Probabilistic Limit Cycle Oscillator Reservoir Computer without Delay and Related Methods

Abstract

Various examples related to reservoir computing are provided. In one example, a physical reservoir computer includes a processing circuit, the processing circuit having an input layer, a reservoir implemented with a forced limit cycle oscillator and implemented without delay or feedback, and a readout layer. The forced limit cycle oscillator can include a Hopf oscillator or a Lorenz oscillator. The processing circuit can include an analog processing circuit, an optoelectronic circuit, or other suitable processing circuit.

Description

Technical Field

[0001] Cross - Reference to Related Applications This application claims the benefit and priority of U.S. Provisional Application No. 63 / 388,420, titled "Probabilistic Limit Cycle Oscillator Reservoir Computer Without Delay and Related Methods," filed on July 12, 2022, which is incorporated herein by reference in its entirety.

[0002] Description of Research and Development Sponsored by the Federal Government This invention was made with government support under Contract No. W911NF - 20 - 1 - 0336 awarded by the Army Research Laboratory - Army Research Office. The government has certain rights in this invention.

Background Art

[0003] Reservoir computing (RC) is an unconventional computing technique that utilizes the physics of non - linear dynamical systems for computation. The RC approach differs from the principle of the Turing machine because the computations performed by RC do not depend on static memory. Computations are obtained by mapping the transient dynamics of non - linear physical systems into a high - dimensional space. Some common applications of RC include logical operations, speech and handwritten digit recognition, wireless communication, complex and chaotic time - series prediction, long - term chaotic time - series prediction, image recognition, and morphological computation.

Summary of the Invention

[0004] Aspects of the present disclosure relate to reservoir computing. In one aspect, among other things, a physical reservoir computer comprises a processing circuit, the processing circuit comprising an input layer, a reservoir comprising a forced limit - cycle oscillator and implemented without delay or feedback, and a readout layer. In one or more aspects, the forced limit - cycle oscillator can comprise a Hopf oscillator or a Lorenz oscillator. The forced limit - cycle oscillator can comprise a two - state forced Hopf oscillator.

[0005] In various aspects, the processing circuit can comprise an analog processing circuit. The analog processing circuit can comprise an operational amplifier and a multiplier. The reservoir computer can utilize an aperiodic probabilistic mask. The aperiodic probabilistic mask can be defined by white Gaussian noise. The processing circuit can comprise an optoelectronic circuit. A vibration signal can be applied to the input layer. The vibration signal can be an audio signal. In some aspects, the readout layer can be trained to map the state of a forced limit cycle oscillator to a desired output. The training of the readout layer can include linear regression or ridge regression. The desired output can be a logical output. The logical output can be an XOR output, a NOT output, an AND output, or an OR output. The input layer can encode an applied signal for input to the reservoir. The applied signal can be encoded as a continuous input function.

[0006] Other systems, methods, features, and advantages of the present disclosure will become apparent to those skilled in the art upon examination of the following drawings and detailed description. All such additional systems, methods, features, and advantages are included within this description, are within the scope of the present disclosure, and are intended to be protected by the accompanying claims. Additionally, all optional preferred features and modifications of the described embodiments are usable in all aspects of the disclosure taught herein. Further, the individual features of the dependent claims, as well as all optional preferred features and modifications of the described embodiments, can be combined with one another and are interchangeable.

[0007] Many aspects of the present disclosure can be better understood with reference to the following drawings. The components of the drawings are not necessarily to scale relative to each other, and instead emphasis has been placed on clearly illustrating the principles of the present disclosure. Further, in the drawings, like reference numerals denote corresponding parts throughout the several views.

Brief Description of the Drawings

[0008]

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Embodiments for Carrying Out the Invention

[0009] This specification discloses various examples related to reservoir computing. By discarding delay lines and feedback lines while relying on the physical processes of oscillators to create and connect virtual nodes, a simplified version of a time-multiplexed reservoir computer is provided. Forced limit cycle oscillators (e.g., Hopf oscillators or Lorenz oscillators) can be used as basic non-linear dynamic systems that can be manufactured as circuits. To avoid sectional and node-dependent masks, node-independent probabilistic masking signals generated from white Gaussian noise can be used. By adjusting the parameters of a Hopf oscillator, nodes can be connected to have rich dynamics used in the RC method. Here, referring in detail to the description of the embodiments shown in the drawings, like reference numerals indicate like parts throughout several figures.

[0010] Due to echo state network structures, many physical systems are used as reservoirs, which are generally known as physical reservoir computers (PRCs). Some of the classical PRCs include arrays of Duffing oscillators, limit cycle-based Hopf oscillators, soft robot bodies, tensegrity structures, and origami structures. In addition to systems from classical physics, quantum physical systems can be used as RCs to perform tasks from both the classical and quantum domains. By utilizing the naturally disordered quantum dynamics of an ensemble system, non-linear time series including chaotic systems can be emulated. A Kerr non-linear oscillator can be used for sine wave phase estimation that uses its complex amplitude as a computing node. By implementing a spatial multiplexing technique to enhance computing power, a nuclear magnetic resonance spin ensemble system can be used for non-linear dynamics emulation tasks. A quantum reservoir computer for non-linear time tasks can be constructed using dissipative quantum dynamics. Statistical physics has played an important role in the theoretical development of neural networks that form a connection between information processing and physics.

[0011] Generally, a delay dynamical system can be used as a reservoir from a single non-linear node. A coupled delay system can also be used for computation by creating a deep neural network and a signal processor. A simpler implementation can also be achieved by excluding delay lines or feedback lines. Here, a reservoir computer is constructed by implementing a two-state Hopf oscillator. It has been shown that a Hopf oscillator reservoir computer has been previously studied and can successfully complete several benchmark tasks. A Hopf oscillator has the ability to store and learn information due to the existence of a stable limit cycle and is also suitable for constructing an adaptive oscillator. Conventionally, a binary mask has been used in a time multiplexing procedure to create virtual nodes for computation. Additionally, noise can also be used as a mask. Previously, eigenvalue analysis has been associated with non-resonance conditions for designing a reservoir computer operating near a stable equilibrium. However, since a Hopf reservoir is a limit cycle-based reservoir, the focus of this disclosure is different, so the analysis cannot be applied here. It should also be noted that the general concept of "the edge of chaos" is not used in this disclosure to optimize reservoir performance. The edge of chaos is not a necessary condition to achieve good computational ability of a reservoir computer. Therefore, staying away from chaotic regions and adjusting a set of network parameters can also be a path to constructing a reservoir computer with good performance.

[0012] To enhance the reservoir computing performance, microwave-based magnetic forced synchronization was implemented in a spintronic oscillator. The spin dynamics of magnetic tunnel junctions was also used to construct a reservoir system. Furthermore, based on magnetization dynamics, a nanoscale spintronic oscillator was optimized as a reservoir. These spintronic oscillators may be further optimized by utilizing the relationship between the period (pseudo-frequency) of the input and the forcing force as described.

[0013] In one example, a driven Hopf oscillator was studied as a reservoir computer excluding both the masking function and the commonly used delay line, and the masking function and the delay line were discarded to focus on the dynamics of the computational oscillator. Resonance phenomena, Arnold tongues, and Farey sequences all contribute to the performance of the Hopf oscillator as a reservoir computer. Arnold tongues refer to phase-locked or synchronous regions within the parameter space and have a strong effect on this Hopf oscillator reservoir. To perform a parametric study of this Hopf oscillator computer, parity and chaotic laser time-series benchmarks are used. This oscillator was experimentally realized as an analog electrical circuit to investigate the information processing ability of the reservoir. A modified version of Shannon's information rate can be used as a performance metric for the parity task.

[0014] Furthermore, the reservoir computer can be developed from a network of virtual nodes that utilize the non-linearity of a single Lorenz system while excluding feedback. This can lead to a simpler and cheaper way to construct a virtual-node-based reservoir computer. These dynamically coupled virtual nodes can be time-multiplexed using a probabilistic masking procedure. Since noise exists in the system, the Euler-Maruyama method can be used to simulate the equation system of this Lorenz reservoir computer. The resulting reservoir computer was found to succeed in performing second-order, third-order, and fourth-order parity tasks and fail in fifth-order tasks. Since these are logical tasks, Shannon's information metric can be used to quantify the performance of the reservoir computer.

[0015] Hopf Physical Reservoir Computer Reservoir computing (RC) is a bio-inspired supervised machine learning computing framework based on artificial recurrent neural networks (RNNs) that utilizes the natural dynamics of physical resources. Conventional machine learning methods use backpropagation through time to train the entire recurrent neural network. This method has a high computational cost because it is necessary to update all the weights of the network to mimic the target function. Echo state networks and liquid state machines are two concepts that addressed this problem in the early 2000s. Reservoir computing combines these concepts. In reservoir computing, the neural network is formed from a set of interconnected non-linear nodes, and the network is divided into three parts: an input layer, a reservoir, and a readout layer. Unlike conventional RNNs, only the readout layer requires training with a simpler training algorithm such as linear or ridge regression. Therefore, the RC architecture is much faster and more stable than the conventional RNN method, which is an advantage of this information processing framework.

[0016] There are many real-world applications of reservoir computing, including bitwise logical operations, speech recognition, handwritten digit recognition, wireless communication, complex and chaotic time series prediction, image recognition, emulation of non-linear time series, and morphological computation. The echo state architecture of the reservoir enables the use of physical systems as reservoir computers, also known as physical reservoir computers (PRCs). Many physical systems, including non-linear mechanical oscillators, soft robot bodies, tensegrity structures, and arrays of origami structures, have been shown to function as PRCs.

[0017] Importantly, a quantum system can be used as a PRC. By utilizing the natural chaotic quantum dynamics of an ensemble system, non-linear time series including chaotic systems were emulated. A Kerr non-linear oscillator was used for sine wave phase estimation using its complex amplitude as a computational node. A nuclear magnetic resonance spin ensemble system was used for non-linear dynamics emulation tasks by implementing a spatial multiplexing technique to enhance computational power. A quantum reservoir computer (QRC) for non-linear time tasks was constructed using dissipative quantum dynamics.

[0018] Physical reservoir computers were initially constructed from only coupled actual dynamic nodes. Subsequently, a virtual node-based reservoir computing method was proposed by implementing a time multiplexing technique that uses delay feedback as a single non-linear dynamic node to perform calculations. This method simplifies the complexity of the reservoir constructed from an array of physical non-linear nodes. This technique can be used to construct physical reservoir computers for various tasks such as optoelectronic oscillators for optical information processing, photonics-based passive linear fiber reservoirs for signal processing, FPGA implementations using a single autonomous Boolean logic element for pattern recognition, time delay reservoirs for probabilistic non-linear time series prediction, delay Duffing silicon beams for parity tasks, and / or semiconductor lasers with delay optical feedback for non-linear time series prediction. These reservoirs can use delay lines to create the nodes required for computation. A simpler approach can be taken by creating nodes that do not have delay lines or feedback lines.

[0019] Here, a Hopf oscillator is used as a physical reservoir. The Hopf oscillator can also be used as a component of an adaptive oscillator that can naturally learn information without any training. The Hopf oscillator can exhibit a limit cycle motion that provides a source of memory by storing information in its dynamic state. A binary periodic masking function can be used for the time-multiplexed reservoir, but noise can also be used as a periodic mask. Here, a Hopf oscillator PRC using an aperiodic probabilistic mask is constructed. The Hopf oscillator physical reservoir computer is manufactured as an analog circuit and compared with Euler-Maruyama simulation. This Hopf PRC can successfully complete benchmark machine learning tasks including parity tasks, basic logic gate tasks, non-linear dynamic emulation tasks, and various time series prediction tasks. The information rate can be used as a performance metric for logical tasks, and the normalized mean squared error (NMSE) can be used for emulation and time series tasks.

[0020] Next, the equation of motion of the probabilistic Hopf oscillator PRC is presented, followed by a methodology for mapping the dynamics of the oscillator to an information processing scheme, illustrated for exemplary tasks using Euler-Maruyama simulation. The effects of quasi-periodicity and noise on computational power are considered, and analog circuit experiments are described. Different benchmark tasks are performed using numerical and experimental Hopf PRCs including logical tasks, time series emulation tasks, and prediction tasks. A list of parameters, states, and functions used in the following description is shown below.

Table 1

[0021] The system equation of the Hopf physical reservoir computer The equation of motion of the Hopf oscillator is as follows.

Equation

Number

Number

Number

[0022] To send information to the PRC to be processed, an external forcing function containing the information signal u(t) and the stochastic mask m(t) can be constructed as follows. f(t) = 1 + u(t)m(t) (3) This external forcing function is injected into both the amplitude A of the sinusoidal forcing and the parameters that affect the limit cycle radius μ. Including this force, the equation of the Hopf PRC is described as follows.

Number

[0023] Mapping Method To use the dynamics of a Hopf oscillator as a physical reservoir computer, it is necessary to map the dynamics. To explain this mapping, an exclusive OR (XOR) logic task is used as an example. Since the mask is probabilistic, the Euler-Maruyama method can be used to simulate the Hopf PRC. The Shannon information metric can be used to quantify the performance of the reservoir when performing logic tasks such as the XOR operation.

[0024] For this task, the binary "false" and "true" values are encoded as discrete negative and positive values, respectively, in the discrete signal r(z). r(z) is defined as z ∈ Z + and r(z) ∈ {-1, +1}, as shown in Figure 1. Plot (a) in Figure 1 shows the discrete random binary signal r(z), and plot (b) shows the continuous input signal u(t). Plot (c) in Figure 1 shows an example of the probabilistic masking function m(t), plot (d) shows the time history of x(t), and plot (e) shows the rescaled time history X(t). Plot (f) in Figure 1 shows an example of 20 equally spaced nodes for a single pseudo-period T p shown as circles. Plot (g) in Figure 1 shows an example of the node states collected from the nodes of a machine learning input dataset, with different colors indicating different nodes. In the simulations shown here, the parameters were set to μ = 5, A = 0.5, Ω = 40π rad / s, ω0 = 40π rad / s, T p = 0.1 s, N = 20 nodes, φ = π / 3 rad, σ = 100, β = 1.0.

[0025] To input these into a continuous dynamic system, these values are first mapped to a continuous input function u(t) as follows. (n - 1)T p ≤ t < (n)T p in which case u(t) = r(z), n ∈ Z + (5) This function is shown in plot (b) of Figure 1. T pis a constant pseudo-period, and the value of u(t) does not change. Therefore, in the case of the XOR logic task, the input function u(t) ∈ {-1, +1} is a random square wave with a pseudo-period T p which is. This means that each value of "true" (e.g., +1) or "false" (e.g., -1) affects the system over a certain time T p . The mask function m(t) is shown in plot (c) of Figure 1.

[0026] The Hopf PRC system represented by Equation (4) can be numerically integrated using the Euler-Maruyama (EM) method because the PRC is stochastic. In these simulations, the integration time step dt = 10 -5 seconds, and the total simulation time in this case is 3000T p = 300 seconds, with T p = 0.1 second. This simulation example is shown in Figure 1.

[0027] The time history of the x state obtained from the simulation is shown in plot (d) of Figure 1. Next, x(t) can be re-scaled by subtracting the mean μ x using Equation (6) and dividing by the standard deviation σ x using Equation. :

Number

Number

[0028] Next, as shown in plot (f) of Figure 1, equally spaced nodes can be created by equally dividing each pseudo-period T p into N (= 20) nodes. Each pseudo-period T pOver N node values are called node states and are shown in plot (g) of FIG. 1.

[0029] The node matrix S is an N×K matrix. In this example, N = 20 is the number of nodes over the pseudo-period, and K = 3000 is the total number of pseudo-periods. The last 20% of this S matrix (600T p ) is discarded to form a new matrix L (480T p ), which is used in the training process. The reservoir computer can be trained using ridge regression as follows. [Number] (7) The target signal (M vector) can be created from the encoded input based on a benchmark task, which is the XOR task in this case. For each pseudo-period, there is one target value obtained by performing an XOR operation between the input r(z) and r(z - 1). In this way, the target vector M for the XOR task is found. Then, linear regression-based training can be applied to the node state matrix L to map it to the desired output using Equation (7). In Equation (7), w is the weight vector found after training, I is the identity matrix, λ = 10 -1 is the regularization parameter used to avoid overfitting, and o(k) is the prediction of the reservoir computer at the k-th pseudo-period.

[0030] The discrete random binary input signal r(z) and the continuous input signal u(t) are shown in plots (a) and (b) of FIG. 2, respectively. In the simulation shown here, the parameters are μ = 5, A = 0.5, Ω = 40π rad / s, ω0 = 40π rad / s, T pIt was set such that = 0.1 s, N = 20 nodes, φ = π / 3 rad, σ = 100, and β = 1.0. Plot (c) in Fig. 2 shows this continuous prediction along with the corresponding target signal. In the last step, since XOR is a binary task, the prediction is binarized, which is shown as the discretized target and prediction in plot (d) of Fig. 2. Note that the non-linear dynamic emulation task does not require this last step of discretization.

[0031] For logical tasks, the effectiveness of the reservoir computer is quantified using Shannon's information rate. The information rate R can be defined as follows. R = H(x) - H y (x) (8) Here, H(x) is the Shannon entropy, which indicates how much information is encoded in the signal. It can be defined as follows. [Number] (9) In this equation, p i is the probability of obtaining a specific bit i. H y (x) is the conditional entropy, which indicates the probability of obtaining an incorrect bit in the target signal: [Number] (10) Here, [Number] And p(i, j) is the joint probability distribution of two variables j, which can take values of "1" or "-1" for each logical task. i is the bit from the target and j is the bit from the prediction. In this case, the information rate R was calculated to be 0.98 based on the prediction from the verification part (not included in the training process). Due to the nature of this binary target signal, the Shannon entropy is 1.0, which indicates the maximum value of the information rate for this task. The lower limit of R is 0, which is achieved when all predictions are incorrect, but note that the upper limit of R depends on the task. For the parity task considered here, the upper limit of R is equal to 1.

[0032] Pseudo - period and noise Next, the effects of pseudo - period and noise on the computing power of the reservoir are investigated. For the purpose of this explanation, several parity tasks (defined by the following equation (12)) are used to understand the effects of pseudo - period and noise on the computing power of the reservoir.

[0033] Pseudo - period T p The relationship between and the natural frequency ω0 of the oscillator is investigated in Figure 3 using second - order and fourth - order parity tasks. Figure 3 shows, for (a) T p = 0.05 s, (b) T p = 0.1 s, and (c) T p = 0.15 s, a comparison of the computing performance R of the reservoir regarding the selection of the pseudo - period T p and the natural frequency ω0. Different ratios of the natural period and the pseudo - period (e.g.,

Number

[0034] In the plot of Figure 3, the natural period

Number

[0035] Noise is ubiquitous in physical systems. For this reason, noise was introduced into this system using a probabilistic masking function. Figure 4 shows the relationship between the computing power measured by R, the noise amplitude σ, and the noise bias β. The simulations shown in Figure 4 were performed for the 4th-order parity task (left) and the 6th-order parity task (right), showing the effects of σ and β. The parameters were set such that μ = 5, A = 0.5, Ω = 40π rad / s, ω0 = 40π rad / s, T p = 0.1 s, N = 1000 nodes, φ = π / 3 rad. The reservoir was found to be robust against a certain level of noise intensity, demonstrating its potential to be implemented under the effects of environmental noise. However, increasing the noise intensity causes the computing power of the reservoir to decrease. This effect can be observed in higher-order tasks that require longer memory (e.g., the 6th-order parity task in Figure 4). When β = 0, the computing power was the lowest. Note that it is also possible to construct a Hopf reservoir computer by excluding the noise mask (σ = 0) since the aperiodic noise mask that increases the noise intensity decreases the computing power.

[0036] Analog Circuit Experiment To construct a Physical Reservoir Computer (PRC), the analog circuit implementation of Equation (4) was designed, fabricated, and tested. The circuit equations are given by Equation (11) below:

Number

Number

Number

Number

[0037] V x The state can be processed in the same way as the x state was processed in the "Mapping Method" section. That is, the V x state is rescaled using Equation (6), and then the rescaled state is used to form the node state matrix L. The target signal vector M is created according to the same process as described in the "Mapping Method" section. Finally, the PRC can be trained to map the input data to the desired output values using Equation (7). As an example, an analog circuit Hopf PRC was used to solve the XOR task as in the previous section shown in Figure 6. Plot (a) in Figure 6 shows the input voltage signal V u and plot (b) in Figure 6 shows the time history of V x . Plot (c) in Figure 6 shows the XOR target signal M and the prediction, and plot (d) in Figure 6 shows the discretized prediction. The calculated information metric is R = 1.0. For the experimental results shown here, the parameters were set such that V μ = 5 volts, A = 0.5 volts, Ω = 40π rad / s, [Number] volts T p = 0.1 seconds, N = 20 nodes, φ = π / 3 rad, σ = 10 volts, β = 1.0 volts. The information rate R in this case was calculated to be 1.0 based on the prediction from the verification part (not included in the training process).

[0038] Benchmark tasks for the Hopf PRC The Hopf PRC was numerically and experimentally tested using three benchmark tasks: (1) logical tasks, (2) time series emulation tasks, and (3) prediction tasks. The logical tasks include basic logic gate tasks and parity tasks in different orders. The time series emulation tasks test the ability of the PRC to reproduce non - linear autoregressive moving average (NARMA) tasks in different orders. The prediction tasks include the Santa Fe time series and sunspot prediction tasks.

[0039] The computational efficiency of the logical benchmark task parity task reservoir was first evaluated on the parity benchmark task. Since the parity benchmark task is a logical task, the input function u(t) is generated by a random binary signal r(z) as described in the "Mapping Method" section. The n-th parity function P n can be defined by the following formula. [Number] (12) As n increases, this task utilizes more memory and non-linearity from the reservoir. As described in the "Mapping Method" section, the performance of the PRC of the logical task can be measured using Shannon's information metric. When n = 1, the first-order task is linear because it does not require any memory from the previous pseudo-periodic input. When n > 1, the task is non-linear and requires the reservoir computer to also have memory and non-linear separation capabilities. In Figure 7, the ability of the Hopf PRC to follow second- to fifth-order parity tasks is shown both experimentally and through simulation. Figure 7 shows a comparison of the PRC performance for the parity tasks. Plot (a) in Figure 7 shows the discrete input function r(z). Plot (b) in Figure 7 shows the second-order parity task (information metric: R exp = 1.00, R sim = 0.98), plot (c) in Figure 7 shows the third-order parity task (information metric: R exp = 1.00, R sim = 0.98), plot (d) in Figure 7 shows the fourth-order parity task (information metric: R exp = 0.68, R sim = 0.93), and plot (e) in Figure 7 shows the fifth-order parity task (information metric: R exp = 0.31, R sim = 0.74). The parameters are V μ = μ = 5, A = 0.5, Ω = 40π rad / s, [Number] and T p = 0.1 sec, N = 1000 nodes, φ = π / 3 rad, σ = 15, β = 1.0, total time 5000T p = 500 seconds (only showing part of the discrete prediction) was set. The first 4000T p = 400 seconds was used for training, and the last 1000T p = 100 seconds was used for testing. The performance difference between the PRC experiment and the simulation may be due to the presence of non - linear circuit components in the analog circuit, which is not represented by Equation (11). For example, V u has to jump between - 1 and + 1, but this instantaneous change requires a finite time within the circuit.

[0040] The computational performance of the basic logic gate task reservoir is also evaluated with the basic logic gates: NOT (number), AND (∧), and OR (∨). The input function u(t) can be generated with a random binary signal as described in the "Mapping Method" section, and the Shannon information metric is used again to measure the performance of this PRC. Figure 8 shows the responses of the Hopf PRC acting as a basic logic gate in both experimental and simulation cases. Figure 8 shows a comparison of the performance of the PRC for the parity task. Plot (a) in Figure 8 shows the input function u(t), plot (b) shows the NOT (￢) gate, plot (c) shows the AND (∧) gate, and plot (d) shows the OR (∨) gate. For all numerical and experimental results, the information rate was the theoretical maximum. The Hopf PRC can function as any of the basic logic gates. The parameters are V μ = μ = 5, A = 0.5, Ω = 40π rad / s,

Number

[0041] The emulation task reservoir was also evaluated with emulation tasks. Using the non-linear autoregressive moving average (NARMA) time series, it is possible to test whether the reservoir has appropriate non-linearity and long time lags. These tasks demonstrate the multitasking ability of the reservoir. To test the reservoir, NARMA tasks from order 2 to order 20 are used. The nth order NARMA task is given by Equation (13), and the initial target value is set to 0.19.

Equation

Equation

[0042] In the simulations and experiments, Δt = 0.1 second, and the sampling rate was 10 5 samples / second. Figure 9 shows several NARMA tasks. Instead of the information rate, the normalized mean squared error (NMSE) is used to evaluate the performance of the reservoir computer for the NARMA tasks.

Number

Number

Number

[0043] The prediction task, the Santa Fe task time series prediction, is an important benchmark for the reservoir. The Santa Fe time series was first used in the time series prediction competition as a benchmark test. The Santa Fe time series dataset A is a univariate time series found from the recorded intensity of a chaotic far-infrared laser. The target signal can be generated to predict the value at the next time step based on the values at the current and previous time steps. Fig. 11 shows a comparison of the performance of the PRC for the Santa Fe prediction task, which shows the performance of the Hopf PRC for this laser time series for both the experiment and numerical simulation. The NMSE is used as the performance metric. Plot (a) in Fig. 11 shows the Santa Fe chaotic time series of the laser intensity prediction task (performance metric: NMSE exp = 0.0615, NMSE sim = 0.02), plot (b) shows the Santa Fe heart rate prediction task (NMSE exp = 6.0258×10 -4 , NMSE sim = 6.5060×10 -4 ), plot (c) shows the Santa Fe respiratory force prediction task (NMSE exp = 0.1826, NMSE sim = 0.1753), and plot (d) shows the Santa Fe blood oxygen concentration prediction task (NMSE exp = 3.3287×10 -4 , NMSE sim = 1.7×10 -4 ). The parameters are V μ = μ = 5 volts, A = 0.5 volts, Ω = 40π rad / s,

Number

[0044] The Santa Fe time series dataset B is a multivariate time series obtained from the sleep laboratory of Beth Israel Hospital (current name: Beth Israel Deaconess Medical Center) in Boston, Massachusetts. This dataset was acquired from the MIT - BIH polysomnography database record (slp60) and submitted to the 1991 Santa Fe time series competition. Heart rate, chest volume (respiratory effort), and blood oxygen concentration constitute the target.

[0045] For each of these time series, target signals were regenerated to predict the next step based on the values of the current and previous time steps. In both cases, the original time series was normalized and used as input. Plots (b), (c), and (d) show the performance of the reservoir computer in predicting subsequent values of heart rate, respiratory effort, and blood oxygen concentration, respectively, by both experiment and numerical simulation. The NMSE was calculated for each case to evaluate the performance of the reservoir.

[0046] Solar sunspot prediction task The prediction of the total number of solar sunspots (S n ) is also a one - step time series prediction task similar to the Santa Fe time series. The daily and monthly total solar sunspot numbers were used for one - step prediction purposes by the reservoir computer. The required dataset was obtained from WDC - SILSO, Royal Observatory of Belgium, Brussels. Here too, for each of the time series, target signals were generated to predict the next value based on the values of the current and previous time steps, and the original time series was normalized for use as input to the oscillator. Figure 12 shows the solar sunspot prediction (S n)Shows the comparison of task performance. The upper plot in Figure 12 shows the performance of the reservoir when predicting the next step of the sunspots counted daily, and the lower plot in Figure 12 shows the performance when predicting the sunspots counted monthly. Again, the NMSE is used to evaluate the effectiveness of the reservoir for this task. The total number of daily sunspot prediction tasks for both experiments and numerical simulations is shown in the upper plot of Figure 12 (performance metric: NMSE exp =0.0548, NMSE sim =0.0534). The monthly average total number of sunspot prediction tasks is shown in the lower plot of Figure 12 (performance metric: NMSE exp =0.0595, NMSE sim =0.0455). The parameters were set such that V μ =μ = 5 volts, A = 0.5 volts, Ω = 40π rad / s,

Number

[0047] Here, the Hopf oscillator was considered as a physical reservoir computer by adopting a time-division multiplexed node-based architecture with a stochastic masking function. When the regularly used delay line is discarded, this Hopf PRC is a simple and inexpensive way to create a physical reservoir computer. Since the quantum system can limit the cyclic motion, this Hopf PRC formulation may be applicable to the quantum PRC. The Euler-Maruyama method was used for the numerical simulation of this Hopf PRC. An analog circuit of this Hopf PRC was developed, fabricated, and tested. Since the Hopf PRC has been shown to perform logical operations, emulation tasks, and time series prediction tasks, it was found to have multitasking capabilities. Inspired by the adaptive oscillator, the input signal was injected at multiple positions including parameters that affect the limit cycle radius and the amplitude of the sine wave forcing. Furthermore, the masking function used in this PRC is stochastic. Since this PRC architecture has been tested using noise, it also suggests that this reservoir computer should be robust to environmental noise in actual implementation.

[0048] Reservoir Computing by Lorenz System Examine the information processing capabilities of the Lorenz system, which was one of the first systems shown to exhibit chaos, from the perspective of reservoir computing. The Lorenz system can be used as a reservoir computer that has an echo state network method for separating superimposed chaotic signals and a small-world and scale-free network method for reproducing the long-term nonlinear characteristics of the system. The coupled Lorenz oscillators can also be used as a reservoir computer for signal reconstruction by utilizing continuous transient oscillation dynamics. Here, the Lorenz system is implemented as a reservoir computer by coupling several virtual nodes using a time-division multiplexing method. The silicon beam is also used as a reservoir computer by performing an audio recognition task using a Mackey-Glass oscillator and performing a parity task using Duffing nonlinearity.

[0049] A series of equations for the Lorenz reservoir computer used in the following description are given by:

Number

Number

Number

[0050] The reservoir computer is trained using the y - states (L - matrix) of 100 nodes for each pseudo - period τ. The target signal (M - vector) is created from the time history of the input following a benchmark task. Linear readout training is applied to the node states and mapped to the desired output using Equation 16, where w is the weight vector found after training, I is the identity matrix, and λ = 10 -1 is the regularization parameter used to avoid overfitting, and o(k) is the prediction of the reservoir computer at the k - th time step.

[0051] The n - th parity function P n is [Number] defined by. As n increases, this task utilizes more memory and non - linearity of the reservoir computer to execute the task. Since these parity tasks are essentially logical, Shannon's information metric is used to measure the effectiveness of the reservoir computer. The information metric R is defined as follows. [Number] (17) Here, H(x) is the Shannon entropy, and H y (x) is the conditional entropy. The first 80% of the time history is used for training the reservoir, and the remaining 20% is used for testing. Figure 13 shows an example of a parity task using a Lorenz reservoir computer. The simulation time is 1440 seconds, with dt = 0.0001 sec, N = 100, τ = 0.48 sec, θ = 0.0048 sec, b = 1.5, a = 10, ρ = 2, σ = 10, β = 8 / 3, Ω = 100π rad / s, and φ = π / 3 rad. Plot (a) in Figure 13 shows the input, plot (b) shows the second-order parity task, plot (c) shows the third-order parity task, plot (d) shows the fourth-order parity task, and plot (e) shows the fifth-order parity task. Figure 13 shows that for each time step, the per-bit prediction accuracy of the reservoir computer for the second-, third-, and fourth-order parity tasks is 100%, which corresponds to R = 1.0. However, the fifth-order parity task has an information metric close to 0.

[0052] Here, the Lorenz system is studied to process information using a time-division multiplexing virtual node-based reservoir computing framework. In this study, a probabilistic masking function is used. A simpler and cheaper way to develop a reservoir computer by eliminating feedback and delay is shown.

[0053] Dynamic effects on reservoir computing using a Hopf oscillator The non-linear system is perturbed by the input signal that carries the information to be processed. The input u(t) is embedded into the reservoir dynamics using a single non-linear node as follows. (n - 1)T p ≦ t < (n)T p For f(t) = 1 + u(t) with respect to (18), u(t) = r(z). Here, r(z) is a discrete signal that sequentially encodes logical values. This discrete signal is then mapped to a continuous function as described by Equation (18), where n, z ∈ Z + Since r(z) is a random sequence of logical statements, u(t) has a pseudo-period T p and pseudo-frequency

Number

[0054] The Hopf reservoir computer can be described by the equation of motion in Equation (4). This system is a two-state forced Hopf oscillator, where x and y are the states, Ω is the harmonic forcing frequency, ω0 is the resonance constant, and μ is the parameter that controls the limit cycle radius. The governing equation of the Hopf RC in Equation (4) is numerically integrated, and then the x state is scaled by subtracting the mean and dividing by the standard deviation. Next, each pseudo-period is evenly divided, and N virtual nodes are collected from each pseudo-period T p The nodal state is then nonlinearly scaled using the nonlinear activation function tanh -1 x. Approximately 80% of the scaled node states are used in the training process, and the remaining 20% are used to test the performance of the RC. These virtual nodes, extracted by downsampling the time history, are similar to the nodes found in a delay-based reservoir. Since there is only one actual node, which is the oscillator itself, the other nodes are called virtual according to the terminology of the delay-based reservoir. However, the Hopf RC studied in this paper does not include delay lines or masking functions (time multiplexing), simplifying the system.

[0055] The reservoir computer is trained using ridge regression with Tikhonov regularization as shown in Equation (7). Here, M is the target vector that the reservoir should match, X is the scaled node state, and L is the matrix containing the node states of the reservoir. λ is 10 -1is a regularization parameter set to avoid overfitting, I is the identity matrix, N is the number of nodes, w is the weight vector obtained from the training procedure, o(k) is the prediction of the reservoir, and k ∈ Z + is as follows.

[0056] The δ-delayed n-th parity function P n is defined by the following equation:[[]] [Number] (19) where δ ∈ Z + is the delay. For the parity task tested here, δ = 0 is used. When n = 2, Equation (19) is quadratic parity, which is the exclusive or (XOR) task [Number] is. Since this test only deals with the parity benchmark which is a logical task, the input u(t) = {-1, +1} is randomly selected for each pseudo-period. Therefore, the final prediction of the reservoir is also binarized and becomes the high bit (+1) or the low bit (-1). The performance of the reservoir can be evaluated using Shannon's information rate. The logical bits are used to calculate the information metric R as defined in Equation (8). Here, H(x) is the Shannon entropy, which is an indicator of the encoded information in the signal. This can be defined as in Equation (9), where p i is the probability of obtaining a specific bit i. H y (x) is the conditional entropy, which is the probability of obtaining an incorrect bit in the target signal as defined by Equation (10).

[0057] Here,[[]] [Number] And p(i, j) is the joint probability distribution of two variables i and j, which can be evaluated as "1" or "-1" for a logical task. i is associated with the target signal, and j is the relevant bit value from the prediction signal of the RC. Note that in these parity tasks, the maximum value of R is 1.0. Using Equation (4), a Hopf RC was fabricated as an analog circuit. The circuit was constructed using TL082 operational amplifiers and AD633 multipliers in a standard integrator network configuration. National Instrument cDAQ-9174 was used as the data acquisition device.

[0058] Parametric study The dynamic limits of the Hopf reservoir computer are next investigated by studying the effects of different parameters on its information processing ability. Performance is quantified using the information rate for various parameter combinations and the order of the parity tasks. The results provide a deeper understanding of the interaction of the oscillator dynamics and its computational ability as a reservoir computer. Some of these results can be used as guidelines for tuning other virtual node-based reservoir computers.

[0059] The input signal to the Hopf RC is embedded in the oscillator via the limit cycle radius μf(t) and the harmonic forcing amplitude Af(t). Thus, adjusting the parameters μ and A potentially controls the amount of information sent to the oscillator. A parametric sweep of this two-parameter space is shown in Figure 14, which shows the parity task (ω p = 20π(T p = 0.1 sec), Ω = ω0 = 40π rad / s, N = 1000 nodes, φ = π / 3, 4000T pAn example of parametric study of the limit cycle radius constant μ and the harmonic forcing amplitude A of Hopf RC based on the simulation time = 400 seconds is shown. Plot (a) in Fig. 14 shows the second parity, plot (b) in Fig. 14 shows the third parity, plot (c) in Fig. 14 shows the fourth parity, and plot (d) in Fig. 14 shows the fifth parity. The color bar shows the information metric R.

[0060] It is observed that small limit cycles with low forcing amplitudes are not effective for calculations as expected. However, bands are observed in higher-order tasks, which is counterintuitive. This relationship can be used to maximize the computing power of the oscillator. By simply changing the magnitude of the input of the spintronic reservoir, the information processing ability can be changed. The magnitude of the input can be varied to obtain different limit cycle responses in the spintronic oscillator. Similarly, the current Hopf RC changes the magnitude of the input by changing A to optimize the RC performance.

[0061] The external forcing frequency has been found to be important in determining the performance of the reservoir computer. Adjustments of the forcing frequencies of RCs created from Duffing oscillators, Arnold tongue structures, and arrays of topological mixing have been observed. Similarly, parametric studies were carried out using the resonance constant ω0 and the harmonic forcing frequency Ω.

[0062] The resonance condition of the Hopf oscillator is achieved when the resonance constant and the harmonic forcing frequency are equal (ω0 = Ω). Fig. 15 shows the parity task (ω p = 20π(T p = 0.1 sec), μ = 5, A = 0.5, N = 1000 nodes, φ = π / 3 rad, 4000T pAn example of a parametric study of the resonance constant ω0 and the harmonic forcing frequency Ω of a Hopf RC with a simulation time of 400 seconds is shown. Plot (a) in Fig. 15 shows the second parity, plot (b) in Fig. 15 shows the third parity, plot (c) in Fig. 15 shows the fourth parity, and plot (d) in Fig. 15 shows the fifth parity. The color bar indicates the information metric R. In plots (a)–(d), the bands along each 45° angle of the ω0-Ω parametric plot correspond to this resonance condition. However, when the resonance condition (ω0 = Ω) is achieved, the performance of the reservoir suddenly jumps from poor performance to successful calculation. Therefore, the resonance phenomenon is a necessary condition for the Hopf oscillator to function as an effective RC.

[0063] Another matching condition is that ω0 is a multiple of the quasi-period ω p by an integer (ω0 = zω p , where [Number] ) and is satisfied. This can be observed for all parity tasks presented in Figs. 15 and 16. The ω0-Ω parametric space was also investigated experimentally, which is presented in Fig. 16 by constructing an analog circuit that can be modeled by Eq. (4). Due to experimental limitations, a relatively coarse parametric space is shown in Fig. 16, which shows the parity task (ω p = 20π(T p = 0.1 sec), μ = 5, A = 0.5, N = 1000 nodes, φ = π / 3 rad, 4000T pAn example of an experimental and numerical parametric qualitative study of the resonance constant ω0 and the harmonic forcing frequency Ω of the Hopf RC with a coarse resolution based on a simulation time of 400 seconds is shown. The top row of Figure 16 illustrates an analog circuit experiment where plot (a) in Figure 16 shows second parity, plot (b) in Figure 16 shows third parity, plot (c) in Figure 16 shows fourth parity, and plot (d) in Figure 16 shows fifth parity. The bottom row of Figure 16 illustrates a numerical simulation where plot (e) in Figure 16 shows second parity, plot (f) in Figure 16 shows third parity, plot (g) in Figure 16 shows fourth parity, and plot (h) in Figure 16 shows fifth parity. The color bar indicates the information metric R. The experiments show similar trends near resonance and in the coincidence condition. From Figure 16, the best performance of the numerical reservoir can be found for the resonance condition when [Number] and the best performance of the experimental reservoir can be found when [Number] and [Number] . The deviation between the experiment and the simulation can be attributed to the accuracy of the circuit elements and the non-linear effects of the circuit (e.g., parasitic effects).

[0064] Next, the effects of the frequency ratio [Number] and the harmonic forcing amplitude A on the computational ability of the Hopf RC are shown in Figure 17. While keeping the system parameters the same, using the second parity and the chaotic laser intensity prediction task, [Number] The parametric space was studied. In the case of the chaotic time series benchmark, the task of RC was to predict one step ahead based on the previous step. The RMSE was used as the performance metric for this task. To verify that the frequency ratio from the Farey sequence is important for other tasks, the chaotic time series benchmark can also be used to compare the performance with non-binary tasks. The harmonic frequency Ω is set to 40π so that the Hopf oscillator experiences resonance when

Number

[0065] In Figure 17,

Number

Number

[0066] From Figure 17,

Number

Number

[0067] Figure 18 shows how synchronization plays an important role in the computing power of the RC, which is

Number

[0068] Plots (b), (d), and (f) in Fig. 18 show the phase difference between the x state of the Hopf oscillator and the harmonic forcing sin(Ωt + φ) for the parity and chaotic time series prediction tasks.

Number

Number

Number

[0069] Fig. 19 shows the time history of the response of the oscillator when locked by a forcing force and when not phase-locked. Fig. 19 shows that the time series of the x state of Hopf RC is shown for a part of the chaotic time series task. The upper plot of Fig. 19 shows that Hopf RC is at resonance (ω0 = Ω) and the vibration is locked to the external forcing force. The lower plot of Fig. 19 shows that Hopf RC is not in resonance (ω0 = 1.7391*Ω) and the vibration is not locked to the external forcing force. Here, ω p = 20π, ω p = 20π rad / s (T p = 0.1 sec), μ = 5, A = 0.5, N = 1000 nodes, φ = π / 3 rad.

[0070] The tongue region can be particularly important in experimental design. In this tongue region, there is a range of frequency ratios centered around the resonance frequency, which can lead to better calculations. Therefore, this is the only region where the reservoir has some tolerance to mistuning. This means that synchronization in the Arnold tongue region causes robust calculations. There is also a relatively wide range of amplitudes and frequency ratios that can adjust the resonance constant, and the Hopf RC has high computational power but can be adjusted to stay inside the Arnold tongue. Note that the presented research was conducted without considering masking in the system. The existence of robust computational power despite the absence of a masking function suggests that it is also possible to discard conventional periodic or aperiodic masks and reliably construct a single non-linear node-based reservoir.

[0071] To understand the effect of resonance on calculations, the memory capacity of a Hopf oscillator reservoir is calculated. For the δ-delayed n-th parity function given by Equation (19), the memory capacity of the system can be calculated as follows.

Number

Number

Number

[0072] The effect of the non-linear activation function (tanh -1 X) on the performance of the Hopf oscillator RC was also studied, and the results are presented in Figure 21 showing the effect of the non-linear activation function on the performance of the Hopf RC (ω p = 20π(T p = 0.1 sec), ω0 = 40π, μ = 5, N = 1000 nodes, φ = π / 3 rad, simulation time 4000T p = 400 seconds). Plot (a) in Figure 21 shows second-order parity, plot (b) in Figure 21 shows third-order parity, plot (c) in Figure 21 shows fourth-order parity, and plot (d) in Figure 21 shows a chaotic time series. It can be seen that without non-linear activation, the Hopf oscillator RC shows similar performance for low-order tasks (e.g., second- and third-order parity tasks). However, the non-linear activation function becomes important when performing higher-order tasks. Therefore, the base Hopf oscillator dynamics have a certain computational ability. Furthermore, the linear oscillator is also tested as a reservoir computer in the presence of a non-linear activation function. In this case, the non-linear activation function cannot make the linear oscillator function as a reservoir computer. This is similar to the effect of non-linearity in acoustic conversion for the digit recognition task.

[0073] ESP is one of the fundamental properties found in successful reservoir computing frameworks. Previous limit cycle-based systems have been found not to meet the ESP requirements, but the Hopf RC formulation described in this disclosure is different because the limit cycle radius continues to change according to the forcing used to encode information. Furthermore, a reservoir system was constructed considering the resonance phenomenon. In the literature, generalized synchronization or common signal-induced synchronization has been used to verify the presence of ESP in a reservoir. ESP has been empirically studied to measure the stability of the input-driven reservoir dynamics. ESP means that the reservoir has echo state properties when the asymptotic trajectory of the reservoir state clearly depends on the input and is independent of the initial conditions. Therefore, in order to obtain echo state properties, the effect of the initial conditions on the reservoir dynamics should fade over time.

[0074] The echo state properties of the Hopf reservoir were studied when the reservoir is encoded with inputs of chaotic laser time series tasks and parity tasks. According to the algorithm for estimating the ESP of the reservoir, the ESP index was calculated for two benchmarks and averaged over 20 initial conditions randomly generated within the range of {-3, 3}. Using the same input sequence for all different initial conditions, the deviation of the reservoir state trajectories of different initial conditions was calculated. To calculate the deviation, the initial condition of (x0, y0) = (0, 0) was used as the common trajectory, while the other trajectories were derived from each of the different initial conditions, discarding the initial transient on the one hand. Finally, the average of the deviations was calculated to obtain the ESP index. When the ESP index of the reservoir dynamics becomes 0, the reservoir is said to have echo state properties. These results are given in Figure 22, where it is observed that the reservoir has echo state properties for the resonance condition (ω0 = Ω), but does not have this property when the resonance is not satisfied (ω0 ≠ Ω). This can also explain the reason why resonance is useful for calculations. It is also important to note that the reservoir can still have information processing capabilities when ESP is not satisfied.

[0075] Figure 22 shows the calculation of the echo state property (ESP) index for the chaotic laser time series task and the parity task with respect to the Hopf RC. Plot (a) in Figure 20 shows the ESP indices for the resonant and non-resonant conditions of the chaotic time series task, plot (b) in Figure 20 shows a set of random initial conditions (x0 and y0 selected from {-3, 3}) for the chaotic time series task, plot (c) in Figure 20 shows the ESP indices for the resonant and non-resonant conditions of the parity task, and plot (d) in Figure 20 shows a set of random initial conditions (x0 and y0 selected from {-3, 3}) for the parity task. Here, ω p = 20π(T p = 0.1 sec), ω0 = 80π, φ = π / 3 rad, A = 0.5, and the simulation time is 3000T p = 300 seconds. For the case of the resonant condition Ω = ω0 and the non-resonant condition Ω = 1.7391ω0

[0076] In the present disclosure, the Hopf oscillator was constructed as a reservoir computer to gain insights into the relationship between the dynamics of the oscillator and the computational ability of the RC. This implementation of the Hopf reservoir computer provides a simpler design by abandoning the commonly used delay feedback line and masking function. An analog electrical circuit was used as the physical realization of the reservoir. The input ω pWhen the ratio of the pseudo-frequency to the natural frequency of the oscillator ω0 is obtained from the fairy sequence, the reservoir demonstrates high computational ability. Furthermore, a resonance phenomenon occurs when the harmonic forcing frequency of the oscillator is equal to the natural frequency, which provides a favorable condition for constructing a reservoir computer. When the limit cycle radius is relatively small while the forcing amplitude is relatively large, an improvement in computational ability can be achieved. An Arnold tongue structure is observed in the information metric space of the reservoir near the resonance position, which is correlated with the Arnold tongue shown in the phase deviation space. The reservoir is also found to have both a maximum memory capacity and echo state characteristics when the resonance condition is satisfied, which indicates better computational performance in principle. Finally, the results also suggest that a reservoir computer can be constructed using only a single non-linear node without using a time multiplexing process or delay feedback. By utilizing some of the underlying dynamics of the system, a simple and robust limit cycle reservoir computer can be constructed.

[0077] The delay-free probabilistic limit cycle oscillator reservoir computer can be utilized with various techniques for recognition tasks. The limit cycle oscillator can be paired with, but not limited to, vibration signals such as speech recognition or other sound fields. Since the limit cycle oscillator is already a vibration system, this pairing is beneficial in applications such as those implemented on edge devices. For example, the delay-free probabilistic limit cycle oscillator reservoir computer can be paired with microphone technology for speech recognition tasks. FIGS. 23A and 23B show an example of a PRC implementing microphone technology for speech recognition tasks. The system of FIG. 23A includes a PRC and a readout. The PRC provides processing immediately after the microphone, eliminates data preprocessing, and enables high-speed analog computing and an adjustable physical layer for different tasks. The readout can be reconfigured for different tasks, can be directly deployed on an edge device, and can include a feedback loop to enhance the computing performance of the PRC. Similarly, the system of FIG. 23B includes a PRC for perception-based processing that eliminates data preprocessing and enables high-speed analog computing and an adjustable physical layer on different tasks, and a readout that can be reconfigured on different tasks, is directly deployed on the edge device, and can use a feedback loop to enhance the computing performance of the PRC.

[0078] The probabilistic limit cycle oscillator reservoir computer without delay can also be incorporated into some MEMS designs, including but not limited to, cantilever beams, condenser microphones, parallel plate actuators, mechanical Duffing oscillators, and quantum sensing. The limit cycle oscillator reservoir computer can be incorporated into optoelectronic designs (e.g., "Theoretical and experimental study of slow-scale Hopf limit-cycle in laser-based wideband optoelectronic oscillators" by G.R.G. Chengui et al., J. Opt. Soc. Am. B, Vol. 31, No. 10, p. 2310-2316, October 2014), nanoelectromechanical designs (NEMS, e.g., "Limit Cycle Oscillations in CW Laser-Driven NEMS" by K. Aubins et al., J. of Microelectromechanical Systems, Vol. 13, No. 6, pp. 1018-1026, Dec 2004), semiconductor laser designs (e.g., "Limit-Cycle Dynamics with Reduced Sensitivity to Perturbations" by T.B. Simpson et al., Physical Review Letters 112, 023901, January 2014), thermoacoustic designs (e.g., "Effect of amplitude and frequency of limit cycle oscillators on their coupled and forced dynamics" by D. Premraj et al., Nonlinear Dyn 103:1439-1452, Feb 2021), and quantum dot designs (e.g., "Excitability in a Quantum Dot Semiconductor Laser with Optical Injection" by D. Goulding et al., Physical Review Letters 98, 153903 April 2007). To explore the Hopf bifurcation as a processing mechanism, several speech recognition benchmark tasks have been completed.Machine learning in the readout layer can use ridge regression with a training time of less than one minute. Both the MEMS branch point and the excitation signal can be adjusted to enhance its processing ability. An example of the results obtained using pseudo-RC is shown in the following table. Benchmark tests such as these (see also, for example, FIGS. 9-12) verify the application of the limit cycle oscillator reservoir computer.

Table 2

[0079] It should be emphasized that the above-described embodiments of the present disclosure are only possible examples of the implementations described for a clear understanding of the principles of the present disclosure. Many variations and modifications can be made to the above-described embodiments without substantially departing from the spirit and principles of the present disclosure. All such modifications and variations are intended to be included within the scope of the present disclosure and are intended to be protected by the following claims.

[0080] The term "substantially" means allowing a deviation from a descriptive term that does not adversely affect the intended purpose. A descriptive term is implicitly understood to be substantially modified by a word even if the term is not explicitly modified by the word.

[0081] It should be noted that ratios, concentrations, amounts, and other numerical data may be expressed in range format in this specification. Such range format is used for convenience and brevity, and thus should be interpreted flexibly to include not only the numerically explicitly listed limits of the range, but also all individual numerical values or sub-ranges subsumed within that range as if each numerical value and sub-range were explicitly listed. By way of illustration, a concentration range of "about 0.1% to about 5%" should be interpreted to include not only the explicitly listed concentrations of about 0.1 wt% to about 5 wt%, but also the individual concentrations (e.g., 1%, 2%, 3%, and 4%) and sub-ranges (e.g., 0.5%, 1.1%, 2.2%, 3.3%, and 4.4%) within the indicated range. The term "about" can include conventional rounding by significant digits of the numerical value. Also, "about 'x' to 'y'" includes "about 'x' to about 'y'".

Claims

1. A physical reservoir computer comprising a processing circuit, the processing circuit comprising: an input layer, a reservoir comprising a forced limit cycle oscillator implemented without delay or feedback, a readout layer, and a physical reservoir computer.

2. The physical reservoir computer according to claim 1, wherein the forced limit cycle oscillator comprises a Hopf oscillator or a Lorenz oscillator.

3. The physical reservoir computer according to claim 2, wherein the forced limit cycle oscillator comprises a two-state forced Hopf oscillator.

4. The physical reservoir computer according to any one of claims 1 to 3, wherein the processing circuit comprises an analog processing circuit.

5. The physical reservoir computer according to claim 4, wherein the analog processing circuit comprises an operational amplifier and a multiplier.

6. The physical reservoir computer according to any one of claims 1 to 5, wherein the reservoir computer utilizes an aperiodic probabilistic mask.

7. The physical reservoir computer according to claim 6, wherein the aperiodic probabilistic mask is defined by white Gaussian noise.

8. The physical reservoir computer according to any one of claims 1 to 3, wherein the processing circuit comprises an optoelectronic circuit.

9. The physical reservoir computer according to any one of claims 1 to 8, wherein a vibration signal is applied to the input layer.

10. The physical reservoir computer according to claim 9, wherein the vibration signal is an audio signal.

11. The physical reservoir computer according to any one of claims 1 to 10, wherein the readout layer is trained to map the state of the forced limit cycle oscillator to a desired output.

12. The physical reservoir computer according to claim 11, wherein the training of the readout layer includes linear regression or ridge regression.

13. The physical reservoir computer according to claim 11, wherein the desired output is a logical output.

14. The physical reservoir computer according to claim 13, wherein the logical output is an XOR output.

15. The physical reservoir computer according to any one of claims 1 to 14, wherein the input layer encodes an applied signal for input to the reservoir.

16. The physical reservoir computer according to claim 15, wherein the applied signal is encoded as a continuous input function.

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