Delay-free, stochastic limit ring oscillator reserve computer and related methods

Abstract

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

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to and the benefit of the co-pending U.S. provisional application entitled “No-Delay, Stochastic Limit Cycle Oscillator Reservior Computer and Related Methods,” serial number 63 / 388,420, filed on July 12, 2022, which is hereby incorporated herein by reference in its entirety.

[0003] STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0004] This invention was made with Government support under W911NF-20-1-0336 awarded by the Army Research Laboratory - Army Research Office. The Government has certain rights in this invention. Background Art

[0005] Reservoir computing (RC) is an unconventional computing technique that exploits the physics of nonlinear dynamic systems to perform computations. The RC scheme differs from the Turing machine principle because the computations performed by RC do not rely on static memory. The computations are obtained by mapping the transient dynamics of nonlinear physical systems into a higher dimensional space. Some popular applications of RC include logical operations, spoken and handwritten digit recognition, wireless communications, complex and chaotic time series prediction, long-term chaotic time series prediction, image recognition, and morphological computing. Summary of the invention

[0006] Aspects of the present disclosure relate to reservoir computing. In one aspect, a physical reservoir computer includes a processing circuit, the processing circuit includes: an input layer, a reservoir, and a readout layer, the reservoir includes a forced limit cycle oscillator, and the reservoir is implemented without delay or feedback. In one or more aspects, the forced limit cycle oscillator may include a Hopf oscillator or a Lorenz oscillator. The forced limit cycle oscillator may include a two-state forced Hopf oscillator.

[0007] In various aspects, the processing circuit may include an analog processing circuit. The analog processing circuit may include an operational amplifier and a multiplier. The reserve pool computer may utilize a non-periodic random mask. The non-periodic random mask may be defined by white Gaussian noise. The processing circuit may include a photoelectric circuit. The vibration signal may be applied to the input layer. The vibration signal may be a speech signal. In some aspects, the readout layer may be trained to map the state of the forced limit cycle oscillator to the desired output. The training of the readout layer may include linear regression or ridge regression. The desired output may be a logical output. The logical output may be an XOR (exclusive or) output, a NOT (non) output, an AND (and) output, or an OR (or) output. The input layer may encode an applied signal for input to the reserve pool. The applied signal may be encoded as a continuous input function.

[0008] Other systems, methods, features and advantages of the present disclosure will be or 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 intended to be included in this specification, within the scope of the present disclosure and protected by the appended claims. In addition, all optional and preferred features and modifications of the described embodiments may be used in all aspects of the present disclosure taught herein. In addition, the various features of the dependent claims and all optional and preferred features and modifications of the described embodiments may be combined and interchanged with each other. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Many aspects of the present disclosure may be better understood with reference to the following drawings. The components in the drawings are not necessarily drawn to scale, but emphasis is placed on clearly illustrating the principles of the present disclosure. In addition, in the drawings, the same reference numerals refer to corresponding parts throughout the several views.

[0010] Figure 1 An example of mapping dynamics of a Hopf oscillator of a reservoir computer according to various embodiments of the present disclosure is shown.

[0011] Figure 2 An example of continuous prediction together with a corresponding target signal according to various embodiments of the present disclosure is shown.

[0012] Figure 3 The natural frequency ω0 and the pseudo period T of the oscillator according to various embodiments of the present disclosure are shown. p An example of the relationship between .

[0013] Figure 4 An example of the relationship between computing power, noise amplitude σ, and noise offset β according to various embodiments of the present disclosure is shown.

[0014] Figure 5An example of a physical reservoir computer (PRC) according to various embodiments of the present disclosure is shown.

[0015] Figure 6 The use of various embodiments according to the present disclosure is shown Figure 5 Example of a solution to the XOR task for PRC.

[0016] Figure 7 The use of various embodiments according to the present disclosure is shown Figure 5 Example of a solution to the parity task of the PRC.

[0017] Figure 8 1 and 2 show a basic logic gate according to various embodiments of the present disclosure. Figure 5 An example of a response from the PRC.

[0018] Fig. 9 An example of a nonlinear auto-regressive moving average (NARMA) task solution according to various embodiments of the present disclosure is shown.

[0019] Fig.10 An example of normalized mean square error (NMSE) for evaluating PRC performance for a NARMA task according to various embodiments of the present disclosure is shown.

[0020] Fig.11 An example of comparison of PRC performance for the Santa Fe prediction task according to various embodiments of the present disclosure is shown.

[0021] Fig.12 An example of comparison of PRC performance for a sunspot prediction task according to various embodiments of the present disclosure is shown.

[0022] Fig.13 An example of a parity check task utilizing a Lorenz reservoir computer according to various embodiments of the present disclosure is shown.

[0023] Fig.14 Examples of parameter sweeps of limit cycle radius and harmonic forced amplitudes of a Hopf reservoir computer are shown in accordance with various embodiments of the present disclosure.

[0024] Fig.15 An example of a fine resolution parameter study of the resonance constants and harmonic forced frequencies of a Hopf reservoir computer based on a parity task according to various embodiments of the present disclosure is shown.

[0025] Fig.16 An example of a course resolution parameter study based on a parity task according to various embodiments of the present disclosure is shown.

[0026] Fig.17 An example of the effect of frequency ratio and harmonic forced amplitude on the computational power of a Hopf reservoir computer is shown according to various embodiments of the present disclosure.

[0027] Fig.18 An example of the effect of synchronization on the computational power of a Hopf reservoir computer according to various embodiments of the present disclosure is shown.

[0028] Fig.19 An example of a time history of an oscillator response of a Hopf reservoir computer is shown in accordance with various embodiments of the present disclosure.

[0029] Fig. 20 An example of storage capacity calculation of a Hopf reservoir computer according to various embodiments of the present disclosure is shown.

[0030] Fig.21 An example of the effect of a non-linear activation function on the performance of a Hopf reservoir computer according to various embodiments of the present disclosure is shown.

[0031] Fig. 22 Examples of echo state characteristic index calculation for chaotic laser time series tasks and parity check tasks according to various embodiments of the present disclosure are shown.

[0032] Fig.23A and Fig. 23B An example of a PRC for a voice recognition task implemented using microphone technology according to various embodiments of the present disclosure is shown. DETAILED DESCRIPTION

[0033] Various examples related to reservoir computing are disclosed herein. By discarding delay and feedback lines to create and couple virtual nodes while relying on the physics of the oscillator, a simplified version of a time-multiplexed reservoir computer is provided. A forced limit-cycle oscillator (e.g., a Hopf oscillator or a Lorenz oscillator) can be used as a basic nonlinear dynamic system, which can be manufactured as a circuit. In order to avoid segmented and node-related masks, a random masking signal independent of the node generated according to white Gaussian noise can be used. The parameters of the Hopf oscillator are tuned, and the nodes can be coupled so that they have rich dynamics for the RC scheme. Reference will now be made in detail to the description of the embodiments shown in the accompanying drawings, wherein the same reference numerals represent the same parts throughout several views.

[0034] Due to the echo state network structure, many physical systems have been used as reservoirs, which are generally referred to as physical reservoir computers (PRCs). Some classical PRCs include Duffing oscillator arrays, Hopf oscillators based on limit cycles, soft robot bodies, tensile structures, and origami structures. In addition to systems from classical physics, quantum physics systems can be used as RCs to perform tasks from both classical and quantum domains. The natural disordered quantum dynamics of the ensemble system can be used to simulate nonlinear time series, including chaotic systems. Kerr (Kerr) nonlinear oscillators can be used for sine wave phase estimation, using their complex amplitudes as computing nodes. Nuclear magnetic resonance spin ensemble systems can be used for nonlinear dynamics simulation tasks by implementing spatial multiplexing methods to increase computing power. Dissipative quantum dynamics can be used to build quantum reservoir computers for nonlinear time tasks. Statistical physics plays an important role in the theoretical development of neural networks, and neural networks form connections between information processing and physics.

[0035] Generally, delayed dynamic systems can be used as reservoirs from a single nonlinear node. Coupled delay systems can also be used to perform calculations by making deep neural networks and signal processors. Simpler implementations can also be achieved by excluding delays or feedback lines. In this paper, the reservoir computer is constructed by implementing a two-state Hopf oscillator. The Hopf oscillator reservoir computer has been previously studied and shown to successfully complete several benchmark tasks. Due to the existence of stable limit cycles, the Hopf oscillator has the ability to store and learn information, which also makes it suitable for building adaptive oscillators. Conventionally, binary masks are used for time-division multiplexing processes to create virtual nodes for calculations. In addition, noise can also be used as a mask. Previously, eigenvalue analysis was linked to non-resonant conditions to design a reservoir computer that operates close to stable equilibrium. However, the focus of the present disclosure is different because the Hopf reservoir is a reservoir based on limit cycles, so the analysis will not apply in this article. It should also be noted that the popular concept of "edge of chaos" is not used in this disclosure to optimize reservoir performance. The edge of chaos is not a necessary condition for achieving good computing power of the reservoir computer. So, staying away from chaotic regions and tuning a set of network parameters can also be a route to constructing a reservoir computer with good performance.

[0036] Microwave-based magnetically forced synchronization is implemented in spintronic oscillators to improve reservoir computing performance. The spin dynamics of magnetic tunnel junctions are also used to build reservoir systems. In addition, nanoscale spintronic oscillators are optimized as reservoirs based on magnetization dynamics. These spintronic oscillators can be further optimized by exploiting the relationship between the input period (pseudo-frequency) and forced.

[0037] In one example, a driven Hopf oscillator is studied as a reservoir computer, excluding both the masking function and the commonly used delay line; the masking function and the delay line are discarded to focus on the dynamic calculation of the oscillator. Resonance phenomena, Arnold tongue, and Farey sequences all contribute to the performance of the Hopf oscillator as a reservoir computer. The Arnold tongue refers to the phase-locked or synchronized region in the parameter space, which has a strong impact on this Hopf oscillator reservoir. The parameter study of this Hopf oscillator computer is performed using parity and chaotic laser time series benchmarks. The oscillator is experimentally implemented as an analog circuit to study the information processing capabilities of the reservoir. A modified version of Shannon's information rate can be used as a performance measure for parity tasks.

[0038] In addition, a reservoir computer can be developed from a virtual node network that simultaneously excludes feedback while utilizing the nonlinearity of a single Lorenz system. This may lead to a simpler and cheaper way to build a reservoir computer based on virtual nodes. These dynamically coupled virtual nodes can be reused in time using a random masking process. Due to the presence of noise in the system, the Euler-Maruyama method can be used to simulate the system of equations of the Lorenz reservoir computer. It was found that the resulting reservoir computer successfully performed 2nd, 3rd and 4th order parity check tasks, while the 5th order task failed. Since these are logical tasks, the Shannon information metric can be used to quantify the performance of the reservoir computer.

[0039] Hopf Physical Reserve Computer

[0040] Reservoir computing (RC) is a bio-inspired, supervised machine learning computational framework based on artificial recurrent neural networks (RNNs) that exploits the natural emergent dynamics of physical resources. Conventional machine learning schemes use backpropagation through time to train the entire recursive neural network. This approach is computationally expensive because all weights of the network need to be updated to mimic the target function. Echo state networks and liquid machines are two concepts that addressed this problem in the early 2000s. Reservoir computing merges these concepts. In reservoir computing, the neural network is formed by a set of coupled nonlinear nodes, where the network is divided into three parts: the input layer, the reservoir, and the readout layer. Unlike conventional RNNs, only the readout layer needs to be trained by a simpler training algorithm (such as linear or ridge regression). As a result, the RC architecture is much faster and more stable than conventional RNN methods, which is an advantage of this information processing framework.

[0041] There are many real-world applications of reservoir computing, including bit-by-bit logic operations, speech recognition, handwritten digit recognition, wireless communications, complex and chaotic time series prediction, image recognition, simulation of nonlinear time series, and morphological computing. The echo state architecture of the reservoir allows the use of physical systems as reservoir computers, also known as physical reservoir computers (PRCs). Many physical systems have been shown to perform as PRCs, including arrays of nonlinear mechanical oscillators, soft robotic bodies, tensegrity structures, and origami structures.

[0042] Importantly, quantum systems can be used as PRCs. The quantum dynamics of the natural disorder of an ensemble system is exploited to simulate nonlinear time series, including chaotic systems. Kerr nonlinear oscillators are used for sine wave phase estimation using their complex amplitudes as computational nodes. Nuclear magnetic resonance spin ensemble systems are used for nonlinear dynamics simulation tasks by implementing spatial multiplexing methods to increase computational power. Dissipative quantum dynamics is used to build quantum reservoir computers (QRCs) for nonlinear time tasks.

[0043] Physical reservoir computers were initially constructed from coupled actual dynamic nodes only. Later, a virtual node-based reservoir computing method was proposed, which implemented a time multiplexing method in which delayed feedback was used as a single nonlinear dynamic node to perform calculations. This method simplifies the complexity of the reservoir built from a physical nonlinear node array. This method can be used to construct physical reservoir computers for different tasks, such as optoelectronic oscillators for optical information processing, photonics-based passive linear fiber reservoirs for signal processing, FPGA implementations of pattern recognition using single autonomous Boolean logic elements, time-delayed reservoirs for predicting random nonlinear time series, delayed Duffing silicon beams for parity check tasks, and / or semiconductor lasers with delayed optical feedback for nonlinear time series prediction. These reservoirs can use delay lines to create the necessary nodes for calculations. By creating nodes without any delay or feedback lines, a simpler approach can be adopted.

[0044] In this paper, the Hopf oscillator is used as a physical reservoir. The Hopf oscillator can also be used as a building block for an adaptive oscillator that can learn information locally without any training. The Hopf oscillator can exhibit limit cycle motions, which provide a memory source by storing information in its dynamic state. Although binary periodic masking functions can be used for time-multiplexed reservoirs, noise can also be used as a periodic mask. In this paper, the Hopf oscillator PRC is constructed to use a non-periodic random mask. The Hopf oscillator physical reservoir computer is fabricated as an analog circuit, which is compared with the Euler-Maruyama simulation. The Hopf PRC can successfully complete benchmark machine learning tasks, including parity check tasks, basic logic gate tasks, nonlinear dynamic simulation tasks, and various time series prediction tasks. The information rate can be used as a performance measure for logic tasks, and the normalized mean square error (NMSE) can be used for simulation tasks and time series tasks.

[0045] Next, the equations of motion for a random Hopf oscillator PRC are presented, followed by a discussion of methods for mapping oscillator dynamics to information processing schemes for example tasks using Euler-Maruyama simulations. The effects of pseudoperiodicity and noise on computational performance are discussed and simulation circuit experiments are described. Different benchmark tasks are performed with numerical and experimental Hopf PRCs, including logic tasks, simulation tasks of time series, and prediction tasks. A list of parameters, states, and functions used in the following discussion is provided below.

[0046]

[0047] The system equations for the Hopf physics reservoir computer. The equation of motion for the Hopf oscillator is:

[0048]

[0049] For this Hopf oscillator, x and y are the first and second states respectively, and the sinusoidal forcing is given by A sin(Ωt+φ). The list of parameters is given in the table above. The information is first encoded as the input u(t), which will depend on the benchmark task being performed. The mask can be defined by white Gaussian noise:

[0050]

[0051] Here, σ is the noise amplitude, is white Gaussian noise, and β is a positive bias. It should be noted that does not exist, but its differential form dW does exist.

[0052] To send information to the PRC for processing, an external forced function containing the information signal u(t) and the random mask m(t) can be constructed as:

[0053] f(t)=1+u(t)m(t) (3)

[0054] This external forcing function is injected into both the amplitude A of the sinusoidal forcing and the parameter μ that affects the radius of the limit cycle. Including this forcing, the equation for the Hopf PRC is written as:

[0055]

[0056] Mapping method. In order to use the dynamics of the Hopf oscillator as a physical reservoir computer, the dynamics needs to be mapped. To describe the mapping, the exclusive OR (XOR) logic task is used as an example. The Hopf PRC can be simulated using the Euler-Maruyama scheme because the mask is random. The Shannon information metric can be used to quantify the performance of the reservoir when performing a logic task (such as an XOR operation).

[0057] For this task, binary “false” and “true” values ​​are encoded as discrete negative and discrete positive values, respectively, in a discrete signal r(z). r(z) is defined such that z∈Z + and r(z)∈{-1,+1}, which is Figure 1 Depicted in. Figure 1 The graph (a) shows a discrete random binary signal r(z), while Figure 1 The graph (b) shows the continuous input signal u(t). Figure 1 Graph (c) shows an example of a random masking function m(t), graph (d) shows the time history x(t), and graph (e) shows the rescaled time history X(t). Figure 1 The graph (f) shows a single pseudo-period T p An example of 20 equally spaced nodes, represented by circles. Figure 1 The graph (g) shows an example of node states collected from nodes for a machine learning input dataset, with different colors representing different nodes. For the simulations depicted in this article, the parameters are set such that: μ = 5, A = 0.5, Ω = 40πrad / s, ω0 = 40πrad / s, T p =0.1 seconds, N=20 nodes, φ=π / 3rad, σ=100 and β=1.0.

[0058] To input them into a continuous dynamical system, these values ​​are first mapped to a continuous input function u(t) as follows:

[0059] u(t)=r(z) for (n-1)T p ≤t<(n)T p , n∈Z + (5)

[0060] exist Figure 1 This function is depicted in the graph (b) of T p is a constant pseudo-period in which u(t) does not change its value. Therefore, for the XOR logic task, the input function u(t)∈{-1,+1} is a pseudo-period T p This implies that each "true" (e.g., +1) or "false" (e.g., -1) value is p Influence the system. Figure 1 The mask function m(t) is depicted in graph (c).

[0061] The Hopf PRC system described in equation (4) can be integrated numerically using the Euler-Maruyama (EM) method since the PRC is stochastic. For these simulations, the integration time step dt = 10 -5 seconds, the total simulation time in this case is 3000T p = 300 seconds, T p = 0.1 seconds. This example simulates Figure 1 Shown in.

[0062] exist Figure 1 The time history of the x state obtained from the simulation is depicted in Figure (d). Next, x(t) can be calculated using equation (6) by subtracting the average value μ x and divided by the standard deviation σ x To rescale:

[0063]

[0064] In this equation, the inverse hyperbolic tangent function is used as the nonlinear activation function. The real part of is used in the subsequent steps. Figure 1 The time history of the X state is depicted in graph (e).

[0065] Next, the equidistant nodes can be obtained by replacing each pseudo-period T p Divide it equally into N (=20) nodes to create, such as Figure 1 As shown in the curve diagram (f). In each pseudo cycle T p In the example, the N node values ​​are called node states. Figure 1 The curve diagram (g) is depicted in FIG.

[0066] The node matrix S is an N×K matrix; for this example, N=20 is the number of nodes in the pseudocycle, and K=3000 is the total number of pseudocycles. Truncating this S matrix (600T p ) to form the new matrix L(480T p ), which will be used in the training process. The reservoir computer can be trained using ridge regression, as in:

[0067]

[0068] The target signal (M vector) can be created from the encoded input based on the reference task, which in this case is the XOR task. For each pseudo-cycle, there will be a target value that is found by performing an XOR operation between the inputs r(z) and r(z-1). In this way, the target vector M is found for the XOR task. Linear regression-based training can then be applied to the node state matrix L in order 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, and λ=10 -1 is a regularization parameter used to avoid overfitting, and o(k) is the prediction of the reserve pool computer at the kth pseudo-period.

[0069] The discrete random binary input signal r(z) and the continuous input signal u(t) are shown in Figure 2 For the simulations described in this article, these parameters were set so that μ = 5, A = 0.5, Ω = 40πrad / s, ω0 = 40πrad / s, T p =0.1s, N=20 nodes, φ=π / 3rad, σ=100 and β=1.0. Figure 2 The curve graph (c) of shows the continuous prediction and the corresponding target signal. In the final step, the prediction is binarized because XOR is a binary task. Figure 2 The discretized target and prediction are depicted in graph (d). Note that nonlinear dynamic simulation tasks will not require this final step of discretization.

[0070] For logical tasks, the power of the reservoir computer is quantified using the Shannon information rate. The information rate, R, can be defined as follows:

[0071] R=H(x)-H y (x) (8)

[0072] Here, H(x) is the Shannon entropy, which indicates how much information is encoded in the signal. This can be defined as follows:

[0073]

[0074] In this equation, p i is the probability of getting a specific bit i. y (x) is the conditional entropy, which represents the probability of getting an incorrect bit in the target signal:

[0075]

[0076] Here, and p(i,j) is the joint probability distribution of two variables i and j, each of which can take the value of "1" or "-1" for a logical task. i is the bit from the target and j is the bit from the prediction. The information rate R for this case is calculated to be 0.98 based on the predictions from the validation part (not included in the training process). Due to the nature of this binary target signal, the Shannon entropy is 1.0, which marks the maximum value of the information rate for this task. It should be noted that the lower limit of R is 0, which will be achieved if every prediction is incorrect, while the upper limit of R depends on the task. For the parity check task considered in this paper, the upper limit of R is equal to 1.

[0077] Pseudo-cycles and noise. Next, we explore the impact of pseudo-cycles and noise on the computational capacity of the reservoir. For this discussion, several parity tasks (defined in equation (12) below) are used to understand the impact of pseudo-cycles and noise on the computational capacity of the reservoir.

[0078] exist Figure 3 The pseudoperiod T is explored using second-order and fourth-order parity check tasks. p The relationship between the oscillator's natural frequency ω0. Figure 3 shows the use of (a) T p = 0.05 seconds, (b) T p = 0.1 sec and (c) T p = 0.15 seconds of second-order and fourth-order parity check tasks in the pseudo-period T p Comparison of the computational performance R of the reservoir on the choice of the natural frequency ω0. Simulating different ratios of natural and pseudoperiods (e.g. T p ), and the ratio is plotted for the peak in the information metric. ω0=Ω=50πrad / s is the resonance case. The parameters are set such that: μ=5, A=0.5, Ω=50πrad / s, N=1000 nodes, φ=π / 3rad, σ=15, and β=1.0.

[0079] exist Figure 3 In the curve diagram, when changing the natural cycle When three different T pThe performance of the reserve pool computer was measured. It was found that the reserve pool has better performance when the pseudo period is an integer multiple of the natural period of the oscillator. The performance of the reserve pool was studied using both resonant (ω0=Ω) and non-resonant (ω0≠Ω) conditions. It was found that both cases can lead to strong or weak computing power depending on the fractional relationship between the natural period and the pseudo period. However, when T p Very low (e.g., T p = 0.05 seconds), maintaining this design still may not make it possible to make a robust reservoir computer. For the remainder of this discussion, choose T p The combination of and ω0 makes the pseudo-period an integer multiple of the natural period of the oscillator.

[0080] Noise is ubiquitous in physical systems. For this reason, a random masking function is used to introduce noise into the system. Figure 4 The relationship between computing power as measured using R, noise amplitude σ, and noise offset β is shown. Figure 4 The simulations presented in Figure 1 are performed for a 4th order parity check task (left) and a 6th order parity check task (right), showing the effects of σ and β. The parameters are set such that: μ = 5, A = 0.5, Ω = 40πrad / s, ω0 = 40πrad / s, T p = 0.1 seconds, N = 1000 nodes, and φ = π / 3rad. The reservoir is found to be robust against a certain level of noise intensity, which demonstrates its potential for implementation under the influence of environmental noise. However, increasing the noise intensity does reduce the computational power of the reservoir. For high-order tasks that require longer memory (e.g., Figure 4 This effect can be observed for the 6th order parity check task of β = 0. The computational power is lowest when β = 0. Since aperiodic noise masks with increasing noise strength reduce the computational power, it should be noted that the Hopf reservoir computer can also be constructed by excluding the noise mask (σ = 0).

[0081] Analog circuit experiment

[0082] To build a physical reservoir computer (PRC), an analog circuit implementation of equation (4) was designed, fabricated, and tested. The equation for the circuit is given in equation (11):

[0083]

[0084] Here, V u is the input voltage, V m is the random masking voltage, V μ is the limit cycle radius voltage, is the resonant constant voltage. V x and V yThe states corresponding to states x and y in equation (4) are implemented using a TL082 op amp and an AD633 multiplier in a standard integrator network configuration. The error tolerance for the resistors is 1% and for the capacitors is 2%. The continuous input function V is created in MATLAB u , Random masking function V m and sin(Ωt+φ) are forced and sent to the circuit via a National Instrument (NI) cDAQ-9174. This cDAQ-9174 also collects V x Status and V y Status. Use 10 5 Data for all experiments were collected at a sampling frequency of 10 samples / s. The resistor values ​​were chosen such that R1 = 10 kΩ and R2 = 100 kΩ, and the capacitor value was chosen such that C = 0.1 μF. Figure 5 shows a state V x and V y A simplified schematic diagram of the HopfPRC, where

[0085] V can be treated in the same way as the x state in the "Mapping Method" section. x state. That is, V x The states will be rescaled using equation (6), and the rescaled states will then be used to form the node state matrix L. The target signal vector M will be created following the same process discussed in the "Mapping Method" section. Finally, equation (7) can be used to train the PRC to map the input data to the desired output values. As an example, the analog circuit Hopf PRC is used to solve the XOR task in the previous section, which is Figure 6 Depicted in. Figure 6 The graph (a) shows the input voltage signal V u , Figure 6 The curve (b) shows that V x The time history, Figure 6 The graph (c) shows the XOR target signal M and the prediction, Figure 6 The plot (d) of shows the discretized predictions. The information measure calculated is R = 1.0. For the experimental results depicted here, the parameters are set so that: V μ =5V, A = 0.5V, Ω = 40πrad / s, Volt, T p = 0.1 seconds, N = 20 nodes, φ = π / 3 rad, σ = 10 volts, and β = 1.0 volts. For this case, the information rate R is calculated to be 1.0 based on the predictions from the validation part (not included in the training process).

[0086] Hopf PRC Benchmark Tasks. Hopf PRC is numerically and experimentally tested using three benchmark tasks: (1) logic tasks, (2) time series simulation tasks, and (3) prediction tasks. The logic tasks include tasks of basic logic gates of different orders and parity check tasks. The time series simulation tasks test the PRC's ability to reproduce nonlinear autoregressive moving average (NARMA) tasks of different orders. The prediction tasks include the Santa Fe time series and the sunspot prediction task.

[0087] Logical benchmark task. Parity check task. First, the parity check benchmark task is used to evaluate the computational efficiency of the reserve pool. Since it is a logical task, the input function u(t) is generated by a random binary signal r(z) as discussed in the "Mapping Method" section. The n-th order parity check function P n It can be defined by the following equation:

[0088]

[0089] As n increases, the task will utilize more memory and nonlinearity from the reservoir. As given in the "Mapping Methods" section, the Shannon Information Metric can be used to measure the performance of the PRC for logic tasks. For n=1, the task is linear because the first-order task does not require any memory for the input from the previous pseudocycle. For n>1, the task is nonlinear, which requires that the reservoir computer must also have memory and nonlinear separation capabilities. Figure 7 In , the ability of HopfPRC to follow parity check tasks of 2nd to 5th order is shown experimentally and in simulation. Figure 7 A comparison of the performance of PRC for the parity check task is shown. Figure 7 The graph (a) shows the discrete input function r(z). Figure 7 The curve (b) shows the second-order parity check task (information metric: R exp =1.00,R sim =0.98), Figure 7 The curve (c) shows the third-order parity check task (information metric: R exp =1.00,R sim =0.98), Figure 7 The curve (d) shows the fourth-order parity check task (information metric: R exp =0.68,R sim =0.93), Figure 7 The curve (e) shows the fifth-order parity check task (information metric: R exp =0.31,R sim=0.74). Set the parameters so that: V μ =μ=5, A=0.5, Ω=40πrad / s, T p = 0.1 seconds, N = 1000 nodes, φ = π / 3rad, σ = 15 and β = 1.0, total time 5000T p = 500 seconds (only a portion of the discrete prediction is shown). Initial 4000T p = 400 seconds for training, and finally 1000T p = 100 seconds for testing. The performance difference between PRC experiments and simulations can be attributed to the presence of nonlinear circuit components in the simulated circuit that are not included in Eq. (11). For example, V u The jump between -1 and +1 must be made, but this instantaneous change takes a finite amount of time in the circuit.

[0090] Basic logic gate tasks. The computational performance of the reservoir is also evaluated using basic logic gates: AND(∧), OR(∨). As discussed in the “Mapping Method” section, the input function u(t) can be generated with a random binary signal, and the Shannon information metric is again used to measure the performance of this PRC. Figure 8 The response of a Hopf PRC used as a basic logic gate is described experimentally and by simulation. Figure 8 A comparison of the performance of PRC for even and odd tasks is shown. Figure 8 The graph (a) shows the input function u(t), and the graph (b) shows The information rate is at the theoretical maximum for all numerical and experimental results. The Hopf PRC can act as any basic logic gate. The parameters are set so that: V μ =μ=5, A=0.5, Ω=40πrad / s, T p = 0.1 seconds, N = 1000 nodes, φ = π / 3rad, σ = 15 and β = 1.0, total time 3000T p = 300 seconds (only a portion of the discrete prediction is shown). In all cases, the Hopf PRC achieves the maximum information rate.

[0091] Simulation tasks. The reservoir is also evaluated using simulation tasks. Nonlinear autoregressive moving average (NARMA) time series can be used to test whether the reservoir has sufficient nonlinearity and long time lag. These tasks show the multi-tasking capability of the reservoir. NARMA tasks from order 2 to order 20 are used to test the reservoir. The NARMA task of order n is given in equation (13), where the initial target value is set to 0.19:

[0092]

[0093] In equation (13), M n is the goal of the system, n is the order of the NARMA task, and (α, ζ, γ, δ) = (0.3, 0.05, 1.5, 0.1). u(t) is the continuous input for the forced Hopf PRC, which is a function of three sine functions. It should be noted that this formula for the NARMA simulation task is non-standard. The u(t) given in equation (13) is used for other dynamic systems where inertia plays a large role. Similarly, the non-standard NARMA task can be used here to evaluate the analog circuit reservoir. The reservoir simulates the nonlinear function, but it should be noted that the correlations present in equation (13) do not allow an explicit assessment of the long-term memory properties of the reservoir.

[0094] In the simulation and experiment, Δt = 0.1 s and the sampling rate is 10 5 samples / second. Fig. 9 Several NARMA tasks are shown. Instead of information rate, the normalized mean square error (NMSE) is used to evaluate the performance of the reservoir computer for NARMA tasks:

[0095]

[0096] Finally, 20% of the target signal (16,000-20,000 pseudo cycles) is used for verification. n is the goal, and o n is the prediction from the reservoir computer. In equation (14), j0 is the starting time step, j f is the ending time step from the test segment. Fig. 9 A comparison of the performance of PRC on the NARMA task is shown. Fig. 9 The graph (a) shows the input function u(t). Fig. 9 The curve (b) shows the performance of the 2nd-order NARMA task (performance metric: NMSE exp =2.8181×10 -6 ,NMSE sim =7.8199×10 -7 ), Fig. 9 The curve (c) shows the 10th-order NARMA task (NMSE exp =0.0037,NMSE sim =5.0362×10 -4 ), and graph (d) shows the 20th-order NARMA task (NMSE exp =0.0060,NMSE sim =0.0033). Only a portion of the results are shown in each graph. The parameters are set so that: V μ =μ=5, A=0.5, Ω=40πrad / s, T p =0.1 seconds, N=1000 nodes, φ=π / 3rad, σ=15 and β=1.0. Fig.10 Plots of NMSE for 2nd to 20th order NARMA tasks for simulation and experiment are shown. The parameters are set such that: V μ =μ=5, A=0.5, Ω=40πrad / s, T p = 0.1 seconds, N = 1000 nodes, φ = π / 3 rad, σ = 15 and β = 1.0. Fig. 9 and Fig.10 The numerical simulation of Hopf PRC shows a much superior performance compared to the experiment. However, up to the 20th order task, both have acceptable performance. PRC can perform NARMA tasks of much higher order than parity tasks.

[0097] Forecasting Tasks. Santa Fe Tasks. Time series forecasting is an important benchmark for the pool. The Santa Fe time series was first used as a benchmark in the Time Series Forecasting Competition. The Santa Fe Time Series Dataset A is a univariate time series discovered from the intensity of recorded chaotic far-infrared lasers. A target signal can be generated to predict the value of the next time step based on the values ​​of the current and previous time steps. Fig.11 A comparison of the PRC performance for the Santa Fe prediction task is shown, which shows the performance of HopfPRC on laser time series in both experiments and numerical simulations. NMSE is used as the performance metric. Fig.11 The graph (a) shows the Santa Fe chaotic time series for the laser intensity prediction task (performance metric: NMSE exp =0.0615,NMSE sim =0.02), and graph (b) shows the Santa Fe heart rate prediction task (NMSE exp =6.0258×10 -4 ,NMSE sim =6.5060×10 -4), and graph (c) shows the Santa Fe respiratory effort prediction task (NMSE exp =0.1826,NMSE sim =0.1753), and curve (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 set so that: V μ =μ=5V, A=0.5V, Ω=40πrad / s, Volt, T p = 0.1 sec, N = 1000 nodes, φ = π / 3 rad, σ = 15 volts, and β = 1.0 volt. Only a portion of the results is shown in each figure.

[0098] The Santa Fe Time Series Dataset B is a multivariate time series found in the sleep laboratory of Beth Israel Hospital in Boston, Massachusetts. The dataset was taken from the MIT-BIH Polysomnography Database Records (slp60) and submitted to the Santa Fe Time Series Competition in 1991. Heart rate, chest volume (respiratory effort), and blood oxygen concentration constitute the targets.

[0099] For each of these time series, a target signal is generated again to predict the next time step based on the values ​​of the current and previous time steps. In each case, the original time series is normalized for use as input. Graphs (b), (c) and (d) show the performance of the reserve pool computer to predict subsequent values ​​of heart rate, respiratory effort and blood oxygen concentration through experiments and numerical simulations, respectively. NMSE is calculated in each case to evaluate the performance of the reserve pool.

[0100] Sunspot prediction task. Prediction of the total number of sunspots (S n ) is also a one-step time series forecasting task similar to the Santa Fe time series. The reservoir computer uses daily and monthly total sunspot numbers for one-step forecasting purposes. The required dataset is taken from WDC-SILSO at the Royal Observatory in Brussels, Belgium. Again, for each time series, a target signal is generated to predict the next value based on the values ​​of the current and previous time steps, and the original time series is normalized to be used as the input to the oscillator. Fig.12 Shows the sunspot forecast (S n ) task performance comparison. Fig.12The top graph shows the performance of the pool in predicting the next step growth of the total daily count of sunspots. Fig.12 The bottom plot of shows the performance in predicting the monthly count of sunspots. Again, NMSE is used to evaluate the efficacy of the reserve pool for this task. For both the experiment and the numerical simulation, the daily total counts for the sunspot prediction task are Fig.12 The top graph shows (performance metric: NMSE exp =0.0548,NMSE sim =0.0534). The monthly average total number of sunspot prediction tasks is Fig.12 The bottom graph shows (performance metric: NMSE exp =0.0595,NMSE sim =0.0455). The parameters are set so that: V μ =μ=5V, A=0.5V, Ω=40πrad / s, N = 1000 nodes, φ = π / 3 rad, σ = 15 volts, and β = 1.0 volts.

[0101] In this paper, the Hopf oscillator is explored as a physical reservoir computer by adopting a time-division multiplexed node-based architecture with a random masking function. Discarding the conventionally used delay line, this Hopf PRC is a simple and cheap way to create a physical reservoir computer. Since quantum systems are capable of limit cycle motion, the Hopf PRC formula can be applied to quantum PRC. The Euler-Maruyama method is used for numerical simulation of this Hopf PRC. The analog circuit of this Hopf PRC is developed, fabricated and tested. The Hopf PRC is found to have multi-tasking capabilities as it is shown to perform logical operations, simulation tasks and time series prediction tasks. Inspired by adaptive oscillators, the input signal is injected into multiple locations, including parameters that affect the limit cycle radius and the amplitude of the sinusoidal forced. In addition, the masking function used in this PRC is random. Since this PRC architecture is tested with noise, it is also suggested that this reservoir computer is robust to environmental noise in practical implementation.

[0102] Reservoir Calculation with Lorenz System

[0103] The information processing capabilities of the Lorenz system, one of the first systems shown to exhibit chaos, are explored from the perspective of reservoir computing. The Lorenz system can be used as a reservoir computer with an echo state network approach for separating superimposed chaotic signals and with small-world and scale-free network approaches for reproducing the long-term nonlinear properties of the system. Coupled Lorenz oscillators can also be used as a reservoir computer by signal reconstruction exploiting continuous transient oscillatory dynamics. In this paper, the Lorenz system is implemented as a reservoir computer by coupling multiple virtual nodes via a time-division multiplexing method. Mackey-Glass oscillators are used to perform speech recognition tasks, and silicon beams are also used as reservoir computers by exploiting Duffing nonlinearities to perform parity check tasks.

[0104] The equation for the Lorenz reservoir computer used in the following discussion is given by:

[0105]

[0106] Among them, y new =y+u(t)m(t) and ρ new =ρ(1+u(t)m(t))sin(Ωt+φ), where u(t) is the input of the reservoir computer, is a masking function based on random forced noise, a is the noise amplitude, is white Gaussian noise, and b is a positive constant bias. The input can be defined as u(t)-r(z) for (n-1)τ≤t<(n)τ, where r(z) is a discrete signal with values ​​of {-1,1}, n,z∈Z + The performance of this pool computer has been evaluated by using the parity check task, where u(t) = [-1, +1] is a random square wave with pseudo-period τ. λ is the time difference between two consecutive nodes. By numerically integrating Equation 15 with the Euler-Maruyama method, equidistant virtual nodes (N=100) were formed. Next, the reservoir computer was trained using the following set of equations:

[0107]

[0108] For each pseudocycle τ, the reservoir computer is trained using the y states of 100 nodes (L matrix). After the benchmark task, the target signal (M vector) is generated from the time history of the input. Linear readout training is applied to the node states to map them to the desired output using Equation 16, where w is the weight vector found after training, I is the identity matrix, and θ=10 -1is a regularization parameter used to avoid overfitting, and o(k) is the prediction of the reservoir computer at the kth time step.

[0109] n-th order parity check function P n Depend on Definition. As n increases, the task exploits more memory and nonlinearity of the reservoir computer to perform the task. Since these parity tasks are logical in nature, the Shannon information metric is used to measure the performance of the reservoir computer. The information metric R is defined as follows:

[0110]

[0111] Where H(x) is the Shannon entropy, H y (x) is the conditional entropy. The first 80% of the time history is used to train the reserve pool, and the remaining 20% ​​is used for testing. Fig.13 An example of a parity check task for a Lorenz reservoir computer is shown. The simulation time is 1440 seconds, where dt = 0.0001 seconds, N = 100, τ = 0.48 seconds, θ = 0.0048 seconds, b = 1.5, a = 10, ρ = 2, σ = 10, β = 8 / 3, Ω = 100πrad / s, and φ = π / 3rad. Fig.13 The graph (a) shows the input u(t), Fig.13 The curve (b) shows the second-order parity check task, Fig.13 The curve (c) shows the third-order parity check task, Fig.13 The curve (d) shows the fourth-order parity check task, Fig.13 Graph (e) shows a fifth-order parity check task. Fig.13 It is shown that for each time step, the bit-by-bit prediction accuracy of the reservoir computer is 100% for the 2nd, 3rd, and 4th order parity check tasks, which corresponds to R = 1.0. However, the 5th order parity check task has an information metric close to zero.

[0112] This paper studies the Lorenz system to process information using a time-division multiplexed virtual node-based reservoir computing framework. In this work, random masking functions are used. Feedback and delay are discarded, showing a simpler and cheaper way to develop a reservoir computer.

[0113] Dynamic effects of reservoir calculations using Hopf oscillators

[0114] The nonlinear system is perturbed by an input signal that carries the information to be processed. The input u(t) is embedded into the reservoir dynamics using a single nonlinear node as follows:

[0115] u(t)=r(z) for (n-1)T p ≤t<(n)Tp ,

[0116] f(t)=1+u(t). (18)

[0117] Here, r(z) is a discrete signal that encodes logical values ​​in sequence. This discrete signal is then mapped to a continuous function as described in equation (18), where n,z∈Z + Since r(z) is a random sequence of logical statements, u(t) is a pseudo-period T p and pseudo-frequency A random square wave.

[0118] 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 forced frequency, ω0 is the resonance constant, and μ is the parameter that controls the radius of the limit cycle. The governing equations of the Hopf RC in equation (4) are numerically integrated and then scaled by subtracting the mean and dividing by the standard deviation. Next, each pseudocycle is averaged and the number of states from each pseudocycle T is 0. p Collect N virtual nodes. Then use the nonlinear activation function tanh -1 x is used to nonlinearly scale the node states. About 80% of the scaled node states are used for the training process, while 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 the delay-based reserve pool. Since there is only one real node, the oscillator itself, the other nodes are called virtual in the terminology of the delay-based reserve pool. However, the Hopf RC studied in this paper does not include any delay lines or masking functions (time multiplexing), which simplifies the system.

[0119] The reservoir computer is trained using ridge regression with Tikhonov regularization as shown in equation (7). In this paper, 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. It is set to 10- 1 λ is the regularization parameter to avoid overfitting, I is the identity matrix, N is the number of nodes, w is the weight vector obtained from the training process, and o(k) is the prediction of the reserve pool, where k∈Z + .

[0120] nth-order parity check function P with delta delay n It is defined by the following equation:

[0121]

[0122] Among them, δ∈Z +is the delay. For the parity check task tested here, δ=0 is used. For n=2, equation (19) is the second-order parity check, which is an exclusive OR (XOR) task ∨ . Since this check only deals with the parity check reference as a logical task, the input u(t) = {-1, +1} is randomly selected for each pseudo-cycle. Therefore, the final prediction of the reserve pool is also binarized, producing a high (+1) or low (-1) bit. The Shannon information rate can be used to evaluate the performance of the reserve pool. 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 a measure of the encoded information in the signal. This can be defined as in equation (9), where p i is the probability of getting a specific bit i. y (x) is the conditional entropy, which measures the probability of obtaining an incorrect bit in the target signal, as defined in equation (10).

[0123] Here, and p(i,j) is the joint probability distribution of two variables i and j, which can take the value of "1" or "-1" for logic tasks. i is associated with the target signal, while j is the associated bit value of the predicted signal from the RC. It should be noted that for these parity check tasks, the maximum value of R is 1.0. Using equation (4), the Hopf RC is fabricated as an analog circuit. The circuit is constructed using a TL082 operational amplifier and an AD633 multiplier in a standard integrator network configuration. A National Instruments cDAQ-9174 was used as a data acquisition device.

[0124] Parameter study. Next we will explore the dynamic limits of the Hopf reservoir computer by studying the effect of different parameters on its information processing capabilities. The performance is quantified using different parameter combinations of the parity task and the information rate of the order. The results allow a deeper understanding of the interplay between the dynamics of the oscillator and its computational power as a reservoir computer. Some of these results can be used as guiding principles for tuning other reservoir computers based on virtual nodes.

[0125] The input signal of the Hopf RC is embedded into the oscillator through the limit cycle radius μf(t) and the harmonic forced amplitude Af(t). Therefore, the tuning parameters μ and A potentially control the amount of information sent into the oscillator. Fig.14 The parameter sweep of this dual parameter space is shown in Figure 2, which shows the p =20π(T p = 0.1 seconds), Ω = ω0 = 40πrad / s, N = 1000 nodes, φ = π / 3rad, and simulation time 4000T p =400 seconds) of the Hopf RC limit cycle radius constant μ and the forced amplitude A. Fig.14 The graph (a) shows the second order parity check, Fig.14 The graph (b) shows the third-order parity check, Fig.14 The graph (c) shows the fourth-order parity check, Fig.14 The curve (d) shows the fifth-order parity check. The color bar represents the information metric R.

[0126] It is observed that small limit cycles with low forced amplitudes are inefficient for computation as expected. However, bands are observed for higher order tasks, which is not intuitive. This relationship can be used to maximize the computational power of an oscillator. By simply changing the input amplitude to the spintronic reservoir, the information processing power can be varied. The input amplitude can be varied to obtain different limit cycle responses in a spintronic oscillator. Similarly, the current Hopf RC varies the input amplitude by varying A to optimize the RC performance.

[0127] It has been found that the external forced frequency is important in determining the performance of the reservoir computer. It has been observed that the forced frequency of RCs created from Duffing oscillators, Arnold tongue structures, and topological hybrid arrays are tuned. In a similar manner, parameter studies were performed using the resonance constant ω0 and the harmonic forced frequency Ω.

[0128] When the resonance constant and the harmonic forced frequency are equal (ω0=Ω), the resonance condition of the Hopf oscillator is reached. Fig.15 It shows that the parity check task (ω p =20π(T p = 0.1 seconds), μ = 5, A = 0.5, N = 1000 nodes, φ = π / 3rad, and simulation time 4000T p Example of a fine resolution parametric study of the resonance constant ω0 and the harmonic forced frequency Ω of the Hopf RC with 100 Hz to 400 s). Fig.15 The graph (a) shows the second order parity check, Fig.15 The graph (b) shows the third-order parity check, Fig.15 The graph (c) shows the fourth-order parity check, Fig.15 The graph (d) shows a fifth-order parity check. The color bar represents the information measure R. In graphs (a)-(d), the band along the 45° angle in each ω0-Ω parameter graph corresponds to the resonance condition. However, if the resonance condition (ω0=Ω) is reached, 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.

[0129] When ω0 is a pseudo-periodic ω p When the integer multiple of (ω0=zω p ,in ), meet another matching condition. This can be targeted Fig.15 and Fig.16 We also experimentally explore the ω0-Ω parameter space by constructing an analog circuit that can be modeled by Eq. (4). Fig.16 Due to experimental limitations, the coarse parameter space is Fig.16 It is shown in Fig.16 It shows that the parity check task (ω p =20π(T p = 0.1 seconds), μ = 5, A = 0.5, N = 1000 nodes, φ = π / 3rad and simulation time 4000T p = 400 s), an example of a coarse-resolution experimental and numerical parameter qualitative study of the resonance constant ω0 and the harmonic forced frequency Ω of the Hopf RC. Fig.16 The top row shows a simulated circuit experiment where Fig.16 The graph (a) shows the second order parity check; Fig.16 The graph (b) shows the third-order parity check; Fig.16 The graph (c) shows a fourth-order parity check; Fig.16 Graph (d) shows a fifth-order parity check. Fig.16 The bottom row shows numerical simulations where Fig.16 The graph (e) shows the second order parity check; Fig.16 The graph (f) shows the third-order parity check; Fig.16 Graph (g) shows a fourth-order parity check; Fig.16 The graph (h) shows a fifth-order parity check. The color bar represents the information metric R. The experiment shows similar trends near resonance and under matching conditions. Fig.16 It can be seen that when The best performance of the numerical reservoir can be found for the resonance condition when and The best performance of the experimental reservoir is found when . The deviations between the experiments and simulations can be attributed to the precision of the circuit elements and the nonlinear effects of the circuit (e.g., parasitic effects).

[0130] Next, the frequency ratio The influence of the harmonic forced amplitude A on the computational power of Hopf RC is Fig.17 The second-order parity check and chaotic laser intensity prediction tasks are used to study parameter space while keeping the system parameters the same. For the chaotic time series benchmark, the RC task is to predict one step ahead based on the previous step. RMSE is used as the performance measure 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 when The Hopf oscillator undergoes resonance when Fig.17 In , it can be observed that the Hopf reservoir can have high computational power even when the resonance condition is not satisfied.

[0131] exist Fig.17 In Varies along the horizontal axis, where the values ​​from the Farey sequence (30th order) are marked with tick marks, and the forced amplitude A varies along the vertical axis. The top graph shows the second-order parity check task, where the color bar represents the information measure. The bottom graph shows the chaotic laser time series prediction, where the color bar represents the performance based on the root mean square error (RMSE), which has been binarized to high (logical 1) if RMSE>0.3 and low (logical 0) for RMSE≤0.3. For these tasks, μ=5, Ω=40πrad / s, N=1000 nodes, φ=0rad, ω p =20πrad / s(T p = 0.1 seconds) and simulation time 3000T p = 300 seconds. In both cases, the computational power of HopfRC is increased by Strongly predicted by alignment with Farey sequence numbers.

[0132] from Fig.17 It was observed that when Hopf RC has high computational power when is a number from a Farey sequence based on number theory. Using a 30th order Farey sequence, it can be observed that the computational power of the reserve pool is limited by the frequency ratio , matching the numbers from the Farey sequence. To see the correlation between the Farey ratio and the performance of the RC for chaotic time series tasks, the RMSE was binarized as high (logical 1) if RMSE>0.3 and low (logical 0) for RMSE≤0.3. The Farey sequence is found in many natural phenomena, such as in the auditory system, resonance graphs of accelerators, mode locking in quantum accelerators, and cardiac arrhythmias. However, this is when the Farey sequence has been reported in the context of reservoir computing.

[0133] Fig.18shows how synchronization plays an important role in the computational capability of RC, which is observed to be around The resonance zone of the Arnold tongue. Fig.18 The graph (a) depicts a simulation showing the Arnold tongue area in the second-order parity task of RC. The color bar represents the information metric R. Fig.18 The graph (b) depicts a simulation showing the Arnold tongue area in the second-order parity task of the Hopf oscillator. The color bar represents the Hopf oscillator response x to the external forced The phase lag between them (degrees). Fig.18 Graph (c) depicts a simulation showing the Arnold tongue region in the chaotic time series task of RC. The color bar represents RMSE. Fig.18 Graph (d) depicts a simulation showing the Arnold tongue region in the chaotic time series task of a Hopf oscillator. The color bar indicates the phase lag (°). Fig.18 Graph (e) depicts an experiment showing the Arnold tongue region in the chaotic time series task of RC. The color bar represents RMSE. Fig.18 Graph (f) depicts an experiment showing the Arnold tongue region in the chaotic time series task of the Hopf oscillator. The color bar indicates the phase lag (°). In graphs (c) and (e), the RMSE has been binarized to high (logic 1) if RMSE>0.1, and to low (logic 0) for RMSE≤0.1. For these tasks, ω p =20πrad / s(T p =0.1 seconds), μ=5, A=0.5, N=1000 nodes and φ=0 rad.

[0134] Fig.18 The graphs (b), (d) and (f) show the x-state of the Hopf oscillator and the forced harmonics for the parity check task and the chaotic time series prediction task. The phase difference between near When Arnold tongues are formed in the space of the reservoir's performance metrics and in the space of the oscillators' phase deviations. It should be noted that the phase difference is not constant throughout the time history. So, the maximum phase difference is taken into account by binning the response, taking a fast Fourier transform, and then plotting the maximum phase difference. Arnold tongue regions are also found in reservoir computers consisting of arrays of Duffing oscillators. When this resonant condition holds, the Hopf RC performs best. In Fig.18In the graphs (a), (c), and (e) of , an Arnold tongue is observed in the performance space of the reservoir when the Hopf RC is operated near the resonant frequency. Fig.18 Graphs (e) and (f) depict the Arnold tongue of the experimental Hopf RC. Fig.18 Graphs (a)-(d) are found from numerical simulations.

[0135] Fig.19 Shown is the time history of the oscillator's response when forced to lock and when not phase locked. Fig.19 The time series of the x-state of the Hopf RC shown for a portion of the chaotic time series task is shown. Fig.19 The top graph of shows that the Hopf RC is at resonance (ω0=Ω) and the oscillation is locked by external forcing. Fig.19 The bottom graph of shows that the Hopf RC is not at resonance (ω0=1.7391*Ω) and the oscillation is not locked by external forced. p =20πrad / s(T p =0.1 seconds), μ=5, A=0.5, N=1000 nodes and φ=π / 3rad.

[0136] The tongue region can be particularly important in experimental design. For this tongue region, there is a range of frequency ratios centered on the resonant frequency, which can lead to better calculations. Therefore, this is the only area where the reserve pool is found to have a certain tolerance to detuning. This means that synchronization in the Arnold tongue region leads to robust calculations. There is also a relatively wider range of amplitudes and frequency ratios available, and the resonance constant can be tuned so that the Hopf RC has high computational power while remaining within the Arnold tongue. It should be noted that the proposed study was conducted considering the absence of masking in the system. Although there is no masking function, the presence of robust computational power suggests that a reserve pool based on a single nonlinear node can also be reliably constructed, thereby discarding conventional periodic or non-periodic masks.

[0137] To understand the effect of resonance on computation, the storage capacity of the Hopf oscillator reservoir is calculated. For the delta-delayed nth-order parity check function given in equation (19), the storage capacity of the system can be calculated as follows:

[0138]

[0139] Fig. 20 The storage capacity calculations of Hopf RC for first, second, third, and fourth order parity check tasks with δ delays are shown (δ = 0 to 10). p =20π(Tp = 0.1 seconds), Ω = 40π, A = 0.5, μ = 5, N = 1000 nodes and φ = π / 3rad, simulation time 4000T p = 400 seconds. Fig. 20 In the data center, the storage capacity (bits) is relatively A drawing showing that The storage pool has the highest memory under the resonance condition. Therefore, due to the presence of sufficient nonlinearity under this condition, the Hopf oscillator should perform better than other parameter combinations. This explains the Figure 15-18 Excellent computational performance for the resonance condition shown in .

[0140] We also studied nonlinear activation functions (tanh -1 X) has an impact on the performance of the Hopf oscillator RC, and Fig.21 The results are presented in Fig.21 The effect of nonlinear activation function on the performance of Hopf RC is shown in (ω p =20π(T p = 0.1 seconds), ω0 = 40π, μ = 5, N = 1000 nodes, φ = π / 3rad, and simulation time 4000T p =400 seconds). Fig.21 The graph (a) shows the second order parity check, Fig.21 The graph (b) shows the third-order parity check, Fig.21 The graph (c) shows the fourth-order parity check, Fig.21 The curve graph (d) shows the chaotic time series. It was found that in the absence of nonlinear activation, the Hopf oscillator RC showed similar performance for low-order tasks (e.g., second-order and third-order parity tasks). However, the nonlinear activation function becomes important in performing higher-order tasks. Therefore, the basic Hopf oscillator dynamics has a certain level of computing power. In addition, the linear oscillator was also tested as a reservoir computer in the presence of a nonlinear activation function. In this case, the nonlinear activation function cannot enable the linear oscillator to act as a reservoir computer. This is similar to the nonlinear effect in acoustic transformation of digital recognition tasks.

[0141] ESP is one of the fundamental properties found in successful reservoir computing frameworks. No previous limit cycle-based systems have been found to satisfy the ESP requirements, however, the Hopf RC formula described in this disclosure is different because the limit cycle radius varies according to the forced hold used to encode information. In addition, the reservoir system is constructed considering resonance phenomena. In the literature, universal synchronization or common signal-induced synchronization is used to verify the existence of ESP in the reservoir. ESP has been empirically studied to measure the stability of input-driven reservoir dynamics. It is implied that if the asymptotic trajectory of the reservoir state depends differentially on the input and is independent of the initial conditions, the reservoir possesses echo state characteristics. Therefore, in order to obtain the echo state characteristics, the effect of the initial conditions on the reservoir dynamics should decay over time.

[0142] The echo state properties of the Hopf reservoir are studied when it is encoded with the input of the chaotic laser time series task and the parity check task. Following the algorithm for estimating the ESP of the reservoir, the ESP indices of the two benchmarks are calculated and averaged over the range {-3,3} of 20 randomly generated initial conditions. The deviations of the reservoir state trajectories for different initial conditions are calculated using the same input sequence for all different initial conditions. To calculate the deviations, the initial condition (x0, y0) = (0, 0) is used as the common trajectory, while the other trajectories are from each of the different initial conditions, while the initial transients are discarded. Finally, the average of the deviations is calculated to find the ESP index. If the ESP index becomes zero for the dynamics of the reservoir, the reservoir is considered to have echo state properties. These results are presented in Fig. 22 , where it is observed that the reservoir has an echo state property for the resonance condition (ω0=Ω), while the reservoir does not have this property when resonance (ω0≠Ω) is not satisfied. This can also explain why resonance helps in computation. It is also important to note that the reservoir can still have information processing capabilities when ESP is not satisfied.

[0143] Fig. 22 The calculation of the echo state property (ESP) index of the Hopf RC for the chaotic laser time series task and the parity check task is shown. Fig. 20 The graph (a) shows the ESP index for the resonant and non-resonant conditions of the chaotic time series task. Fig. 20 The graph (b) shows a set of random initial conditions (x0 and y0 selected from {-3,3}) for the chaotic time series task. Fig. 20 The graph (c) shows the ESP index for the resonant and non-resonant conditions for the parity check task, Fig. 20The graph (d) of shows a set of random initial conditions (x0 and y0 selected from {-3,3}) for the parity check task. Here, ω p =20π(T p = 0.1 seconds), ω0 = 80π, φ = π / 3rad, A = 0.5, and simulation time 3000T p = 300 seconds. For the resonant condition, Ω = ω0, and for the non-resonant condition, Ω = 1.7391ω0.

[0144] In the present disclosure, a Hopf oscillator is constructed as a reservoir computer to gain insight into the relationship between the dynamics of the oscillator and the computational power of the RC. This implementation of the Hopf reservoir computer provides a simpler design by discarding the commonly used delay feedback lines and masking functions. An analog circuit is used as the physical implementation of the reservoir. When the input pseudo frequency ω p The reservoir exhibits high computational power when the ratio to the natural frequency ω0 of the oscillator is taken from the Farey sequence. In addition, the resonance phenomenon occurs when the harmonic forced frequency and the natural frequency of the oscillator are equal, which provides favorable conditions for constructing a reservoir computer. Enhanced computational power can be achieved when the limit cycle radius is relatively small and the forced amplitude is relatively large. An Arnold tongue structure is observed in the information metric space of the reservoir near the resonance position, which is related to the Arnold tongue presented in the phase deviation space. When the resonance condition is met, the reservoir is also found to possess maximum storage capacity and echo state characteristics, which in principle indicates better computational performance. Finally, the results also show that the reservoir computer can be constructed with only a single nonlinear node and is not a time-division multiplexing process nor delayed feedback. By exploiting some of the underlying dynamics of the system, a limit cycle reservoir computer that is both simple and robust can be constructed.

[0145] The no-delay, random limit cycle oscillator reservoir computer can be used with a range of technologies for recognition tasks. The limit cycle oscillator can be paired with a vibration signal, such as but not limited to speech recognition or other soundscapes. Since the limit cycle oscillator is already a vibration system, this pairing is beneficial in applications such as those implemented in edge devices. For example, the no-delay, random limit cycle oscillator reservoir computer can be paired with microphone technology for voice recognition tasks. Fig.23A and Fig. 23B An example of a PRC implemented using microphone technology for a voice recognition task is shown. Fig.23A The system includes a PRC and a reader. The PRC provides processing immediately after the microphone to eliminate data preprocessing, enable fast analog computing on different tasks, and tunable physical layer. The reader can be reconfigured for different tasks, is deployed directly on the edge device, and can include a feedback loop to improve the computational performance of the PRC. Similarly, Fig. 23B The system includes: a PRC and a reader for processing using sensing, the PRC eliminates data preprocessing, enables fast analog computing and a tunable physical layer on different tasks, the reader can be reconfigured for different tasks, directly deployed on edge devices, and feedback loops can be used to improve the computing performance of the PRC.

[0146] The delay-free, random limit cycle oscillator reservoir computer can also be integrated into multiple MEMS (Micro-electromechanical Systems) designs, such as but not limited to cantilever beams, capacitive microphones, parallel plate actuators, mechanical load-reducing oscillators, quantum sensing, etc. Limit ring oscillator reservoir computers can also be integrated into optoelectronic designs (such as, GRG Chemgui et al., "Theoretical and experimental study of slow-scale Hopf limit-cycles in laser-based wideband optoelectronic oscillators", Journal of the Optical Society of America, Vol. 31, No. 10, pp. 2310-2316, October 2014), nanoelectromechanical designs (NEMS, such as K. Aubins et al., "Limit Cycle Oscillations in CW Laser-Driven NEMS", Journal of Microelectromechanical Systems, Vol. 13, No. 6, pp. 1018-1026, December 2004), semiconductor laser designs (such as, in TBS Impson et al., "Limit-Cycle Dynamics with Reduced Sensitivity to Perturbations (Limit cycle dynamics with reduced sensitivity to perturbations), Physical Review Letters 112, 023901, January 2014), thermoacoustic design (such as, in D. Premraj et al., “Effect of amplitude and frequency of limit cycle oscillator on their coupled and forced dynamics,” Nonlinear Dynamics 103:1439-1452, February 2021), and quantum dot design (such as, in D. Goulding et al., “Excitability in a Quantum Dot Semiconductor Laser with Optical Injection,” Physical Review Letters 98, 153903, April 2007). Several sound recognition benchmark tasks have been completed to explore Hopf bifurcation as a processing mechanism.Machine learning at the readout layer can use ridge regression with a training time of less than 1 minute. Both the MEMS bifurcation point and the excitation signal can be tuned to enhance its processing capabilities. Examples of results obtained using simulated RC are shown in the table below. Benchmarks such as these (see also e.g. Figure 9-12 ) verified the application of limit cycle oscillator reservoir computer.

[0147]

[0148] It should be emphasized that the above-mentioned embodiments of the present disclosure are only possible examples of implementation methods set forth for a clear understanding of the principles of the present disclosure. Many changes and modifications may be made to the above-mentioned embodiments without departing substantially from the spirit and principles of the present disclosure. All these modifications and changes are intended to be included within the scope of the present disclosure and to be protected by the appended claims.

[0149] The term "substantially" is intended to allow deviations from a descriptive term that do not adversely affect the intended purpose. A descriptive term is implicitly understood to be modified by the word substantially, even if the term is not explicitly modified by the word substantially.

[0150] It should be noted that ratio, concentration, amount and other numerical data can be expressed in range format herein.It should be understood that, for the sake of convenience and brevity, such range format is used, and therefore, should be interpreted in a flexible manner as not only including the numerical value clearly recorded as the limit of the range, but also including all individual numerical values ​​or sub-ranges contained within the range, as if each numerical value and sub-range are clearly recorded.For illustration, the concentration range of "about 0.1% to about 5%" should be interpreted as not only including the concentration of about 0.1wt% to about 5wt% clearly recorded, but also including a single concentration (e.g., 1%, 2%, 3% and 4%) and a sub-range (e.g., 0.5%, 1.1%, 2.2%, 3.3% and 4.4%) within the indicated range. The term "about" can include traditional rounding according to the significant figures of the numerical value. In addition, the phrase "about 'x' to 'y'" includes "about 'x' to about 'y'".

Claims

1. A physical reserve pool computer, comprising: Processing circuit, including: Input layer; a reservoir comprising a forced limit cycle oscillator, the reservoir being implemented without delay or feedback; and Read out layer.

2. The physical reserve pool computer according to claim 1, wherein: The forced limit cycle oscillator includes: a Hopf oscillator or a Lorenz oscillator.

3. The physical reserve pool computer according to claim 2, wherein: The forced limit cycle oscillator includes: a two-state forced Hopf oscillator.

4. The physical reserve pool computer according to any one of claims 1 to 3, wherein: The processing circuit includes: an analog processing circuit.

5. The physical reserve pool computer according to claim 4, wherein: The analog processing circuit includes: an operational amplifier and a multiplier.

6. The physical reserve pool computer according to any one of claims 1 to 5, wherein: The reservoir computer utilizes a non-periodic random mask.

7. The physical reserve pool computer according to claim 6, wherein: The non-periodic random mask is defined by white Gaussian noise.

8. The physical reserve pool computer according to any one of claims 1 to 3, wherein: The processing circuit includes: a photoelectric circuit.

9. The physical reserve pool computer according to any one of claims 1 to 8, wherein: A vibration signal is applied to the input layer.

10. The physical reserve pool computer according to claim 9, wherein: The vibration signal is a voice signal.

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

13. The physical reserve pool computer according to claim 11, wherein: The desired output is a logic output.

14. The physical reserve pool computer according to claim 13, wherein: The logic output is an XOR output.

15. The physical reserve pool 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 reserve pool computer according to claim 15, wherein: The applied signal is encoded as a continuous input function.

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