Multi-dimensional time series prediction method and system based on hybrid higher-order reservoir computing

Abstract

This invention relates to a multidimensional time series prediction method and system based on hybrid higher-order reservoir computation, belonging to the field of reservoir computation technology. This invention utilizes the updated reservoir node states to output the prediction results of the target variable, and selects a matching combination of input node block ratio, input weight scaling factor, and higher-order coupling strength based on error analysis. First, higher-order neighborhood information related to the target variable is distinguished and organized at the input layer. Then, within the reservoir, higher-order node coupling is used to nonlinearly propagate and reorganize this structured input information, thus more fully expressing the dynamic relationships generated by the combined effects of multiple variables. Simultaneously, this invention retains the characteristic of reservoir computation primarily training the readout layer weights, offering advantages such as simple training process and low computational cost. Furthermore, by matching the input structural parameters with the higher-order coupling strength, the prediction error fluctuations caused by simply increasing the higher-order coupling strength can be avoided.

Description

Technical Field

[0001] This invention relates to the field of reservoir computing technology, and in particular to a multidimensional time series forecasting method and system based on hybrid high-order reservoir computing. Background Technology

[0002] Reservoir Computing (RC) is a type of recurrent neural network method for modeling nonlinear dynamical systems and predicting time series. It typically includes an input layer, a fixed recursive reservoir, and a linear readout layer. The input layer maps the external multivariate time series u(t) to a high-dimensional reservoir state space. The reservoir layer consists of numerous nonlinear nodes and their fixed connections, used to generate rich transient responses and memory effects to the input signal. The readout layer obtains the predicted output based on a linear combination of the reservoir states. Unlike recurrent neural networks that require training all network parameters, the core feature of RC is that the input weights and internal connections of the reservoir are usually unchanged after initialization. The training phase mainly requires reading out the weights Wout, thus offering advantages such as small training load, simple implementation, fast convergence speed, and suitability for online prediction. In tasks such as predicting the state of chaotic systems, neurodynamics, traffic flow, power networks, and complex network nodes, the objects to be predicted often exhibit strong nonlinearity, coexistence of long short-term memory, and complex coupling relationships between variables. The reservoir can expand the original input signal to a certain extent through the high-dimensional nonlinear state response, enabling the linear readout layer to fit complex mapping relationships. Therefore, it has been widely used in Lorenz systems, Rossler systems, multidimensional sensor sequences, and networked dynamic systems for prediction.

[0003] However, the state evolution of complex dynamical systems is often not a single-variable independent change, but rather formed by the combined effects of multiple variables, multiple nodes, or multiple local structures. For example, in the Lorenz system, the evolution of the y-component and z-component involves nonlinear combinations of terms such as xz and xy, respectively. In networked power systems or coupled oscillator networks, the dynamics of a target node are not only affected by its own state, but also by neighboring nodes, edge connections, and local topology. These relationships essentially involve the combined effects of multiple variables and higher-order neighborhood information, and it is difficult for models to explicitly express these structural relationships using only ordinary random inputs and paired reservoir connections. Summary of the Invention

[0004] The purpose of this invention is to propose a multidimensional time series forecasting method based on hybrid high-order reservoir computation. On one hand, a block-structured input matrix is ​​constructed based on the high-order neighborhood set of the target variable, enabling the target variable itself, linearly correlated variables, and nonlinearly combined correlated variables to enter the reservoir according to different structural relationships. On the other hand, a high-order connectivity tensor and corresponding high-order coupling terms are introduced into the reservoir state update equation, allowing the reservoir to express the nonlinear dynamic response generated by the combined action of multiple nodes. Through the synergy between input organization and high-order reservoir evolution, this invention can improve the accuracy, stability, and adaptability to high-order dynamic structures in complex multidimensional time series forecasting.

[0005] This invention overcomes the shortcomings of the prior art and provides a multidimensional time series forecasting method and system based on hybrid high-order reservoir calculation.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows: The first aspect of this invention provides a multidimensional time series forecasting method based on hybrid higher-order reservoir calculations, comprising the following steps: Obtain multivariate time series data, normalize them, divide them into training and prediction data, select the q-th state variable as the prediction target, and determine the high-order neighborhood set based on the dynamic equation, variable correlation, or network topology. Based on the higher-order neighborhood set, a block-based structured input matrix oriented towards the target variable is constructed, and the input variables belonging to the local higher-order neighborhood set are injected into the corresponding structure blocks with random weights, while the columns of irrelevant variables are set to zero. Establish a reservoir containing paired connection matrices and higher-order connection tensors, and set the lower-order coupling strength, higher-order coupling strength, leakage rate, damping coefficient, input coupling strength, and bias. A state update equation is constructed, and the state of the reserve pool nodes is updated using the state update equation. Based on the updated state of the reserve pool nodes, the prediction result of the target variable is output. The matching combination of input node block ratio, input weight scaling factor and high-order coupling strength is selected according to error analysis.

[0007] Furthermore, in the multidimensional time series prediction method based on hybrid high-order reservoir computation, multivariate time series are obtained, normalized, and training and prediction data are separated. The q-th state variable is selected as the prediction target, and the high-order neighborhood set is determined according to the dynamic equation, variable correlation, or network topology. Specifically: Obtain multivariate time series data, normalize them, and divide them into training and prediction data. When it is necessary to predict a certain state variable, the q-th state variable is recorded as the target variable. For the target variable, if the evolution of the target variable includes linearly correlated variables or nonlinear combination terms, the corresponding variables or combinations of variables are used as local high-order neighborhood information of the target variable. Finally, a set of higher-order neighborhoods related to the target variable and the dynamic equation, variable correlation, or network topology is determined.

[0008] Furthermore, in the multidimensional time series prediction method based on hybrid high-order reservoir calculation, a block-based structured input matrix oriented towards the target variable is constructed according to the high-order neighborhood set. Input variables belonging to the local high-order neighborhood set are injected into the corresponding structure blocks with random weights, while columns corresponding to irrelevant variables are set to zero. Specifically: A block-structured input matrix is ​​constructed based on the higher-order neighborhood set. The input matrix is ​​a matrix composed of multiple input structure blocks, and each structure block corresponds to a local higher-order neighborhood set. Determine whether an input variable belongs to the current higher-order neighborhood set. If an input variable belongs to the current higher-order neighborhood set, set a random weight in the corresponding column of the current structure block. If the input variable is unrelated to the current structure block, the corresponding column is set to zero or set as a weak connection. Finally, the target variable's own information, linearly related variable information, and nonlinear combination related variable information are injected into the reserve pool in a block manner.

[0009] Furthermore, in the multidimensional time series prediction method based on hybrid high-order reservoir calculation, a reservoir containing pairwise connection matrices and high-order connection tensors is established, and low-order coupling strength, high-order coupling strength, leakage rate, damping coefficient, input coupling strength, and bias are set, specifically as follows: Establish a reservoir containing pairwise connection matrices and higher-order connection tensors, and configure N nonlinear reservoir nodes in the reservoir containing pairwise connection matrices and higher-order connection tensors; Damping, natural frequency or external drive and nonlinear response term are treated as node self-dynamic terms, and the structured input matrix, input vector, input coupling strength and bias are used as the external input drive term. The state of the reserve pool nodes is constructed based on the pairwise node coupling terms described by the ordinary pairwise connection matrix, the node self-dynamic terms, and the higher-order node coupling terms described by the higher-order connection tensor.

[0010] Furthermore, in the multidimensional time series prediction method based on hybrid high-order reservoir computation, a state update equation is constructed, which is used to update the state of the reservoir nodes. The prediction result of the target variable is output based on the updated reservoir node state. A matching combination of input node block ratio, input weight scaling factor, and high-order coupling strength is selected according to error analysis, specifically: The state of the reservoir nodes is updated based on the state update equation, which includes node self-dynamics, pair coupling, higher-order coupling, and structured input-driven processes. The transient state matrix of the reservoir is collected and the target output matrix is ​​constructed using the training output sequence of the target variable. The solution is obtained by ridge regression. The prediction results of the target variable are output based on the solution results, and the matching combination of input node block ratio, input weight scaling factor and high-order coupling strength is selected based on error analysis.

[0011] A second aspect of the present invention provides a multidimensional time series forecasting system based on hybrid higher-order reservoir calculation, including a memory and a processor. The memory includes a multidimensional time series forecasting method program based on hybrid higher-order reservoir calculation. When the multidimensional time series forecasting method program based on hybrid higher-order reservoir calculation is executed by the processor, it implements the steps of the multidimensional time series forecasting method based on hybrid higher-order reservoir calculation as described in any one of the present invention.

[0012] This invention addresses the shortcomings of the prior art and has the following beneficial effects: This invention first distinguishes and organizes higher-order neighborhood information related to the target variable at the input layer, and then uses higher-order node coupling within the buffer pool to nonlinearly propagate and reorganize this structured input information, thus more fully expressing the dynamic relationships generated by the combined effects of multiple variables. Simultaneously, this invention retains the characteristic of primarily training readout layer weights in the buffer pool calculation, offering advantages such as simple training process and low computational cost. Furthermore, by matching the input structure parameters with the higher-order coupling strength, it can avoid prediction error fluctuations caused by simply increasing the higher-order coupling strength. Attached Figure Description

[0013] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other embodiments can be obtained from these drawings without creative effort.

[0014] Figure 1 Schematic diagram of traditional reservoir calculation structure; Figure 2 This is a schematic diagram of the multidimensional time series prediction method based on hybrid high-order reservoir calculation according to the present invention; Figure 3 A schematic diagram comparing the errors of traditional reservoir calculations and the method of this invention in the Lorenz system; Figure 4 A schematic diagram illustrating the impact of the input node block ratio on the prediction error of different state variables. Figure 5 This is a schematic diagram illustrating the impact of higher-order coupling strength on the prediction error of different state variables. Detailed Implementation

[0015] To better understand the above-mentioned objectives, features, and advantages of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments and features described in these embodiments can be combined with each other.

[0016] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and therefore the scope of protection of the invention is not limited to the specific embodiments disclosed below.

[0017] In this invention, the closest existing technology is the traditional reservoir computation method. This method typically first constructs a random input matrix Win, injecting each input variable into the reservoir nodes according to random weights; then, a fixed reservoir connection matrix A describes the recursive connections between the reservoir nodes, and a nonlinear activation function is used to update the reservoir state; finally, the reservoir states from the training phase are collected, and the linear readout weights Wout are solved using ridge regression or least squares methods to obtain the predicted output. Its basic structure can be summarized as "the input signal is randomly mapped into a fixed recursive reservoir, and then the predicted value is output by the linear readout layer."

[0018] like Figure 1 As shown, the input matrix Win of a traditional RC (Recursive Conversion) system typically involves random mixing of all input variables, and the internal connection matrix A of the reservoir usually represents pairwise connections between nodes. This structure can enhance the forecasting ability of ordinary time series data by utilizing the high-dimensional nonlinear response of the reservoir, but it lacks explicit distinction of the physical, topological, or dynamic relationships between input variables, making it difficult to ensure that neighboring variables closely related to the target variable are preferentially and stably injected into the reservoir. Some improved schemes perform local structuring on the input matrix, giving input variables related to the target variable different injection weights, or allowing unrelated variables to enter the reservoir with zero connections. These schemes can enhance the relevant information of the target variable at the input stage, but the internal state evolution of the reservoir still mainly relies on ordinary pairwise connection matrices, lacking corresponding high-order node coupling mechanisms. Therefore, when multiple variables act together, multiple neighboring nodes have synergistic effects, or nonlinear combination terms dominate the dynamic evolution, the structured input information may be remixed by ordinary recursive connections after entering the reservoir, failing to fully form an internal state response that matches high-order dynamics.

[0019] Traditional random input matrices cannot distinguish between the target variable itself, its linearly correlated variables, and nonlinear combinations, leading to the random mixing of high-order structural information during the input phase. For prediction tasks involving obvious variable combinations or local network neighborhood relationships, random input methods may treat key variables and weakly correlated variables equally, resulting in insufficient response of the reservoir to input components truly relevant to the target variable, thus reducing prediction accuracy and interpretability. Furthermore, traditional reservoir internal state updates primarily rely on pairwise connection matrices. This mechanism is suitable for describing the interaction between two reservoir nodes but struggles to express high-order coupling effects formed by the combined actions of three or more nodes. For product terms in chaotic systems, multi-neighbor collaborative effects in network nodes, and local topological constraints in complex systems, relying solely on pairwise connections easily leads to insufficient internal state representation, especially in the prediction of strongly nonlinear variables, boundary nodes, or nodes affected by multi-source coupling, where errors are more easily amplified. Furthermore, when only the input matrix is ​​structured, although local correlation information can be selectively introduced into the reservoir at the input end, the state evolution within the reservoir is still mainly determined by the original low-order connections, which limits the characterization of nonlinear dynamic responses under the combined effects of multiple variables. Therefore, whether an effective match is formed between the structured design of the input layer and the dynamics within the reservoir will further affect the model's ability to express the dynamic characteristics of complex systems and its predictive performance.

[0020] The purpose of this invention is to propose a multidimensional time series forecasting method based on hybrid high-order reservoir computation. On one hand, a block-structured input matrix is ​​constructed based on the high-order neighborhood set of the target variable, enabling the target variable itself, linearly correlated variables, and nonlinearly combined correlated variables to enter the reservoir according to different structural relationships. On the other hand, a high-order connectivity tensor and corresponding high-order coupling terms are introduced into the reservoir state update equation, allowing the reservoir to express the nonlinear dynamic response generated by the combined action of multiple nodes. Through the synergy between input organization and high-order reservoir evolution, this invention can improve the accuracy, stability, and adaptability to high-order dynamic structures in complex multidimensional time series forecasting.

[0021] Therefore, the first aspect of the present invention provides a multidimensional time series forecasting method based on hybrid high-order reservoir calculation, comprising the following steps: S1, Obtain the multivariate time series u(t) = [u1(t), u2(t), ..., u m (t)] T Then, normalization and separation of training data and prediction data were performed.

[0022] S2, select the q-th state variable as the prediction target u q (t), and determine its higher-order neighborhood set S based on the dynamic equation, variable correlation, or network topology. q ={sq,1 ,s q,2 ,...,s q,Dq}

[0023] S3, construct a block-structured input matrix oriented towards the target variable uq according to Sq. (q) This allows input variables belonging to the local higher-order neighborhood set to be injected into the corresponding structure block with random weights, while the columns of irrelevant variables are set to zero.

[0024] S4, Construct a matrix containing pairwise connections. and higher-order connection tensors The reservoir is configured with low-order coupling strength k1, high-order coupling strength k2, leakage rate ε, damping coefficient γ, input coupling strength α, and bias β.

[0025] S5 updates the state θ of the reservoir nodes based on the state update equation, which includes node self-dynamics, pairwise coupling, higher-order coupling, and structured input-driven processes. i (t). The structured input driving term is... (q) u(t) is determined by both u(t) and β.

[0026] S6, collect the transient state matrix Θ of the reservoir, and use the target variable u q The training output sequence of (t) constitutes the target output matrix Y, and Wout=YΘ is solved by ridge regression. T (ΘΘ T +λI) -1 Solve for the output layer weights ; It should be noted that this is the regularization coefficient of ridge regression, which is used to control the regularization strength and avoid model overfitting. It is the identity matrix. This term is added to ensure stable inversion even when the matrix is ​​not invertible or the condition number is too large.

[0027] S7, Prediction Phase Based on Output target variable u q The prediction results of (t) are used, and the matching combination of input node block ratio p, input weight scaling factor and higher-order coupling strength k2 is selected based on error analysis.

[0028] It should be noted that the output weights are obtained after training is completed. Then, the prediction phase begins: for any given time... Using the current state vector θ(t) of the reserve pool, calculate the predicted value of the target variable. : Ultimately, the hyperparameters of the selection model need to be adjusted based on the prediction error: finding the input node block ratio. Input weight scaling factor, higher-order coupling strength The optimal combination of these hyperparameters.

[0029] In step S1, as Figure 2 As shown, this invention proposes a multidimensional time series prediction method based on hybrid higher-order reservoir computation. It builds upon the traditional reservoir computation framework by introducing higher-order enhancement mechanisms simultaneously at the input and reservoir layers. The overall structure includes a data acquisition and preprocessing module, a target variable and higher-order neighborhood determination module, a block-based structured input matrix construction module, a higher-order reservoir state evolution module, a linear readout training module, and a prediction output module. These modules sequentially complete the multivariate time series input, selection of target variable-related structural information, structured input injection, higher-order dynamic evolution within the reservoir, readout weight calculation, and target variable prediction output.

[0030] For a dynamical system with m state variables, the input time series at time t is represented as u(t) = [u1(t), u2(t), ..., u...]. m (t)] T , where u m (t) represents the value of the m-th state variable at time t. When it is necessary to predict a certain state variable, the q-th state variable is denoted as the target variable u. q u(t), where q∈{1,2,...,m}. It should be noted that u(t) is the complete multivariate state vector received by the input layer. q (t) is the prediction target selected from the multivariate system; different target variables q can correspond to different local higher-order neighborhood structures and different structured input matrices.

[0031] For the target variable u q This invention determines the higher-order neighborhood set S related to its dynamic evolution. q ={s q,1 ,s q,2 ,...,s q,Dq}. Among them, s q,d Represents the relationship with the target variable u q The relevant set of local variables, D q This indicates the number of higher-order neighborhood sets corresponding to the target variable. For example, in the Lorenz system, if the evolution of the target variable includes linearly correlated variables or nonlinear combinations such as xz and xy, then the corresponding variables or combinations of variables can serve as local higher-order neighborhood information of the target variable; in network node dynamics, the target node itself and its neighboring nodes can also constitute a local structure set related to the target variable.

[0032] In step S2, the input layer no longer uses a completely random input matrix, but instead constructs a block-structured input matrix based on Sq. (q) This matrix can be represented as a matrix composed of multiple input structure blocks ψ(sq,d), each structure block corresponding to a local high-order neighborhood set. If an input variable belongs to the current local set sq,d, a random weight is set in the corresponding column of that structure block; if the input variable is unrelated to the current structure block, the corresponding column is set to zero or a weak connection is established. In this way, the information of the target variable itself, the information of linearly related variables, and the information of nonlinearly combined related variables can be injected into the reserve pool in a block-based manner, avoiding complete random mixing during the input stage.

[0033] In step S3, the reserve pool layer consists of N nonlinear reserve pool nodes, and the state of the i-th reserve pool node is denoted as θ. i (t). This layer also contains ordinary pairwise connection matrices. and higher-order connection tensors According to the state update formula in the paper, the state of the reservoir node is determined by four types of factors: first, the node's own dynamics, including damping, natural frequency or external driving, and nonlinear recovery; second, the dynamics caused by... The description of paired node coupling terms; three are composed of The higher-order node coupling terms described represent the combined effect of multiple reserve pool nodes on the current node state; fourthly, the structured input matrix... (q) The external input driving term is formed by the input vector u(t), the input coupling strength α, and the bias β.

[0034] In step S4, the overall structure of this invention does not involve simply changing the input layer or adding nonlinearity to the reservoir. Instead, it selectively organizes the high-order neighborhood information related to the target variable during the input stage, and further propagates, reorganizes, and amplifies it through high-order node coupling during the reservoir state evolution stage. The training stage retains the fundamental advantages of reservoir computation, namely, the input matrix and the internal connections of the reservoir remain fixed after being set, and the weights Wout are mainly derived through ridge regression. The prediction stage utilizes the trained Wout to evaluate the target variable u. q Output the future state of (t).

[0035] The state of the i-th reserve pool node can be represented as: ,

[0036] , (q)The structured input matrix is ​​constructed for the target variable uq. The state equation shows that the state of the reservoir nodes is simultaneously influenced by the node's own dynamics, paired node coupling, higher-order coupling of multiple nodes, and the structured input related to the target variable.

[0037] Among them, the representation of parameters, For the node's own dynamics, where, To control the leakage rate, the relative impact of "current input / coupling" and "historical reserve pool status" is controlled. The smaller the size, the stronger the "memory" of the historical state; The larger the value, the stronger the impact of the current input / coupling. The damping coefficient describes the "resistance" to changes in the node state (similar to damping in damped vibrations in physics). For the first The natural frequency of each node determines the "reference frequency" of its own oscillation. It is a nonlinear restoring force term (similar to the gravitational restoring force of a pendulum in physics, which makes the nodal state tend to "return to equilibrium").

[0038] For the inter-node coupling term: where k1 is the pairwise coupling strength, controlling "two nodes ( and "State difference" for nodes Impact ( The weights of the pairwise connections describe right The coupling weights). k2 is the higher-order coupling strength, controlling the coupling strength of the three nodes ( ) Combination relationship of states ( ")" for nodes The impact, These are the weights of higher-order connections, describing... right Higher-order coupling weights. Normalization of the summation term ( or ): Avoid coupling strength increasing with the number of nodes Unrestricted growth ensures model stability.

[0039] For the input driving term, where, The input coupling strength controls the magnitude of the influence of external inputs on the node state. To add a "baseline offset" to the input bias term, the model's expressive power is enhanced. The input matrix is ​​a structured matrix consisting of "target variable". "Self-reconstruction" indicates that the input is strongly correlated with the target task and is a customized input mapping. It is a multivariate time series.

[0040] It should be noted that a block-structured input matrix is ​​constructed based on the higher-order neighborhood set of the target variable, enabling relevant information of the target variable to enter the reserve pool according to local structural relationships. Paired connections are also introduced during the state evolution of the reserve pool. and higher-order connection tensors This allows the state of the reservoir nodes to express the nonlinear response generated by the combined action of multiple nodes. By matching the input node block ratio, input weight scaling factor, and high-order coupling strength, the prediction error of complex multidimensional time series is reduced. At the same time, the advantage of the reservoir computation being used to train only the readout layer weights is maintained, while the ability to express nonlinear combination relationships and high-order neighborhood relationships is improved, forming a system architecture that can be used for chaotic systems, complex network node dynamics, and other multidimensional time series prediction tasks.

[0041] The implementation examples were verified using the Lorenz chaotic system. Figure 3 The comparison of prediction performance between traditional RC and the hybrid high-order RC of this invention is shown. Experiments show that this invention reduces the RMS prediction error for all three variables (x, y, and z), with the x component error reduced from 8.74 × 10⁻⁶. -4 Reduced to 2.19×10 -6 The error of the y component is 2.00 × 10⁻⁶. -3 Reduced to 9.83×10 -6 The z-component error is 6.70 × 10⁻⁶. -3 Reduced to 6.46×10 -6 .

[0042] Figure 4 This demonstrates the impact of the input node block ratio on prediction error. For a block-structured input matrix, the input node block ratio determines the node resources occupied by different local high-order neighborhood information in the reservoir. When the distribution of structure blocks corresponding to the target variable itself, linearly correlated variables, or nonlinearly combined correlated variables is relatively balanced, the reservoir can simultaneously obtain sufficient target variable information and coupled variable information, and the prediction error is usually low. If the proportion of a certain input structure block is too large, it will compress the information representation space of other related variables; if the proportion of a certain structure block is too small, the corresponding variable relationship will be difficult to fully inject into the reservoir, both of which may lead to an increase in prediction error. Therefore, Figure 4 This indicates that the proportion of input node blocks should be set in combination with the dynamic characteristics of the target variable and the importance of each input structural block, rather than using a fixed empirical proportion.

[0043] Figure 5This demonstrates the impact of higher-order coupling strength on prediction error. Higher-order coupling strength is used to modulate the influence of the combined effects of multiple nodes within the reservoir on state evolution. Appropriate higher-order coupling can enrich the nonlinear dynamic response within the reservoir, allowing higher-order neighborhood information contained in the structured input to be further propagated and reorganized within the reservoir. However, the effect of this parameter on model performance does not exhibit a simple monotonic change. Figure 5 It is evident that the prediction error fluctuates significantly with changes in higher-order coupling strength, and the model performance alternates at different parameter points. This indicates a matching relationship between the effect of higher-order interaction terms and the input weights, node block ratios, and the higher-order neighborhood structure corresponding to the target variable. The model prediction error only decreases when the second-order coupling strength is consistent with the current input structure and the dynamic state of the reservoir. When these two factors are mismatched, the additional second-order interaction can actually perturb the evolution of the reservoir state, leading to an increase in prediction error.

[0044] Secondly, in this method, the higher-order neighborhood set can be determined based on dynamic equations, network topology, correlation coefficients, mutual information, Granger causality, or other variable dependency analysis methods. Higher-order connectivity tensor It can be generated from random higher-order hyperedges, simplex complexes, hypergraphs, or known higher-order network structures. The sinusoidal coupling function in state updates can be replaced with a hyperbolic tangent function, a polynomial function, a piecewise linear function, or other nonlinear functions achievable by a physical reservoir. Readout layer training can employ ridge regression, recursive least squares, LASSO, or online regression methods.

[0045] In summary, this invention first distinguishes and organizes higher-order neighborhood information related to the target variable at the input layer, and then uses higher-order node coupling within the reservoir to nonlinearly propagate and reorganize this structured input information, thus more fully expressing the dynamic relationships generated by the combined effects of multiple variables. Simultaneously, this invention retains the characteristic of reservoir calculation primarily training readout layer weights, offering advantages such as simple training process and low computational cost. Furthermore, by matching the input structure parameters with the higher-order coupling strength, it can avoid prediction error fluctuations caused by simply increasing the higher-order coupling strength.

[0046] In addition, this method may also include: The original time series is normalized, and the input signal is expanded into a multi-dimensional input vector by a randomly generated mask matrix, which is constructed using a single nonlinear node in conjunction with a delayed feedback line. M virtual nodes are set in the delay line, and the time interval between the virtual nodes is θ. The state of the M virtual nodes is updated based on the time interval between the virtual nodes. The system receives the states of M virtual nodes from the time-delay coupling reservoir as input, constructs a hypergraph structure reservoir, updates the node states of the hypergraph structure reservoir, sets a programmable coupling matrix between the first and second levels, and dynamically adjusts the elements of the coupling matrix. Collect the state vectors of all nodes in the second-level hypergraph reservoir, construct an extended state vector, and train the readout matrix using ridge regression.

[0047] It should be noted that combining the temporal continuity of the time-delay coupled reservoir with the high-order topological interaction capability of the hypergraph reservoir takes into account the nonlinear modeling of both the time and structural dimensions. The hypergraph structure enables group interaction among multiple nodes, breaking through the limitations of traditional binary edges, and can more accurately represent the high-order interactions in real complex systems. The extended state vector includes first-order, second-order, and third-order combined features, which significantly improves the nonlinear mapping capability of the readout layer.

[0048] It should be noted that the states of M virtual nodes from the received delay-coupled reservoir are used as input to construct a hypergraph-structured reservoir. This hypergraph consists of N nodes and K hyperedges, with each hyperedge connecting at least 3 nodes. A programmable coupling matrix C ∈ R^{N×M} is set between the first and second levels. The matrix elements are dynamically adjusted through an online learning algorithm to achieve optimal information flow between the two reservoir levels.

[0049] In addition, this method also includes: We receive high-dimensional multivariate time series data, construct a greedy iterative algorithm based on Granger causality, initialize the candidate higher-order neighbor set to an empty set, and calculate the improvement in the prediction accuracy of each candidate variable after adding the candidate variable, given the current empty set. Select the variable with the largest increase and add it to the candidate higher-order neighbor set. Repeat the operation and output the candidate higher-order neighbor set as the higher-order neighbor set of the variable. Encode it into the input matrix and adjacency matrix of the reserve pool. In the nth row of the input matrix, the column corresponding to the variable in the candidate higher-order neighbor set is set to non-zero weight. The adjacency matrix contains pairwise connections and hyperedge connections. The weight of the hyperedge is determined by the variables in the candidate higher-order neighbor set. A cascaded two-layer reservoir architecture is adopted, with the first layer being a fast dynamics reservoir and the second layer being a slow dynamics reservoir. The two layers interact through a cross-layer connection matrix encoded with a high-order structure. The cascaded two-layer reservoir architecture is used to achieve parallelization of structural inference and dynamics prediction. Within each time window, a portion of the computational resources is used to update the inference results of higher-order structures, while another portion is used to perform dynamic predictions. The two exchange data through shared memory.

[0050] It should be noted that the adjacency matrix W contains not only paired connections (binary edges) but also hyperedge connections (connecting three or more nodes). The weight of the hyperedge is jointly determined by the variables in the candidate higher-order neighbor set. A cascaded two-layer reservoir architecture is adopted—the first layer is a fast dynamic reservoir (using dynamic devices with small time constants), and the second layer is a slow dynamic reservoir (using dynamic devices with large time constants). The two layers interact through a cross-layer connection matrix encoded with higher-order structures. This method introduces the Granger causal inference mechanism into the reservoir computation, solving the modeling problem when higher-order structural information is partially or completely unknown. Moreover, structural inference and dynamic prediction are executed in parallel, realizing an online learning mode of "inference and prediction on the same side." Finally, a two-layer slow-fast hybrid reservoir architecture is used to match the dynamic behavior of different time scales in real-world complex systems.

[0051] A second aspect of the present invention provides a multidimensional time series forecasting system based on hybrid higher-order reservoir calculation, including a memory and a processor. The memory includes a multidimensional time series forecasting method program based on hybrid higher-order reservoir calculation. When the multidimensional time series forecasting method program based on hybrid higher-order reservoir calculation is executed by the processor, it implements the steps of the multidimensional time series forecasting method based on hybrid higher-order reservoir calculation as described in any one of the present invention.

[0052] In the several embodiments provided in this application, it should be understood that the disclosed devices and methods can be implemented in other ways. The device embodiments described above are merely illustrative. For example, the division of units is only a logical functional division, and in actual implementation, there may be other division methods, such as: multiple units or components can be combined, or integrated into another system, or some features can be ignored or not executed. In addition, the coupling, direct coupling, or communication connection between the various components shown or discussed can be through some interfaces, and the indirect coupling or communication connection between devices or units can be electrical, mechanical, or other forms.

[0053] The units described above as separate components may or may not be physically separate. The components shown as units may or may not be physical units. They may be located in one place or distributed across multiple network units. Some or all of the units may be selected to achieve the purpose of this embodiment according to actual needs.

[0054] In addition, in the various embodiments of the present invention, each functional unit can be integrated into one processing unit, or each unit can be a separate unit, or two or more units can be integrated into one unit; the integrated unit can be implemented in hardware or in the form of hardware plus software functional units.

[0055] Those skilled in the art will understand that all or part of the steps of the above method embodiments can be implemented by hardware related to program instructions. The aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it performs the steps of the above method embodiments. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0056] Alternatively, if the integrated units of this invention are implemented as software functional modules and sold or used as independent products, they can also be stored in a computer-readable storage medium. Based on this understanding, the technical solutions of the embodiments of this invention, or the parts that contribute to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the methods of the various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as mobile storage devices, ROM, RAM, magnetic disks, or optical disks.

[0057] The above are merely specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A multidimensional time series forecasting method based on hybrid high-order reservoir calculation, characterized in that, Includes the following steps: Obtain multivariate time series data, normalize them, divide them into training and prediction data, select the q-th state variable as the prediction target, and determine the high-order neighborhood set based on the dynamic equation, variable correlation, or network topology. Based on the higher-order neighborhood set, a block-based structured input matrix oriented towards the target variable is constructed, and the input variables belonging to the local higher-order neighborhood set are injected into the corresponding structure blocks with random weights, while the columns of irrelevant variables are set to zero. Establish a reservoir containing paired connection matrices and higher-order connection tensors, and set the lower-order coupling strength, higher-order coupling strength, leakage rate, damping coefficient, input coupling strength, and bias. A state update equation is constructed, and the state of the reserve pool nodes is updated using the state update equation. Based on the updated state of the reserve pool nodes, the prediction result of the target variable is output. The matching combination of input node block ratio, input weight scaling factor and high-order coupling strength is selected according to error analysis.

2. The multidimensional time series forecasting method based on hybrid higher-order reservoir calculation according to claim 1, characterized in that, Obtain multivariate time series data, normalize them, and separate training and prediction data. Select the q-th state variable as the prediction target, and determine the higher-order neighborhood set based on the dynamic equation, variable correlation, or network topology. Specifically: Obtain multivariate time series data, normalize them, and divide them into training and prediction data. When it is necessary to predict a certain state variable, the q-th state variable is recorded as the target variable. For the target variable, if the evolution of the target variable includes linearly correlated variables or nonlinear combination terms, the corresponding variables or combinations of variables are used as local high-order neighborhood information of the target variable. Finally, a set of higher-order neighborhoods related to the target variable and the dynamic equation, variable correlation, or network topology is determined.

3. The multidimensional time series forecasting method based on hybrid higher-order reservoir calculation according to claim 1, characterized in that, Based on the higher-order neighborhood set, a block-based structured input matrix oriented towards the target variable is constructed. Input variables belonging to the local higher-order neighborhood set are injected into the corresponding structure blocks with random weights, and columns corresponding to irrelevant variables are set to zero. Specifically: A block-structured input matrix is ​​constructed based on the higher-order neighborhood set. The input matrix is ​​a matrix composed of multiple input structure blocks, and each structure block corresponds to a local higher-order neighborhood set. Determine whether an input variable belongs to the current higher-order neighborhood set. If an input variable belongs to the current higher-order neighborhood set, set a random weight in the corresponding column of the current structure block. If the input variable is unrelated to the current structure block, the corresponding column is set to zero or set as a weak connection. Finally, the target variable's own information, linearly related variable information, and nonlinear combination related variable information are injected into the reserve pool in a block manner.

4. The multidimensional time series forecasting method based on hybrid higher-order reservoir calculation according to claim 1, characterized in that, Establish a reservoir containing paired connection matrices and higher-order connection tensors, and set the low-order coupling strength, higher-order coupling strength, leakage rate, damping coefficient, input coupling strength, and bias, specifically as follows: Establish a reservoir containing pairwise connection matrices and higher-order connection tensors, and configure N nonlinear reservoir nodes in the reservoir containing pairwise connection matrices and higher-order connection tensors; Damping, natural frequency or external drive and nonlinear response term are treated as node self-dynamic terms, and the structured input matrix, input vector, input coupling strength and bias are used as the external input drive term. The state of the reserve pool nodes is constructed based on the pairwise node coupling terms described by the ordinary pairwise connection matrix, the node self-dynamic terms, and the higher-order node coupling terms described by the higher-order connection tensor.

5. The multidimensional time series forecasting method based on hybrid higher-order reservoir calculation according to claim 1, characterized in that, A state update equation is constructed, and the state of the reserve pool nodes is updated using this equation. Based on the updated state of the reserve pool nodes, the prediction result of the target variable is output. Furthermore, a matching combination of the input node block ratio, input weight scaling factor, and higher-order coupling strength is selected based on error analysis. Specifically: The state of the reservoir nodes is updated based on the state update equation, which includes node self-dynamics, pair coupling, higher-order coupling, and structured input-driven processes. The transient state matrix of the reservoir is collected and the target output matrix is ​​constructed using the training output sequence of the target variable. The solution is obtained by ridge regression. The prediction results of the target variable are output based on the solution results, and the matching combination of input node block ratio, input weight scaling factor and high-order coupling strength is selected based on error analysis.

6. A multidimensional time series forecasting system based on hybrid higher-order reservoir computation, characterized in that, The system includes a memory and a processor. The memory includes a program for a multidimensional time series forecasting method based on a hybrid higher-order reservoir. When the processor executes the program for the multidimensional time series forecasting method based on a hybrid higher-order reservoir, it implements the steps of the multidimensional time series forecasting method based on a hybrid higher-order reservoir as described in any one of claims 1-5.

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