Laser dynamical reservoir computing with consistency: an approach of a chaos mask signal

Photonic deep residual time-delay reservoir computing

Time-delay reservoir computing (TDRC) represents a simplified variant of recurrent neural networks, employing a nonlinear node with a feedback mechanism to construct virtual nodes. The capabilities of TDRC can be enhanced by transitioning to a deep architecture. In this work, we propose a novel photonic deep residual TDRC (DR-TDRC) with augmented capabilities. The additional time delay added to the residual structure enables DR-TDRC superior to traditional deep structures across various benchmark tasks, especially in memory capability and almost an order of magnitude improvement in nonlinear channel equalization. Additionally, a specifically designed clipping algorithm is utilized to counteract the damage of redundant layers in deep structures, enabling the extension of the deep TDRC to dozens rather than just a few layers, with higher performance. We experimentally demonstrate the proof-of-concept with a 4-layer DR-TDRC containing 960 interrelated neurons (240 neurons per layer), based on four injection-locked distributed feedback lasers. We confirm the potential for scalable deep RC with elevated performance. Our results provide a feasible approach for expanding deep photonic computing to satisfy the boosting demand for artificial intelligence.

Boosting reservoir computer performance with multiple delays

Time delays play a significant role in dynamical systems, as they affect their transient behavior and the dimensionality of their attractors. The number, values, and spacing of these time delays influences the eigenvalues of a nonlinear delay-differential system at its fixed point. Here we explore a multidelay system as the core computational element of a reservoir computer making predictions on its input in the usual regime close to fixed point instability. Variations in the number and separation of time delays are first examined to determine the effect of such parameters of the delay distribution on the effectiveness of time-delay reservoirs for nonlinear time series prediction. We demonstrate computationally that an optoelectronic device with multiple different delays can improve the mapping of scalar input into higher-dimensional dynamics, and thus its memory and prediction capabilities for input time series generated by low- and high-dimensional dynamical systems. In particular, this enhances the suitability of such reservoir computers for predicting input data with temporal correlations. Additionally, we highlight the pronounced harmful resonance condition for reservoir computing when using an electro-optic oscillator model with multiple delays. We illustrate that the resonance point may shift depending on the task at hand, such as cross prediction or multistep ahead prediction, in both single delay and multiple delay cases.

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.

Broadband chaos generation in a distributed-feedback laser by selecting residual side modes

Chaotic dynamics with spectral broadening is experimentally obtained by selective excitation of residual side modes in a distributed-feedback (DFB) laser. For the single-mode laser that emits only at the main mode when free-running, feedback to a residual side mode is introduced via a fiber Bragg grating (FBG). The FBG feedback suppresses the main mode, selectively excites the residual side mode, and generates broadband chaotic dynamics. Such a chaos of the residual side mode has a broad electrical bandwidth reaching at least 26 GHz, which corresponds to a significant broadening by over 50% when compared with the main mode. The dynamics are attributed entirely to the one selected mode without invoking multimode interactions. The wavelength is tunable beyond 10 nm by using different FBGs. Through avoiding multimode interactions, this approach of broadband chaos generation is potentially simple to model and thus promising for applications.

Redox-based ion-gating reservoir consisting of (104) oriented LiCoO2 film, assisted by physical masking

Reservoir computing (RC) is a machine learning framework suitable for processing time series data, and is a computationally inexpensive and fast learning model. A physical reservoir is a hardware implementation of RC using a physical system, which is expected to become the social infrastructure of a data society that needs to process vast amounts of information. Ion-gating reservoirs (IGR) are compact and suitable for integration with various physical reservoirs, but the prediction accuracy and operating speed of redox-IGRs using WO3 as the channel are not sufficient due to irreversible Li+ trapping in the WO3 matrix during operation. Here, in order to enhance the computation performance of redox-IGRs, we developed a redox-based IGR using a (104) oriented LiCoO2 thin film with high electronic and ionic conductivity as a trap-free channel material. The subject IGR utilizes resistance change that is due to a redox reaction (LiCoO2 ⟺ Li1-xCoO2 + xLi+ + xe-) with the insertion and desertion of Li+. The prediction error in the subject IGR was reduced by 72% and the operation speed was increased by 4 times compared to the previously reported WO3, which changes are due to the nonlinear and reversible electrical response of LiCoO2 and the high dimensionality enhanced by a newly developed physical masking technique. This study has demonstrated the possibility of developing high-performance IGRs by utilizing materials with stronger nonlinearity and by increasing output dimensionality.

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.

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.

Optical cascaded reservoir computing for realization of dual-channel high-speed OTDM chaotic secure communication via four optically pumped VCSEL

In this work, we propose a chaotic secure communication system with optical time division multiplexing (OTDM), using two cascaded reservoir computing systems based on multi beams of chaotic polarization components emitted by four optically pumped VCSELs. Here, each level of reservoir layer includes four parallel reservoirs, and each parallel reservoir contains two sub-reservoirs. When the reservoirs in the first-level reservoir layer are well trained and the training errors are far less than 0.1, each group of chaotic masking signals can be effectively separated. When the reservoirs in the second reservoir layer are effectively trained and the training errors are far less than 0.1, the output for each reservoir can be well synchronized with the corresponding original delay chaotic carrier-wave. Here, the synchronization quality between them can be characterized by the correlation coefficients of more than 0.97 in different parameter spaces of the system. Under these high-quality synchronization conditions, we further discuss the performances of dual-channel OTDM with a rate of 4×60 Gb/s. By observing the eye diagram, bit error rate and time-waveform of each decoded message in detail, we find that there is a large eye-openings in the eye diagrams, low bit error rate and higher quality time-waveform for each decoded message. Except that the bit error rate of one decoded message is lower than 7 × 10^-3 in different parameter spaces, and those of the other decoded messages are close to 0, indicating that high-quality data transmissions are expected to be realized in the system. The research results show that the multi-cascaded reservoir computing systems based on multiple optically pumped VCSELs provide an effective method for the realization of multi-channel OTDM chaotic secure communications with high-speed.

Experimental study of spatial and temporal coherence in a laser diode with optical feedback

Optical feedback can reduce the linewidth of a semiconductor laser by several orders of magnitude, but it can also cause line broadening. Although these effects on the temporal coherence of the laser are well known, a good understanding of the effects of feedback on the spatial coherence is still lacking. Here we present an experimental technique that allows discriminating the effects of feedback on temporal and spatial coherence of the laser beam. We analyze the output of a commercial edge-emitting laser diode, comparing the contrast of speckle images recorded using a multimode (MM) or single mode (SM) fiber and an optical diffuser, and also, comparing the optical spectra at the end of the MM or SM fiber. Optical spectra reveal feedback-induced line broadening, while speckle analyses reveal reduced spatial coherence due to feedback-excited spatial modes. These modes reduce the speckle contrast (SC) up to 50% when speckle images are recorded using the MM fiber, but do not affect the SC when the images are recorded using the SM fiber and diffuser, because the spatial modes that are excited by the feedback are filtered out by the SM fiber. This technique is generic and can be used to discriminate spatial and temporal coherence of other types of lasers and under other operating conditions that can induce a chaotic output.

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.

Chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection

Photonic computing has attracted increasing interest for the acceleration of information processing in machine learning applications. The mode-competition dynamics of multimode semiconductor lasers are useful for solving the multi-armed bandit problem in reinforcement learning for computing applications. In this study, we numerically evaluate the chaotic mode-competition dynamics in a multimode semiconductor laser with optical feedback and injection. We observe the chaotic mode-competition dynamics among the longitudinal modes and control them by injecting an external optical signal into one of the longitudinal modes. We define the dominant mode as the mode with the maximum intensity; the dominant mode ratio for the injected mode increases as the optical injection strength increases. We deduce that the characteristics of the dominant mode ratio in terms of the optical injection strength are different among the modes owing to the different optical feedback phases. We propose a control technique for the characteristics of the dominant mode ratio by precisely tuning the initial optical frequency detuning between the optical injection signal and injected mode. We also evaluate the relationship between the region of the large dominant mode ratios and the injection locking range. The region with the large dominant mode ratios does not correspond to the injection-locking range. The control technique of chaotic mode-competition dynamics in multimode lasers is promising for applications in reinforcement learning and reservoir computing in photonic artificial intelligence.

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.

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.

Nanophotonic reservoir computing for COVID-19 pandemic forecasting

The coronavirus disease 2019 (COVID-19) has spread worldwide in unprecedented speed, and diverse negative impacts have seriously endangered human society. Accurately forecasting the number of COVID-19 cases can help governments and public health organizations develop the right prevention strategies in advance to contain outbreaks. In this work, a long-term 6-month COVID-19 pandemic forecast in second half of 2021 and a short-term 30-day daily ahead COVID-19 forecast in December 2021 are successfully implemented via a novel nanophotonic reservoir computing based on silicon optomechanical oscillators with photonic crystal cavities, benefitting from its simpler learning algorithm, abundant nonlinear characteristics, and some unique advantages such as CMOS compatibility, fabrication cost, and monolithic integration. In essence, the nonlinear time series related to COVID-19 are mapped to the high-dimensional nonlinear space by the optical nonlinear properties of nanophotonic reservoir computing. The testing-dataset forecast results of new cases, new deaths, cumulative cases, and cumulative deaths for six countries demonstrate that the forecasted blue curves are awfully close to the real red curves with exceedingly small forecast errors. Moreover, the forecast results commendably reflect the variations of the actual case data, revealing the different epidemic transmission laws in developed and developing countries. More importantly, the daily ahead forecast results during December 2021 of four kinds of cases for six countries illustrate that the daily forecasted values are highly coincident with the real values, while the relevant forecast errors are tiny enough to verify the good forecasting competence of COVID-19 pandemic dominated by Omicron strain. Therefore, the implemented nanophotonic reservoir computing can provide some foreknowledge on prevention strategy and healthcare management for COVID-19 pandemic.

Simultaneous modulation format identification and OSNR monitoring based on optoelectronic reservoir computing

An approach for simultaneous modulation format identification (MFI) and optical signal-to-noise ratio (OSNR) monitoring in digital coherent optical communications is proposed based on optoelectronic reservoir computing (RC) and the signal's amplitude histograms (AHs) obtained after the adaptive post-equalization. The optoelectronic RC is implemented using a Mach-Zehnder modulator and optoelectronic delay feedback loop. We investigate the performance of the proposed model with the number of symbols, bins of AHs and the hyperparameters of optoelectronic RC. The results show that 100% MFI accuracy can be achieved simultaneously with accurate OSNR estimation for different modulation formats under study. The lowest achievable OSNR estimation mean absolute errors for the dual-polarization (DP)-quadrature phase-shift keying signal, the DP-16-ary quadrature amplitude modulation (16QAM) signal, and the DP-64QAM signal are 0.2 dB, 0.32 dB and 0.53 dB, respectively. The robustness of the proposed scheme is also evaluated when the optoelectronic RC is in presence of additive white Gaussian noises. Then, a proof of concept experiment is demonstrated to further verify our proposed method. The proposed approach offers a potential solution for next-generation intelligent optical performance monitoring in the physical layer.

Input-driven chaotic dynamics in vortex spin-torque oscillator

A new research topic in spintronics relating to the operation principles of brain-inspired computing is input-driven magnetization dynamics in nanomagnet. In this paper, the magnetization dynamics in a vortex spin-torque oscillator driven by a series of random magnetic field are studied through a numerical simulation of the Thiele equation. It is found that input-driven synchronization occurs in the weak perturbation limit, as found recently. As well, chaotic behavior is newly found to occur in the vortex core dynamics for a wide range of parameters, where synchronized behavior is disrupted by an intermittency. Ordered and chaotic dynamical phases are examined by evaluating the Lyapunov exponent. The relation between the dynamical phase and the computational capability of physical reservoir computing is also studied.

Edge-of-chaos learning achieved by ion-electron-coupled dynamics in an ion-gating reservoir

Physical reservoir computing has recently been attracting attention for its ability to substantially reduce the computational resources required to process time series data. However, the physical reservoirs that have been reported to date have had insufficient computational capacity, and most of them have a large volume, which makes their practical application difficult. Here, we describe the development of a Li⁺ electrolyte-based ion-gating reservoir (IGR), with ion-electron-coupled dynamics, for use in high-performance physical reservoir computing. A variety of synaptic responses were obtained in response to past experience, which were stored as transient charge density patterns in an electric double layer, at the Li⁺ electrolyte/diamond interface. Performance for a second-order nonlinear dynamical equation task is one order of magnitude higher than memristor-based reservoirs. The edge-of-chaos state of the IGR enabled the best computational capacity. The IGR described here opens the way for high-performance and integrated neural network devices.

Image recognition based on optical reservoir computing

We propose an image recognition approach using a single physical node based optical reservoir computing. Specifically, an optically injected semiconductor laser with self-delayed feedback is used as the reservoir. We perform a handwritten-digit recognition task by greatly increasing the number of virtual nodes in delayed feedback using outputs from multiple delay times. Final simulation results confirm that the recognition accuracy can reach 99% after systematically optimizing the reservoir hyperparameters. Due to its simple architecture, this scheme may provide a resource-efficient alternative approach to image recognition.

Misestimate of the performance in VCSEL-based reservoir computing systems with optical information injection by high surface reflectivity

In reservoir computing (RC) systems based on semiconductor lasers (SLs), the information that must be processed usually enters the reservoir through optical injection. Part of the injection information directly reflected by the front facet of the SLs is inevitably hybridized into the output of the SLs and contributes to the state of virtual nodes. For an RC system based on vertical-cavity surface-emitting lasers (VCSELs), the proportion of the reflected information coupled to the laser output is relatively huge due to the high surface reflectivity. Thus the influence of the directly reflected information will be much more obvious. Using a Santa Fe chaotic time series prediction task and waveform recognition task, we theoretically investigate the influence of high front facet reflectivity on the evaluation of the performance of a VCSEL-based RC system with optical information injection. The simulation results demonstrate that, after taking the directly reflected information into account, a lower error rate is obtained for each benchmark task. The physical mechanism to misestimate the RC performance has been studied through memory correlation and a statistical histogram of virtual node states.

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.

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.

Deep optical reservoir computing and chaotic synchronization predictions based on the cascade coupled optically pumped spin-VCSELs

In this work, we utilize two cascade coupling modes (unidirectional coupling and bidirectional coupling) to construct a four-layer deep reservoir computing (RC) system based on the cascade coupled optically-pumped spin-VCSEL. In such a system, there are double sub-reservoirs in each layer, which are formed by the chaotic x-PC and y-PC emitted by the reservoir spin-VCSEL in each layer. Under these two coupling modes, the chaotic x-PC and y-PC emitted by the driving optically-pumped spin-VCSEL (D-Spin-VCSEL), as two learning targets, are predicted by utilizing the four-layer reservoirs. In different parameter spaces, it is further explored that the outputs of the double sub-reservoirs in each layer are respectively synchronized with the chaotic x-PC and y-PC emitted by the D-Spin-VCSEL. The memory capacities (MCs) for the double sub-reservoirs in each layer are even further investigated. The results show that under two coupling modes, the predictions of the double sub-reservoirs with higher-layer for these two targets have smaller errors, denoting that the higher-layer double sub-reservoirs possess better predictive learning ability. Under the same system parameters, the outputs of the higher-layer dual parallel reservoirs are better synchronized with two chaotic PCs emitted by the D-Spin-VCSEL, respectively. The larger MCs can also be obtained by the higher-layer double reservoirs. In particular, compared with the four-layer reservoir computing system under unidirectional coupling, the four-layer reservoir computing system under bidirectional coupling shows better predictive ability in the same parameter space. The chaotic synchronizations predicted by each layer double sub-reservoirs under bidirectional coupling can be obtained higher qualities than those under unidirectional coupling. By the optimization of the system parameters, the outputs of the fourth-layer double sub-reservoirs are almost completely synchronized with the chaotic x-PC and y-PC emitted by the D-Spin-VCSEL, respectively, due to their correlation coefficient used to measure synchronization quality can be obtained as 0.99. These results have potential applications in chaotic computation, chaotic secure communication and accurate prediction of time series.

Reservoir computing based on an external-cavity semiconductor laser with optical feedback modulation

We numerically and experimentally investigate reservoir computing based on a single semiconductor laser with optical feedback modulation. In this scheme, an input signal is injected into a semiconductor laser via intensity or phase modulation of the optical feedback signal. We perform a chaotic time-series prediction task using the reservoir and compare the performances of intensity and phase modulation schemes. Our results indicate that the feedback signal of the phase modulation scheme outperforms that of the intensity modulation scheme. Further, we investigate the performance dependence of reservoir computing on parameter values and observe that the prediction error improves for large injection currents, unlike the results in a semiconductor laser with an optical injection input. The physical origin of the superior performance of the phase modulation scheme is analyzed using external cavity modes obtained from steady-state analysis in the phase space. The analysis indicates that high-dimensional mapping can be achieved from the input signal to the trajectory of the response laser output by using phase modulation of the feedback signal.

Dynamic Nonlinear Behavior of Ionic Liquid-Based Reservoir Computing Devices

Herein, a physical reservoir device that uses faradaic currents generated by redox reactions of metal ions in ionic liquids was developed. Synthetic time-series data consisting of randomly arranged binary number sequences ("1" and "0") were applied as isosceles-triangular voltage pulses with positive and negative voltage heights, respectively, and the effects of the faradaic current on short-term memory and parity-check task accuracies were verified. The current signal for the first half of the triangular voltage-pulse period, which contained a much higher faradaic current component compared to that of the second half of the triangular voltage-pulse period, enabled higher short-term memory task accuracy. Furthermore, when parity-check tasks were performed using a faradaic current generated by asymmetric triangular voltage-pulse levels of 1 and 0, the parity-check task accuracy was approximately eight times higher than that of the symmetric triangular voltage pulse in terms of the correlation coefficient between the output signal and target data. These results demonstrate the advantage of the faradaic current on both the short-term memory characteristics and nonlinear conversion capabilities and are expected to provide guidance for designing and controlling various physical reservoir devices that utilize electrochemical reactions.

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.

Spintronic reservoir computing without driving current or magnetic field

Recent studies have shown that nonlinear magnetization dynamics excited in nanostructured ferromagnets are applicable to brain-inspired computing such as physical reservoir computing. The previous works have utilized the magnetization dynamics driven by electric current and/or magnetic field. This work proposes a method to apply the magnetization dynamics driven by voltage control of magnetic anisotropy to physical reservoir computing, which will be preferable from the viewpoint of low-power consumption. The computational capabilities of benchmark tasks in single MTJ are evaluated by numerical simulation of the magnetization dynamics and found to be comparable to those of echo-state networks with more than 10 nodes.

Forecasting stock market with nanophotonic reservoir computing system based on silicon optomechanical oscillators

The essence of stock market forecasting is to reveal the intrinsic operation rules of stock market, however it is a terribly arduous challenge for investors. The application of nanophotonic technology in the intelligence field provides a new approach for stock market forecasting with its unique advantages. In this work, a novel nanophotonic reservoir computing (RC) system based on silicon optomechanical oscillators (OMO) with photonic crystal (PhC) cavities for stock market forecasting is implemented. The long-term closing prices of four representative stock indexes are accurately forecast with small prediction errors, and the forecasting results with distinct characteristics are exhibited in the mature stock market and emerging stock market separately. Our work offers solutions and suggestions for surmounting the concept drift problem in stock market environment. The comprehensive influence of RC parameters on forecasting performance are displayed via the mapping diagrams, while some intriguing results indicate that the mature stock markets are more sensitive to the variation of RC parameters than the emerging stock markets. Furthermore, the direction trend forecasting results illustrate that our system has certain direction forecasting ability. Additionally, the stock forecasting problem with short listing time and few data in the stock market is solved through transfer learning (TL) in stock sector. The generalization ability (GA) of our nanophotonic reservoir computing system is also verified via four stocks in the same region and industry. Therefore, our work contributes to a novel RC model for stock market forecasting in the nanophotonic field, and provides a new prototype system for more applications in the intelligent information processing field.

Consistency Hierarchy of Reservoir Computers

We study the propagation and distribution of information-carrying signals injected in dynamical systems serving as reservoir computers. Through different combinations of repeated input signals, a multivariate correlation analysis reveals measures known as the consistency spectrum and consistency capacity. These are high-dimensional portraits of the nonlinear functional dependence between input and reservoir state. For multiple inputs, a hierarchy of capacities characterizes the interference of signals from each source. For an individual input, the time-resolved capacities form a profile of the reservoir's nonlinear fading memory. We illustrate this methodology for a range of echo state networks.

High-Speed Reservoir Computing Based on Circular-Side Hexagonal Resonator Microlaser with Optical Feedback

In the current environment of the explosive growth in the amount of information, the demand for efficient information-processing methods has become increasingly urgent. We propose and numerically investigate a delay-based high-speed reservoir computing (RC) using a circular-side hexagonal resonator (CSHR) microlaser with optical feedback and injection. In this RC system, a smaller time interval can be obtained between virtual nodes, and a higher information processing rate (Rinf) can also be achieved, due to the ultra-short photon lifetime and wide bandwidth of the CSHR microlaser. The performance of the RC system was tested with three benchmark tasks (Santa-Fe chaotic time series prediction task, the 10th order Nonlinear Auto Regressive Moving Average task and Nonlinear channel equalization task). The results show that the system achieves high-accuracy prediction, even with a small number of virtual nodes (25), and is more feasible, with lower requirements for arbitrary waveform generators at the same rate. Significantly, at the high rate of 10 Gbps, low error predictions can be achieved over a large parameter space (e.g., frequency detuning in the interval 80 GHz, injected strength in the range of 0.9 variation and 2% range for feedback strength). Interestingly, it has the potential to achieve Rinf of 25 Gbps under technical advancements. Additionally, its shorter external cavity length and cubic micron scale size make it an excellent choice for large-scale photonic integration reservoir computing.

Influence of the input signal's phase modulation on the performance of optical delay-based reservoir computing using semiconductor lasers

In photonic reservoir computing, semiconductor lasers with delayed feedback have shown to be suited to efficiently solve difficult and time-consuming problems. The input data in this system is often optically injected into the reservoir. Based on numerical simulations, we show that the performance depends heavily on the way that information is encoded in this optical injection signal. In our simulations we compare different input configurations consisting of Mach-Zehnder modulators and phase modulators for injecting the signal. We observe far better performance on a one-step ahead time-series prediction task when modulating the phase of the injected signal rather than only modulating its amplitude.

Adaptive time-delayed photonic reservoir computing based on Kalman-filter training

We propose an adaptive time-delayed photonic reservoir computing (RC) structure by utilizing the Kalman filter (KF) algorithm as training approach. Two benchmark tasks, namely the Santa Fe time-series prediction and the nonlinear channel equalization, are adopted to evaluate the performance of the proposed RC structure. The simulation results indicate that with the contribution of adaptive KF training, the prediction and equalization performance for the benchmark tasks can be significantly enhanced, with respect to the conventional RC using a training approach based on the least-squares (LS). Moreover, by introducing a complex mask derived from a bandwidth and complexity enhanced chaotic signal into the proposed RC, the performance of prediction and equalization can be further improved. In addition, it is demonstrated that the proposed RC system can provide a better equalization performance for the parameter-variant wireless channel equalization task, compared with the conventional RC based on LS training. The work presents a potential way to realize adaptive photonic computing.

Processing-Speed Enhancement in a Delay-Laser-Based Reservoir Computer by Optical Injection

A delay-laser-based reservoir computer (RC) usually has its processing speed limited by the transient response of laser dynamics. Here, we study a simple all-optical approach to enhancing the processing speed by introducing optical injection to the reservoir layer of conventional RC that consists of a semiconductor laser with a delay loop. Using optical injection, the laser’s transient response effectively accelerates due to the speeded carrier-photon resonance. In the chaotic time-series prediction task, the proposed RC achieves good performance in a flexible range of injection detuning frequency under sufficient injection rate. Using proper injection parameters, the prediction error is significantly reduced and stabilized when using high processing speed. For achieving a prediction error below 0.006, the optical injection enhances the processing speed by an order of magnitude of about 5 GSample/s. Moreover, the proposed RC extends the advantage to the handwritten digit recognition task by achieving better word error rate.

Dynamic effects on reservoir computing with a Hopf oscillator

Limit cycle oscillators have the potential to be resourced as reservoir computers due to their rich dynamics. Here, a Hopf oscillator is used as a physical reservoir computer by discarding the delay line and time-multiplexing procedure. A parametric study is used to uncover computational limits imposed by the dynamics of the oscillator using parity and chaotic time-series prediction benchmark tasks. Resonance, frequency ratios from the Farey sequence, and Arnold tongues were found to strongly affect the computation ability of the reservoir. These results provide insights into fabricating physical reservoir computers from limit cycle systems.

Photonic reinforcement learning based on optoelectronic reservoir computing

Reinforcement learning has been intensively investigated and developed in artificial intelligence in the absence of training data, such as autonomous driving vehicles, robot control, internet advertising, and elastic optical networks. However, the computational cost of reinforcement learning with deep neural networks is extremely high and reducing the learning cost is a challenging issue. We propose a photonic on-line implementation of reinforcement learning using optoelectronic delay-based reservoir computing, both experimentally and numerically. In the proposed scheme, we accelerate reinforcement learning at a rate of several megahertz because there is no required learning process for the internal connection weights in reservoir computing. We perform two benchmark tasks, CartPole-v0 and MountanCar-v0 tasks, to evaluate the proposed scheme. Our results represent the first hardware implementation of reinforcement learning based on photonic reservoir computing and pave the way for fast and efficient reinforcement learning as a novel photonic accelerator.

A Hopf physical reservoir computer

Physical reservoir computing utilizes a physical system as a computational resource. This nontraditional computing technique can be computationally powerful, without the need of costly training. Here, a Hopf oscillator is implemented as a reservoir computer by using a node-based architecture; however, this implementation does not use delayed feedback lines. This reservoir computer is still powerful, but it is considerably simpler and cheaper to implement as a physical Hopf oscillator. A non-periodic stochastic masking procedure is applied for this reservoir computer following the time multiplexing method. Due to the presence of noise, the Euler-Maruyama method is used to simulate the resulting stochastic differential equations that represent this reservoir computer. An analog electrical circuit is built to implement this Hopf oscillator reservoir computer experimentally. The information processing capability was tested numerically and experimentally by performing logical tasks, emulation tasks, and time series prediction tasks. This reservoir computer has several attractive features, including a simple design that is easy to implement, noise robustness, and a high computational ability for many different benchmark tasks. Since limit cycle oscillators model many physical systems, this architecture could be relatively easily applied in many contexts.

Optical chaos generated in semiconductor lasers with intensity-modulated optical injection:a numerical study

We numerically report on an optical chaos signal generation scheme based on a semiconductor laser subject to intensity-modulated (IM) optical injection. In this scheme, the characteristics of the chaos signal obtained by destabilizing period-one nonlinear dynamics are numerically investigated. With the aid of bifurcation diagrams and the 0-1 tests for chaos, the chaotic dynamics excited by continuous-wave and IM optical injection are located, and the effects of injection and modulation parameters on chaotic regions are illustrated. Moreover, effective bandwidths and auto-correlation characteristics of chaos signals from the IM optical injection system are systematically investigated. The results show that although chaotic signals under the IM optical injection scenario have a limitation in unambiguous range detection in most parameter regions, wideband chaotic dynamics in large injection and modulation parameter regions can be easily achieved. This study paves the way for potential applications requiring no time-delay signature and broad bandwidth chaos, such as high-speed chaos communications and random bit generation.

Enhancing Performance of Reservoir Computing System Based on Coupled MEMS Resonators

Reservoir computing (RC) is an attractive paradigm of a recurrent neural network (RNN) architecture, owning to the ease of training and existing neuromorphic implementation. Its simulated performance matches other digital algorithms on a series of benchmarking tasks, such as prediction tasks and classification tasks. In this article, we propose a novel RC structure based on the coupled MEMS resonators with the enhanced dynamic richness to optimize the performance of the RC system both on the system level and data set level. Moreover, we first put forward that the dynamic richness of RC comprises linear dynamic richness and nonlinear dynamic richness, which can be enhanced by adding delayed feedbacks and nonlinear nodes, respectively. In order to set forth this point, we compare three typical RC structures, a single-nonlinearity RC structure with single-feedback, a single-nonlinearity RC structure with double-feedbacks, and the couple-nonlinearity RC structure with double-feedbacks. Specifically, four different tasks are enumerated to verify the performance of the three RC structures, and the results show the enhanced dynamic richness by adding delayed feedbacks and nonlinear nodes. These results prove that coupled MEMS resonators offer an interesting platform to implement a complex computing paradigm leveraging their rich dynamical features.

Fast physical repetitive patterns generation for masking in time-delay reservoir computing

Albeit the conceptual simplicity of hardware reservoir computing, the various implementation schemes that have been proposed so far still face versatile challenges. The conceptually simplest implementation uses a time delay approach, where one replaces the ensemble of nonlinear nodes with a unique nonlinear node connected to a delayed feedback loop. This simplification comes at a price in other parts of the implementation; repetitive temporal masking sequences are required to map the input information onto the diverse states of the time delay reservoir. These sequences are commonly introduced by arbitrary waveform generators which is an expensive approach when exploring ultra-fast processing speeds. Here we propose the physical generation of clock-free, sub-nanosecond repetitive patterns, with increased intra-pattern diversity and their use as masking sequences. To that end, we investigate numerically a semiconductor laser with a short optical feedback cavity, a well-studied dynamical system that provides a wide diversity of emitted signals. We focus on those operating conditions that lead to a periodic signal generation, with multiple harmonic frequency tones and sub-nanosecond limit cycle dynamics. By tuning the strength of the different frequency tones in the microwave domain, we access a variety of repetitive patterns and sample them in order to obtain the desired masking sequences. Eventually, we apply them in a time delay reservoir computing approach and test them in a nonlinear time-series prediction task. In a performance comparison with masking sequences that originate from random values, we find that only minor compromises are made while significantly reducing the instrumentation requirements of the time delay reservoir computing system.

Predictive learning of multi-channel isochronal chaotic synchronization by utilizing parallel optical reservoir computers based on three laterally coupled semiconductor lasers with delay-time feedback

In this work, we utilize three parallel optical reservoir computers to model three optical dynamic systems, respectively. Here, the three laser-elements in the response laser array with both delay-time feedback and optical injection are utilized as nonlinear nodes to realize three optical chaotic reservoir computers (RCs). The nonlinear dynamics of three laser-elements in the driving laser array are predictively learned by these three parallel RCs. We show that these three parallel reservoir computers can reproduce the nonlinear dynamics of the three laser-elements in the driving laser array with self-feedback. Very small training errors for their predictions can be realized by the optimization of two key parameters such as the delay-time and the interval of the virtual nodes. Moreover, these three parallel RCs to be trained will well synchronize with three chaotic laser-elements in the driving laser array, respectively, even when there are some parameter mismatches between the response laser array and the driving laser array. Our findings show that optical reservoir computing approach possibly provide a successful path for the realization of the high-quality chaotic synchronization between the driving laser and the response laser when their rate-equations imperfectly match each other.

Parameters optimization method for the time-delayed reservoir computing with a nonlinear duffing mechanical oscillator

Reservoir computing (RC) is a recently introduced bio-inspired computational framework capable of excellent performances in the temporal data processing, owing to its derivation from the recurrent neural network (RNN). It is well-known for the fast and effective training scheme, as well as the ease of the hardware implementation, but also the problematic sensitivity of its performance to the optimizable architecture parameters. In this article, a particular time-delayed RC with a single clamped-clamped silicon beam resonator that exhibits a classical Duffing nonlinearity is presented and its optimization problem is studied. Specifically, we numerically analyze the nonlinear response of the resonator and find a quasi-linear bifurcation point shift of the driving voltage with the driving frequency sweeping, which is called Bifurcation Point Frequency Modulation (BPFM). Furthermore, we first proposed that this method can be used to find the optimal driving frequency of RC with a Duffing mechanical resonator for a given task, and then put forward a comprehensive optimization process. The high performance of RC presented on four typical tasks proves the feasibility of this optimization method. Finally, we envision the potential application of the method based on the BPFM in our future work to implement the RC with other mechanical oscillators.

Performance optimization of a reservoir computing system based on a solitary semiconductor laser under electrical-message injection

A simple reservoir computing (RC) system based on a solitary semiconductor laser under an electrical message injection is proposed, and the performances of the RC are numerically investigated. Considering the lack of memory capacity (MC) in such a system, some auxiliary methods are introduced to enhance the MC and optimize the performances for processing complex tasks. In the pre-existing method, the input information is the current input data combined with some past input data in a weighted sum in the input layer (named as M-input). Another auxiliary method (named as M-output) is proposed to introduce the output layer for optimizing the performances of the RC system. The simulated results demonstrate that the MC of the system can be improved after adopting the auxiliary methods, and the effectiveness under adopting the M-input integrated with the M-output (named as M-both) is the most significant. Furthermore, we analyze the system performances for processing the Santa Fe time series prediction task and the nonlinear channel equalization (NCE) task after adopting the above three auxiliary methods. Results show that the M-input is the most suitable for the prediction task while the M-both is the most appropriate for the NCE task.

Machine learning based on reservoir computing with time-delayed optoelectronic and photonic systems

The concept of reservoir computing emerged from a specific machine learning paradigm characterized by a three-layered architecture (input, reservoir, and output), where only the output layer is trained and optimized for a particular task. In recent years, this approach has been successfully implemented using various hardware platforms based on optoelectronic and photonic systems with time-delayed feedback. In this review, we provide a survey of the latest advances in this field, with some perspectives related to the relationship between reservoir computing, nonlinear dynamics, and network theory.

Photonic reservoir computing based on nonlinear wave dynamics at microscale

High-dimensional nonlinear dynamical systems, including neural networks, can be utilized as computational resources for information processing. In this sense, nonlinear wave systems are good candidates for such computational resources. Here, we propose and numerically demonstrate information processing based on nonlinear wave dynamics in microcavity lasers, i.e., optical spatiotemporal systems at microscale. A remarkable feature is its ability of high-dimensional and nonlinear mapping of input information to the wave states, enabling efficient and fast information processing at microscale. We show that the computational capability for nonlinear/memory tasks is maximized at the edge of dynamical stability. Moreover, we show that computational capability can be enhanced by applying a time-division multiplexing technique to the wave dynamics. Thus, the computational potential of the wave dynamics can sufficiently be extracted even when the number of detectors to monitor the wave states is limited. In addition, we discuss the merging of optical information processing with optical sensing, revealing a novel method for model-free sensing by using a microcavity reservoir as a sensing element. These results pave a way for on-chip photonic computing with high-dimensional dynamics and a model-free sensing method.

Decision making for the multi-armed bandit problem using lag synchronization of chaos in mutually coupled semiconductor lasers

We numerically and experimentally demonstrate the utilization of the synchronization of chaotic lasers for decision making. We perform decision making to solve the multi-armed bandit problem using lag synchronization of chaos in mutually coupled semiconductor lasers. We observe the spontaneous exchanges of the leader-laggard relationship under lag synchronization of chaos, and we find that the leader laser can be controlled by changing the coupling strengths between the two lasers. To solve the multi-armed bandit problem, we select one of the slot machines with unknown hit probabilities based only on the identity of the leader laser while reconfiguring the coupling strength to determine the correct decision. We successfully perform an on-line experimental demonstration of the decision making based on the two-laser coupled architecture. This is the first time that synchronization in chaotic lasers is utilized for decision making, and this study paves the way for novel resources for future photonic intelligence.

Generative adversarial network based on chaotic time series

Generative adversarial networks (GANs) are becoming increasingly important in the artificial construction of natural images and related functionalities, wherein two types of networks called generators and discriminators evolve through adversarial mechanisms. Using deep convolutional neural networks and related techniques, high-resolution and highly realistic scenes, human faces, etc. have been generated. GANs generally require large amounts of genuine training data sets, as well as vast amounts of pseudorandom numbers. In this study, we utilized chaotic time series generated experimentally by semiconductor lasers for the latent variables of a GAN, whereby the inherent nature of chaos could be reflected or transformed into the generated output data. We show that the similarity in proximity, which describes the robustness of the generated images with respect to minute changes in the input latent variables, is enhanced, while the versatility overall is not severely degraded. Furthermore, we demonstrate that the surrogate chaos time series eliminates the signature of the generated images that is originally observed corresponding to the negative autocorrelation inherent in the chaos sequence. We also address the effects of utilizing chaotic time series to retrieve images from the trained generator.

Parallel information processing by a reservoir computing system based on a VCSEL subject to double optical feedback and optical injection

In this work, we propose a scheme of reservoir computing (RC) for processing a Santa-Fe time series prediction task and a signal classification task in parallel, and the performances of the RC have been numerically investigated. For this scheme, a vertical-cavity surface-emitting laser (VCSEL) simultaneously subject to double optical feedback and optical injection is utilized as a nonlinear node, and the parallel information processing of the RC system is implemented based on the dynamical responses of X polarization component (X-PC) and Y polarization component (Y-PC) in the VCSEL. Considering that two different feedback frames (polarization-preserved optical feedback (PP-OF) or polarization-rotated optical feedback (PR-OF)) may be adopted in two feedback loops, four feedback combination cases are numerically analyzed. The simulated results show that the parallel processing ability of the proposed RC system depends on the feedback frames adopted in two loops. After comprehensively evaluating the parallel processing performances of the two tasks under different feedback combinations, the best parallel processing performance can be achieved by adopting PP-OFs in both two feedback loops. Under some optimized operation parameters, this proposed RC system can realize the lowest prediction error of 0.0289 and the lowest signal classification error of 2.78 × 10^-5.

The reservoir's perspective on generalized synchronization

We employ reservoir computing for a reconstruction task in coupled chaotic systems, across a range of dynamical relationships including generalized synchronization. For a drive-response setup, a temporal representation of the synchronized state is discussed as an alternative to the known instantaneous form. The reservoir has access to both representations through its fading memory property, each with advantages in different dynamical regimes. We also extract signatures of the maximal conditional Lyapunov exponent in the performance of variations of the reservoir topology. Moreover, the reservoir model reproduces different levels of consistency where there is no synchronization. In a bidirectional coupling setup, high reconstruction accuracy is achieved despite poor observability and independent of generalized synchronization.

Four-channels reservoir computing based on polarization dynamics in mutually coupled VCSELs system

A novel Four-channels reservoir computing (RC) based on polarization dynamics in mutually coupled vertical cavity surface emitting lasers (MDC-VCSELs) is proposed and demonstrated numerically. Here, the four channels are realized in two orthogonal polarization modes (x-polarization and y-polarization modes) of two VCSELs for the first time. A chaotic time series prediction task is employed to quantitatively evaluated the prediction performance of the proposed system. It is found that the Four-channels RC based on MDC-VCSELs can produce comparable prediction performance with One-channel RC, and it is possible to increase four times information processing rate by using the Four-channels RC. Besides, the effects of injection current, external injection strength, frequency detuning, coupling strength, as well as internal parameters on the prediction performance of the Four-channels RC based on MDC-VCSELs are carefully examined. Moreover, the influences of sampled period of input signal and the number of virtual nodes are also considered. The proposed Four-channels RC based on MDC-VCSELs is valuable for further enhancing the information processing rate of RC-based neuromorphic photonic systems.

Performance optimization research of reservoir computing system based on an optical feedback semiconductor laser under electrical information injection

Via Santa Fe time series prediction and nonlinear channel equalization tasks, the performances of a reservoir computing (RC) system based on an optical feedback semiconductor laser (SL) under electrical information injection are numerically investigated. The simulated results show that the feedback delay time and strength seriously affect the performances of this RC system. By adopting a current-related optimized feedback delay time and strength, the RC can achieve a good performance for an SL biased within a wide region of 1.1-3.5 times its threshold. The prediction errors are smaller than 0.01 when implementing the Santa Fe tests, and the symbol error rates (SERs) are very low on the order of 10^-5 for accomplishing nonlinear channel equalization tests under a signal-to-noise ratio (SNR) of 32 dB. Moreover, under a given RC performance level, the information processing rate of the RC can be improved by increasing the SL current.