimensional, chaotic, Lorenz-like wave-particle dynamical systems one can find diffusive trajectories, which share their appearance with that of laminar chaotic diffusion [Phys. Rev. Lett. 128, 074101 (2022)0031-900710.1103/PhysRevLett.128.074101] known from delay systems with lag-time modulation. Applying, however, to such systems a test for laminar chaos, as proposed in [Phys. Rev. E 101, 032213 (2020)2470-004510.1103/PhysRevE.101.032213], these signals fail such a test, thus leading to the notion of pseudolaminar chaos. The latter can be interpreted as integrated periodically driven on-off intermittency. We demonstrate that, on a signal level, true laminar and pseudolaminar chaos are hardly distinguishable in systems with and without dynamical noise. However, very pronounced differences become apparent when correlations of signals and increments are considered. We compare and contrast these properties of pseudolaminar chaos with true laminar chaos.

Analysis of temporal structure of laser chaos by Allan variance

Allan variance has been widely utilized for evaluating the stability of the time series generated by atomic clocks and lasers, in time regimes ranging from short to extremely long. This multiscale examination capability of the Allan variance may also be beneficial in evaluating the chaotic oscillating dynamics of semiconductor lasers- not just for conventional phase stability analysis. In the present study, we demonstrated Allan variance analysis of the complex time series generated by a semiconductor laser with delayed feedback, including low-frequency fluctuations (LFFs), which exhibit both fast and slow dynamics. While the detection of LFFs is difficult with the conventional power spectrum analysis method in the low-frequency regime, the Allan variance approach clearly captured the appearance of multiple time-scale dynamics, such as LFFs. This study demonstrates that Allan variance can help in understanding and characterizing diverse laser dynamics, including LFFs, spanning a wide range of timescales.

Physical deep learning with biologically inspired training method: gradient-free approach for physical hardware

Ever-growing demand for artificial intelligence has motivated research on unconventional computation based on physical devices. While such computation devices mimic brain-inspired analog information processing, the learning procedures still rely on methods optimized for digital processing such as backpropagation, which is not suitable for physical implementation. Here, we present physical deep learning by extending a biologically inspired training algorithm called direct feedback alignment. Unlike the original algorithm, the proposed method is based on random projection with alternative nonlinear activation. Thus, we can train a physical neural network without knowledge about the physical system and its gradient. In addition, we can emulate the computation for this training on scalable physical hardware. We demonstrate the proof-of-concept using an optoelectronic recurrent neural network called deep reservoir computer. We confirmed the potential for accelerated computation with competitive performance on benchmarks. Our results provide practical solutions for the training and acceleration of neuromorphic computation.

Enhancement of calculation accuracy of the integrated photonic tensor flow processer by global optical power allocation

We present a global optical power allocation architecture, which can enhance the calculation accuracy of the integrated photonic tensor flow processor (PTFP). By adjusting the optical power splitting ratio according to the weight value and loss of each calculating unit, this architecture can efficiently use optical power so that the signal-to-noise ratio of the PTFP is enhanced. In the case of considering the on-chip optical delay line and spectral loss, the calculation accuracy measured in the experiment is enhanced by more than 1 bit compared with the fixed optical power allocation architecture.

Optical computing powers graph neural networks

Graph-based neural networks have promising perspectives but are limited by electronic bottlenecks. Our work explores the advantages of optical neural networks in the graph domain. We propose an optical graph neural network (OGNN) based on inverse-designed optical processing units (OPUs) to classify graphs with optics. The OPUs, combined with two types of optical components, can perform multiply-accumulate, matrix-vector multiplication, and matrix-matrix multiplication operations. The proposed OGNN can classify typical non-Euclidean MiniGCDataset graphs and successfully predict 1000 test graphs with 100% accuracy. The OPU-formed optical-electrical graph attention network is also scalable to handle more complex graph data, such as the Cora dataset, with 89.0% accuracy.

Controlling chaotic itinerancy in laser dynamics for reinforcement learning

Photonic artificial intelligence has attracted considerable interest in accelerating machine learning; however, the unique optical properties have not been fully used for achieving higher-order functionalities. Chaotic itinerancy, with its spontaneous transient dynamics among multiple quasi-attractors, can be used to realize brain-like functionalities. In this study, we numerically and experimentally investigate a method for controlling the chaotic itinerancy in a multimode semiconductor laser to solve a machine learning task, namely, the multiarmed bandit problem, which is fundamental to reinforcement learning. The proposed method uses chaotic itinerant motion in mode competition dynamics controlled via optical injection. We found that the exploration mechanism is completely different from a conventional searching algorithm and is highly scalable, outperforming the conventional approaches for large-scale bandit problems. This study paves the way to use chaotic itinerancy for effectively solving complex machine learning tasks as photonic hardware accelerators.

Classification of hyperchaotic, chaotic, and regular signals using single nonlinear node delay-based reservoir computers

The Lyapunov exponent method is generally used for classifying hyperchaotic, chaotic, and regular dynamics based on the equations modeling the system. However, several systems do not benefit from appropriate modeling underlying their dynamic behaviors. Therefore, having methods for classifying hyperchaotic, chaotic, and regular dynamics using only the observational data generated either by the theoretical or the experimental systems is crucial. In this paper, we use single nonlinear node delay-based reservoir computers to separate hyperchaotic, chaotic, and regular dynamics. We show that their classification capabilities are robust with an accuracy of up to 99.61% and 99.03% using the Mackey-Glass and the optoelectronic oscillator delay-based reservoir computers, respectively. Moreover, we demonstrate that the reservoir computers trained with the two-dimensional Hénon-logistic map can classify the dynamical state of another system (for instance, the two-dimensional sine-logistic modulation map). Our solution extends the state-of-the-art machine learning and deep learning approaches for chaos detection by introducing the detection of hyperchaotic signals.

Discrete-Time Signatures and Randomness in Reservoir Computing

A new explanation of the geometric nature of the reservoir computing (RC) phenomenon is presented. RC is understood in the literature as the possibility of approximating input-output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated, and bounds for the committed approximation error are provided.

Accurate separation of mixed high-dimension optical-chaotic signals using optical reservoir computing based on optically pumped VCSELs

In this work, with the mixing fractions being known in advance or unknown, the schemes and theories for the separations of two groups of the mixed optical chaotic signals are proposed in detail, using the VCSEL-based reservoir computing (RC) systems. Here, two groups of the mixed optical chaotic signals are linearly combined with many beams of the chaotic x-polarization components (X-PCs) and Y-PCs emitted by the optically pumped spin-VCSELs operation alone. Two parallel reservoirs are performed by using the chaotic X-PC and Y-PC output by the optically pumped spin-VCSEL with both optical feedback and optical injection. Moreover, we further demonstrate the separation performances of the mixed chaotic signal linearly combined with no more than three beams of the chaotic X-PC or Y-PC. We find that two groups of the mixed optical chaos signals can be effectively separated by using two reservoirs in single RC system based on optically pumped Spin-VCSEL and their corresponding separated errors characterized by the training errors are no more than 0.093, when the mixing fractions are known as a certain value in advance. If the mixing fractions are unknown, we utilize two cascaded RC systems based on optically pumped Spin-VCSELs to separate each group of the mixed optical signals. The mixing fractions can be accurate predicted by using two parallel reservoirs in the first RC system. Based on the values of the predictive mixing fractions, two groups of the mixed optical chaos signals can be effectively separated by utilizing two parallel reservoirs in the second RC system, and their separated errors also are no more than 0.093. In the same way, the mixed optical chaos signal linearly superimposed with more than three beams of optical chaotic signals can be effectively separated. The method and idea for separation of complex optical chaos signals proposed by this paper may provide an impact to development of novel principles of multiple access and demultiplexing in multi-channel chaotic cryptography communication.

Gel Biter: food texture discriminator based on physical reservoir computing with multiple soft materials

The human oral structure contains organs with distinctly different physical properties, such as teeth, gums, and tongues. When food enters the oral cavity, we can recognize the tactile sensation and shape of the object from multiple perspectives through the texture of the teeth and tongue. Therefore, it is possible to regard oral structures as a group of tactile sensors based on these functions. In this study, we developed a soft-matter artificial mouth that can accurately detect subtle differences in texture by creating and combining oral structural organs using polymer materials with different physical properties and mounting them as end-effectors for a robot arm. The same piezoelectric film sensor was embedded inside each organ, making it possible to acquire tactile sensations from the same object as completely different signal waveforms. We tested whether the sensor data obtained from each soft-matter material could be used for excellent object recognition by applying various machine learning methods. In an actual experiment, we learned the waveform data obtained from chewing sweets and snacks, such as rice crackers, and applied machine learning to classify the data, which led to an accuracy rate of over 90%.

Guiding principle of reservoir computing based on "small-world" network

Reservoir computing is a computational framework of recurrent neural networks and is gaining attentions because of its drastically simplified training process. For a given task to solve, however, the methodology has not yet been established how to construct an optimal reservoir. While, "small-world" network has been known to represent networks in real-world such as biological systems and social community. This network is categorized amongst those that are completely regular and totally disordered, and it is characterized by highly-clustered nodes with a short path length. This study aims at providing a guiding principle of systematic synthesis of desired reservoirs by taking advantage of controllable parameters of the small-world network. We will validate the methodology using two different types of benchmark tests-classification task and prediction task.

Fast physical random bit generation based on a chaotic optical injection system with multi-path optical feedback

Based on the chaotic signal provided by a simple chaotic system, a random bit sequence with a rate of 640 Gb/s is generated through adopting the circulating exclusive-or (CXOR) post-processing method. Such a simple chaotic system is built via a slave semiconductor laser subject to optical injection of a chaotic signal originated from a master semiconductor laser under multi-path optical feedback. First, through inspecting the dependences of the time-delay-signature (TDS) and bandwidth of the chaotic signal on some key operation parameters, optimized parameters are determined for generating a high-quality chaotic signal with a large bandwidth and low TDS. Second, the high-quality chaotic signal is converted to an 8-bit digital signal by sampling with a digital oscilloscope at 80 GSa/s. Next, through adopting the CXOR post-processing method, a bit sequence with a rate of 640 Gb/s is obtained. Finally, the randomness is estimated by the National Institute of Standard Technology (NIST) Special Publication 800-22 statistical tests, and the results demonstrate that the obtained random bit sequence can pass all the NIST tests.

Enhanced prediction performance of a time-delay reservoir computing system based on a VCSEL by dual-training method

In this paper, a new dual-training method for a time-delay reservoir computing (RC) system based on a single vertical-cavity surface-emitting laser (VCSEL) is proposed and demonstrated experimentally for the first time. The prediction performance of the RC system by using the dual-training method has been experimentally and numerically investigated. Here, the dual-training method is defined as performing a further RC based on the difference between the target value and the predicted value of the traditional single training. It is found that enhanced prediction performance of the RC system can be obtained by employing the dual-training method, compared to the traditional single training method. More specifically, the NMSE values of the RC system with the dual-training method applied can be improved to 760% compared with the single training method in experiments. Besides, the effects of injection power, bias currents, feedback strength, and frequency detuning are also considered. The proposed dual-training method is of great significance to the performance enhancement of the RC and has an important promotion effect on the application of the RC in the future.

Time shifts to reduce the size of reservoir computers

A reservoir computer is a type of dynamical system arranged to do computation. Typically, a reservoir computer is constructed by connecting a large number of nonlinear nodes in a network that includes recurrent connections. In order to achieve accurate results, the reservoir usually contains hundreds to thousands of nodes. This high dimensionality makes it difficult to analyze the reservoir computer using tools from the dynamical systems theory. Additionally, the need to create and connect large numbers of nonlinear nodes makes it difficult to design and build analog reservoir computers that can be faster and consume less power than digital reservoir computers. We demonstrate here that a reservoir computer may be divided into two parts: a small set of nonlinear nodes (the reservoir) and a separate set of time-shifted reservoir output signals. The time-shifted output signals serve to increase the rank and memory of the reservoir computer, and the set of nonlinear nodes may create an embedding of the input dynamical system. We use this time-shifting technique to obtain excellent performance from an opto-electronic delay-based reservoir computer with only a small number of virtual nodes. Because only a few nonlinear nodes are required, construction of a reservoir computer becomes much easier, and delay-based reservoir computers can operate at much higher speeds.