Chaotic system synchronous regulation and control method based on reserve pool computing network

Abstract

The invention discloses a chaotic system synchronous regulation and control method based on a reserve pool computing network, and the method specifically comprises the steps: dividing groups based on the symmetry of nodes of the reserve pool computing network, training the nodes of different groups with different system data, and introducing Lorenz and Rossler chaotic systems as RC training data sources; in the training stage, input data are firstly mapped to a reservoir through an input weight, and reservoir neurons update weight state information in time after receiving the input data; after training is completed, output data serve as input data to be fed back to a reservoir, the reservoir operates autonomously, and RC is used for simulating dynamic characteristics of oscillator nodes in a complex network; and introducing a mean square error function RMS to judge the stability of the synchronous group. According to the method, the coupling strength and other parameters among the group nodes are regulated and controlled by analyzing the characteristics of the network topology structure, so that the overall dynamic state of the group nodes is influenced, and stable zero-time-delay synchronization among the calculation nodes of the reserve pools in the group is realized.

Description

Technical Field

[0001] The present invention belongs to the technical field of reservoir computing, and in particular relates to a method for synchronously regulating a chaotic system based on a reservoir computing network. Background Art

[0002] In recent years, neural network algorithms have made remarkable developments and have been widely used in various fields. For some complex and unfamiliar systems, modeling, analysis, and understanding are usually carried out with the help of neural network algorithms to reveal the internal laws and dynamic behaviors of the systems. Reservoir Computing (RC) is a machine learning technology based on neural networks and is a powerful tool for processing tasks such as time series prediction and dynamic system modeling. Its core idea is to use a randomly generated and fixed neural network structure to perform a non-linear mapping on input data and then process related tasks. Compared with traditional neural networks, it greatly simplifies the training process. The high efficiency and flexibility of RC make it an ideal choice for processing dynamic complex tasks.

[0003] In the field of chaotic dynamics, RC is often used for problems such as system prediction, control, and synchronization. Chaotic dynamic systems are sensitive to initial conditions, unpredictable in the long term but predictable in the short term. Traditional linear processing methods are difficult to capture their laws. The RC structure is fixed and consists of three parts: an input layer, an output layer, and a reservoir layer composed of randomly connected non-linear neurons. In RC, input information is mapped to the high-dimensional space of the reservoir layer through the random weight matrix of the input layer. The neuron nodes in the space update their states recursively. By combining the current input and the previous state, the dynamic characteristics of the input information can be obtained. As time goes by, RC will generate complex non-linear behaviors. On the other hand, although the RC structure is fixed, by adjusting the training weights of the output layer, it can flexibly adapt to various chaotic systems.

[0004] Synchronization is an important research direction in chaotic dynamics. The introduction of RC provides a new perspective. In complex networks, it is very intuitive to observe the network through the structure diagram. However, when the structure of the network gradually becomes complex, it is extremely difficult to directly observe the network. Usually, we need to introduce an adjacency matrix to describe the structure of the complex network, A ij describing the connection relationship between nodes. If A ij = 1, it indicates that node i can continuously inject data into node j. In other cases, A ij = 0. Any network has symmetry characteristics. This symmetry can be understood as the permutation of nodes. After the permutation of nodes is completed, the adjacency relationship of the network remains unchanged. This symmetry can also be regarded as a permutation matrix P, which reorders the network nodes in a way that keeps the node dynamic equation unchanged, that is, A ij P = PAij Due to the symmetry in this network structure, the network can be divided into different synchronous groups. The relationships between synchronous groups complement each other. When some groups in the network lose synchronous stability, other groups in the network will be affected more or less. SUMMARY OF THE INVENTION

[0005] Based on this phenomenon, by analyzing the network topology structure to achieve synchronous regulation of the reservoir computing network, the present invention provides a method for synchronous regulation of a chaotic system based on a reservoir computing network.

[0006] A method for synchronous regulation of a chaotic system based on a reservoir computing network according to the present invention is specifically as follows:

[0007] Step 1: Divide groups based on the symmetry of the nodes of the reservoir computing network. Nodes in different groups are trained with different system data, and two chaotic systems, Lorenz and Rossler, are cited as the training data sources for RC.

[0008] Step 2: During the training phase, the input data u(n) is first mapped to the reservoir through the input weight W in ; after receiving the input data, the reservoir neurons will update the weight state information in a timely manner. The specific update rule is as follows:

[0009] x(n) = (1 - α)x(n - 1) + αf(Wx(n - 1) + W in u(n)) (1)

[0010] where x(n) represents the state of the internal nodes of the reservoir at time n, f(·) is the activation function of the internal nodes of the reservoir, here it is the hyperbolic tangent function tanh, α ∈ [0, 1] represents the leakage rate of the reservoir state, W is an N-dimensional matrix representing the connection relationship between nodes, and its elements are randomly and uniformly distributed in [-0.1, 0.1], and its spectral radius is ρ; W in is the mapping weight, which maps the input data to a higher-dimensional space, and its elements are randomly and uniformly distributed in [-1, 1].

[0011] When the input data u(n) enters the reservoir, the states of the neurons in this layer are also updated; the input data u(n) and the updated neuron states need to be stored in matrices X and Y respectively, and the specific storage forms are as follows: [u(k); x(k)], [u(k + 1)]; when all x(n) are obtained, the output weight W out of the reservoir is calculated through ridge regression, and finally the output y out of the reservoir is obtained. The specific calculation formula is as follows:

[0012] W out = YX T (XXT + λI) -1 (2)

[0013] y out = W out [u(n); x(n)] (3)

[0014] Among them, λ is the ridge regression coefficient introduced to prevent overfitting, and I represents the identity matrix.

[0015] Step 3: After the training is completed, output the data y out As the input data x(n) is fed back to the reservoir; at this time, the reservoir operates autonomously, and at this time, RC is used to simulate the dynamic characteristics of oscillator nodes in a complex network.

[0016] In the network, each node can receive various injections, which can be divided into two types: self-injection and external injection; after receiving the injection data, the node feeds these injection data back into RC as the initial data for the next moment; as time goes by, the nodes in the same group will gradually tend to synchronize; the coupling method between nodes is shown in the following formula:

[0017]

[0018]

[0019] Among them, η i is the self-feedback strength coefficient of the i-th node, and β ij represents the strength coefficient of the i-th node injecting into the j-th node. is the output of the j-th node, and N is the number of nodes.

[0020] Step 4: To quantitatively evaluate the stability of the synchronous group, introduce the root mean square function RMS, and determine the stability of the synchronous group by calculating the synchronization error of RC within the same group under different initial conditions, and define it as the following formula:

[0021]

[0022] Among them, N S represents the number of members of the group, that is, the number of nodes within the group, and E m (t) is the output sequence of the m-th node within the group, <·> represents the mean value of the output sequence over time, o represents the start time of the calculation window, and τ represents the window length.

[0023] When RMS < 0.01, stable synchronization is achieved in the reservoir within the same group.

[0024] The beneficial technical effects of the present invention are:

[0025] The trained RC of the present invention operates autonomously and can serve as a node of a complex network. At the same time, by analyzing the characteristics of the complex RC network topology, the working parameters of the system are modulated to achieve synchronous regulation of the reservoir computing nodes. Description of the Drawings

[0026] Figure 1 It is the topology of a complex network based on reservoir computing (where nodes marked with the same shape belong to the same group).

[0027] Figure 2 is Figure 1 the adjacency matrix of the complex network based on reservoir computing.

[0028] Figure 3 It is the dynamic evolution of the synchronous group in the embodiment. a(1)-c(1) When β ij = 0.2, (i,j - 1,..., N); η1 = 0.6, η3 = 0.8, three synchronous groups achieve synchronization. a(2)-c(2) When η1 = 0.8 → η1 = 0.46, the members of group C1 are out of sync; a(3)-c(3) When η3 = 0.6 → η3 = 0.4, the members in groups C2 and C3 lose synchronization.

[0029] Figure 4 It is the synchronous state between members of different groups in the embodiment. (a) The members of all groups are synchronous. (b) The members of group C1 are out of sync, and the remaining groups are still synchronous. (c) The members in all groups lose the synchronous steady state.

[0030] Figure 5 It is the RMS change between members of the synchronous group. Detailed Embodiment

[0031] The following further describes the present invention in detail with reference to the drawings and specific embodiments.

[0032] A method for synchronous regulation of a chaotic system based on a reservoir computing network of the present invention uses the trained RC as an oscillator of a complex network and can serve as each node in the network. By analyzing the characteristics of the network topology, parameters such as the coupling strength between group nodes are regulated, thereby affecting the overall dynamics of the group nodes and achieving stable zero-delay synchronization between the reservoir computing nodes within the group. Specifically:

[0033] Step 1: Divide groups based on the symmetry of the nodes of the reservoir computing network. Nodes in different groups are trained with different system data, and Lorenz and Rossler chaotic systems are cited as the training data sources for the RC.

[0034] The network topology of a complex network based on reservoir computing is as Figure 1 shown, inFigure 1 In it, it can be found that after nodes 6 and 7, nodes 2 and 3, and nodes 4 and 5 are mutually permuted, Figure 1 the network structure in Figure 2 and the adjacency matrix of

[0035] remain unchanged. It can be said that nodes 6 and 7, nodes 2 and 3, and nodes 4 and 5 are mutually mapped in the symmetric arrangement. According to this symmetric characteristic, it can be divided into 4 groups, namely C1 = {6, 7}, C2 = {2, 3}, C3 = {4, 5}, C4 = {1}.

[0036] To ensure the symmetry of the nodes in the group and further increase the difference in the dynamic output, the nodes in different groups are trained with different system data. We introduce two chaotic systems, Lorenz and Rossler, as the training data sources for the RC.

[0037]

[0038] Among them, {a1 = 0.2, b1 = 0.4, c1 = 6.7}, {a2 = 0.2, b2 = 0.4, c2 = 5.7}, n is the nth group; x, y, z are the unknowns of the system equation.

[0039] The Lorenz system equation is as follows:

[0040]

[0041] Among them, {σ3 = 10, r3 = 32, b3 = 8 / 3}, {σ4 = 10, r4 = 28, b4 = 8 / 3}.

[0042] Step 2: In the training stage, the input data u(n) is first mapped to the reservoir through the input weight W in ; after receiving the input data, the reservoir neurons will update the weight state information in a timely manner. The specific update rule is as follows:

[0043] x(n) = (1 - α)x(n - 1) + αf(Wx(n - 1) + W in u(n)) (1)

[0044] Among them, x(n) represents the state of the internal nodes of the reservoir at the nth moment, f(·) is the activation function of the internal nodes of the reservoir pool, here it is the hyperbolic tangent function tanh, α ∈ [0, 1] represents the leakage rate of the reservoir pool state, usually balancing the influence between the past and the current input to adjust the dynamics of the reservoir pool, W is an N-dimensional matrix representing the connection relationship between nodes, and the elements in it are randomly and uniformly distributed in [-0.1, 0.1], and its spectral radius is ρ; W inIt is the mapping weight that maps the input data into a higher-dimensional space, where the elements are randomly and uniformly distributed in [-1, 1].

[0045] When the input data u(n) enters the reservoir, the states of the neurons in this layer are also updated; the input data u(n) and the updated neuron states need to be stored in matrices X and Y respectively, and the specific storage forms are as follows: [u(k); x(k)], [u(k + 1)]; after obtaining all x(n), the output weight W of the reservoir is calculated through ridge regression out , and finally the output y of the reservoir is obtained out , and the specific calculation formula is as follows:

[0046] W out = YX T (XX T + λI) -1 (2)

[0047] y out = W out [u(n); x(n)] (3)

[0048] Among them, λ is the ridge regression coefficient introduced to prevent overfitting, and I represents the identity matrix.

[0049] Step 3: After the training is completed, the output data y out is fed back to the reservoir as the input data x(n); at this time, the reservoir runs autonomously, and at this time, RC is used to simulate the dynamic characteristics of the oscillator nodes in the complex network.

[0050] In the network, each node can receive various injections, which can be divided into two types: self-injection and external injection; after receiving the injection data, the node takes these injection data as the initial data for the next moment and feeds them back into RC again; as time goes by, the nodes in the same group will gradually tend to synchronize; the coupling method between nodes is shown in the following formula:

[0051]

[0052] Among them, η i is the self-feedback strength coefficient of the i-th node, and β ij represents the strength coefficient of the i-th node injecting into the j-th node, is the output of the j-th node, and N is the number of nodes;

[0053] Step 4: In order to quantitatively evaluate the stability of the synchronous group, the root mean square (RMS) function is introduced. By calculating the synchronization error of RC within the same group under different initial conditions, the stability of the synchronous group is determined, and it is defined as the following formula:

[0054]

[0055] Among them, N S represents the number of members of the group, that is, the number of nodes in the group, and E m (t) is the output sequence of the m-th node in the group. <·> represents the mean value of the output sequence over time. o represents the start time of the calculation window, and τ represents the window length.

[0056] When RMS < 0.01, stable synchronization is achieved in the reservoir within the same group.

[0057] Example:

[0058] The topological structure of a complex network based on reservoir computing is as Figure 1 shown. Under this structure, x(t) of the Lorenz and Rossler systems is selected to train the RC of different groups. First, the equations are solved using the fourth-order Runge-Kutta method to generate 10,000 data. The first 1,000 are used as transients, and the remaining data are used as training data. The trained RC can output autonomously and can thus be used as a node in the complex network. At the same time, Figure 1 the structure in can exhibit various synchronization phenomena. Figure 1 When the nodes in operate autonomously, the input has two sources: on the one hand, the output of the previous moment of itself, and on the other hand, the injection from adjacent nodes. Under the influence of these two aspects, the output of the RC will be affected to a certain extent and gradually enter the synchronous state.

[0059] Figure 3 gives Figure 1 the synchronization states of the synchronous groups in under different conditions. Figure 3 .a(1)-c(1) shows the dynamics when three groups achieve synchronization. Members of different groups have different dynamics. Figure 3 .a(2)-c(2), by reducing the proportion η1 of self-injection of the input in Group 1, it can be observed that Figure 3 the members of Group 1 in.a(2) have lost synchronization, but it has no impact on the states of other group members in the network. On this basis, by reducing the proportion η2 of self-injection of the members in Group 3, it can be found that the members of Group 3 have lost synchronization stability. Due to the loss of synchronization stability of the members of Group 3, it further affects the synchronization state of Group 2.

[0060] Figure 4 In , by using the mean square error function to quantitatively evaluate the synchronization states of the members of the synchronous group, it can be observed that in the above two adjustments, the members of different groups are in an asynchronous state.

[0061] Figure 5Among them, corresponding to the synchronous changes among group members after the above two adjustments, by continuously adjusting o, the influence of the RC input source on the RC output was observed. In Figure 5 (a), Groups 2 and 3 entered the synchronous state after a period of correction. Figure 5 (b), due to the decrease in its own injection ratio, Group 1 has been in an asynchronous state, while Groups 2 and 3 have been in a synchronous state after a short correction. Figure 5 (c) It can be seen that all groups are in an asynchronous state due to the two adjustments.

Claims

1. A method for synchronously regulating a chaotic system based on a reservoir computing network, characterized in that Specifically: Step 1: Divide the groups based on the symmetry of the nodes in the reservoir computing network. Nodes in different groups are trained with different system data, and two chaotic systems, Lorenz and Rossler, are cited as the training data sources for the RC. Step 2: During the training phase, the input data u(n) is first mapped to the reservoir through the input weights W in After receiving the input data, the reservoir neurons will update the weight state information in a timely manner. The specific update rules are as follows: x(n) = (1 - α)x(n - 1) + αf(Wx(n - 1) + W in u(n)) (1) Among them, \(x(n)\) represents the state of the internal nodes of the reservoir at time \(n\), \(f(\cdot)\) is the activation function of the internal nodes of the reservoir, which is the hyperbolic tangent function \(\tanh\) here, \(\alpha\in[0,1]\) represents the leakage rate of the reservoir state, \(W\) is an \(N\)-dimensional matrix representing the connection relationship between nodes, and its elements are randomly and uniformly distributed in \([-0.1,0.1]\), and its spectral radius is \(\rho\); \(W\) in is the mapping weight, which maps the input data to a higher-dimensional space, and its elements are randomly and uniformly distributed in \([-1,1]\); When the input data u(n) enters the reservoir, the states of the neurons in this layer are also updated; the input data u(n) and the updated neuron states need to be stored in matrices X and Y respectively, and the specific storage forms are as follows: [u(k); x(k)], [u(k + 1)]; after obtaining all x(n), the output weight W of the reservoir is calculated by ridge regression out , and finally the output y of the reservoir is obtained out , and the specific calculation formula is as follows: W out = YX T (XX T + λI) -1 (2) y out = W out [u(n); x(n)] (3) where λ is the ridge regression coefficient introduced to prevent overfitting, and I represents the identity matrix; Step 3: After training is completed, output the data y out Feed it back to the reservoir as the input data x(n); at this time, the reservoir runs autonomously, and at this time, RC is used to simulate the dynamic characteristics of oscillator nodes in a complex network; In the network, each node can receive various injections, which can be divided into two types: self-injection and external injection. After receiving the injection data, the node uses these injection data as the initial data for the next moment and feeds them back into the RC. As time goes by, the nodes in the same group will gradually tend to synchronize. The coupling method between nodes is shown in the following formula: Among them, η i is the self-feedback strength coefficient of the i-th node, and β ij represents the strength coefficient injected by the i-th node into the j-th node, is the output of the j-th node, and A ij represents the connection relationship between nodes, and N is the number of nodes; Step 4: To quantitatively evaluate the stability of the synchronous group, the root mean square function RMS is introduced. By calculating the synchronization error of the RC within the same group under different initial conditions, the stability of the synchronous group is determined and defined as the following formula: Among them, N S represents the number of members of the group, that is, the number of nodes in the group, and E m (t) is the output sequence of the m-th node in the group, <·> represents the mean value of the output sequence over time, o represents the start time of the calculation window, and τ represents the window length; When RMS < 0.01, stable synchronization is achieved in the reservoir within the same group.

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