Nonlinear multi-scale dynamic system control mechanism inference method based on koopman operator theory

Abstract

The application discloses a nonlinear multi-scale dynamic system control mechanism inference method based on Koopman operator theory, which is based on the Koopman operator theory and establishes a reduced-order mechanism model of a dynamic system from observed time series data. First, the observed time series data is dynamically sampled based on the adaptive sampling strategy designed by the application, the sampling rate is adjusted according to the data change, and the information of different time scales is effectively separated. Then, cubic B-spline interpolation is applied to locally smooth and numerically differentiate the discrete sampling points. On this basis, the nonlinear modeling capability of the Koopman operator theory is used to innovatively construct a Block-Hankel matrix to capture the multi-element dynamic characteristics of the system; and the kernel singular value decomposition technology is used to extract the main dynamic mode, so that the system identification is extended to a nonlinear dynamic system. Finally, the SINDy algorithm is used to identify the reduced-order model, so as to realize accurate modeling of a complex nonlinear multi-scale system. On the basis of the adaptive sampling and multi-scale data processing capability, the application combines the advantages of Koopman embedding and sparse regression, has strong robustness and universality, and has theoretical and practical significance for mechanism modeling of nonlinear and multi-scale systems.

Description

Technical Field

[0001] The present invention belongs to the technical field of nonlinear, multi-scale dynamic system mechanism modeling, and in particular to a nonlinear multi-scale dynamic system control mechanism inference method based on Koopman operator theory. Background Art

[0002] In increasingly complex industrial sectors, systems often exhibit various nonlinear behaviors under varying operating conditions and exhibit unique dynamics across multiple timescales. The nonlinear and multiscale properties of industrial systems pose significant challenges to understanding and predicting their dynamics. Deep learning algorithms based on neural networks have made significant progress in predicting industrial parameters. However, rapid technological advancements have driven the pursuit of more complex and sophisticated analyses in industrial process modeling. Traditional deep learning models, acting as black boxes, struggle to reveal the underlying mechanisms of dynamic systems, limiting their application in applications requiring high-precision interpretability. On the other hand, traditional physical models and mathematical methods based on first principles also face difficulties in simulating complex industrial systems, particularly in the absence of a comprehensive theoretical framework and precise parameters. Therefore, researchers are exploring new approaches to develop models that can describe the essence of industrial processes based on observed time series data, providing more accurate theoretical support for the analysis, design, and optimization of industrial systems.

[0003] In recent years, extensive research has focused on data-driven identification of nonlinear systems, advancing both nonparametric and parametric models. Symbolic regression based on genetic algorithms has been widely studied in various fields, but this approach can overfit the target system with a large number of false positives, leading to high computational costs and identification errors. Methods based on sparse identification of nonlinear dynamics (SINDy) have been widely used to reveal the physical principles of low-dimensional systems, but their application to industrial systems with complex nonlinear characteristics remains exploratory.

[0004] Research has demonstrated that analytical frameworks based on the Koopman operator exhibit great potential for handling nonlinear systems. When combined with the sparsity of SINDy, this approach can identify the most influential low-dimensional variables affecting system dynamics, leading to the construction of more compact reduced-order models. However, this combination still faces many challenges in modeling multiscale systems, particularly in dealing with uneven data sampling and cross-scale dynamics. Therefore, a mechanistic modeling approach for systems that can simultaneously account for nonlinearity and multiscale dynamics is of great research interest. Summary of the Invention

[0005] In view of the defects or deficiencies of the prior art described above, the present invention provides a nonlinear multi-scale dynamic system control mechanism inference method based on Koopman operator theory, which is used for data-driven modeling of dynamic systems and aims to generate a generalizable and interpretable reduced-order model. First, the observed time series data is downsampled based on an adaptive sampling strategy, which dynamically adjusts the sampling rate according to data changes, effectively isolating information of different scales. Subsequently, cubic B-spline interpolation is applied for local data smoothing and numerical differentiation. Utilizing the nonlinear modeling capability of Koopman operator theory, the system identification is extended to nonlinear dynamic systems by implementing kernel SVD on the newly constructed Block-Hankel matrix. In addition, an enhanced SINDy method is integrated to achieve accurate modeling of complex systems.

[0006] The technical solution adopted by the present invention to achieve the above-mentioned purpose is:

[0007] A mechanistic modeling method for nonlinear, multiscale dynamic systems based on Koopman embedding theory is proposed. This method effectively separates multi-timescale information through a designed adaptive sampling strategy and utilizes the nonlinear modeling capabilities of Koopman operator theory to generate a generalizable and interpretable reduced-order model from observed time series data. The method includes the following steps:

[0008] S1. Obtain system observation data;

[0009] S2, performs adaptive sampling according to the dynamic changes of data to effectively capture multi-scale information;

[0010] S3, the sampled data {t i ,x i} Apply cubic B-spline interpolation for local smoothing;

[0011] S4. Analyze the dynamic characteristics of the system, construct the Block-Hankel matrix for the interpolated time series data, and use its high-dimensional structure to extract the spatiotemporal characteristics of the system;

[0012] S5. Decompose the constructed Block-Hankel matrix using kernel singular value decomposition (Kernel SVD) to extract the main dynamic modes and characteristic subspaces of the system.

[0013] S6. Combine the extracted low-dimensional feature subspace with the enhanced SINDy method to construct a sparse dynamic model Used to reduce the order of the model by calculation Obtain an approximate representation of the Koopman operator that characterizes the incremental value of the next time point of the current attribute data, and superimpose the value of the previous time point on it as a predicted value of the future state or dynamic characteristics of the current attribute data;

[0014] Step 7: Use the obtained reduced-order mechanism model to analyze the future state changes of system data information, thereby achieving precise control.

[0015] The observation data is actually collected long-term historical observation information, or time series information of the system evolving over time simulated in a simulation environment under a variety of different initial conditions.

[0016] The adaptive sampling is to dynamically adjust the sampling rate within a range according to the change rate of the data values ​​of the continuous sampling points of the observed data, and obtain the long time series sampling point data of different time scales {t i ,x i}.

[0017] The cubic B-spline interpolation includes: for a given data point {t i ,x i}(i=0,1,...,n), by recursively defining the B-spline basis function N i (t), we get a piecewise cubic B-spline polynomial Use its smooth interpolation data point t i , which is used to maintain the smoothness and stability of the curve while ensuring the interpolation accuracy.

[0018] The recursive definition of B-spline basis function N i (t) specifically include:

[0019]

[0020] where t i are the elements of the knot vector. For cubic B-spline k=3, the above recursive process produces cubic polynomial B-spline basis functions;

[0021] The interpolation function S(t) is a linear combination of basis functions and is expressed as:

[0022]

[0023] where c i is the control point coefficient to be determined, which is solved by introducing boundary conditions;

[0024] To ensure that the interpolation curve S(t) passes through the given data point x i , the following conditions must be met:

[0025] S(t i )=x i ,for i=0,1,...,n.

[0026] Therefore, the first-order derivative of the cubic B-spline basis function is expressed as:

[0027]

[0028] The method of constructing a Block-Hankel matrix for the interpolated time series data includes: constructing a new Block-Hankel matrix different from the traditional standard embedding matrix by introducing a window length L. The Block-Hankel matrix is the Hankel matrix Vertical connection formation; the Block-Hankel matrix It is not only used to capture the dynamic characteristics of each time series, but also preserves the interdependencies between different time series.

[0029] The Hankel matrix for:

[0030]

[0031] Where K = N-L+1, N is the total length of the time series.

[0032] The method of decomposing the constructed Block-Hankel matrix by using kernel singular value decomposition (Kernel SVD) includes: using a Gaussian kernel function Map the Block-Hankel matrix to a high-dimensional feature space:

[0033]

[0034] in and are any two columns of the Block-Hankel matrix;

[0035] Kernel matrix K pq Perform SVD decomposition:

[0036] K=UΣV T

[0037] Then, after KSVD decomposition, the eigenvector matrix V is obtained, which consists of a set of principal singular vectors in the kernel space and is used to represent the principal dynamic mode and the eigensubspace.

[0038] The SINDy algorithm specifically includes:

[0039] Construct a function library Θ(V) that encapsulates the eigenvector matrix V using one of multiple potential dynamical function forms;

[0040]

[0041] Where m is the number of time points and p is the number of functions in the library;

[0042] Obtain a reduced-order model of the system through sparse regression:

[0043]

[0044] Reduced-order model The calculation result of approximately represents the Koopman operator of the incremental value of the next time point of the current attribute data.

[0045] The dynamic function forms include polynomials and trigonometric functions.

[0046] The present invention has the following beneficial effects and advantages:

[0047] A new mechanism modeling method based on Koopman embedding theory is used to establish reduced-order mechanism models for nonlinear, multiscale dynamic systems based on noisy multidimensional observational time series data. From the perspective of separating time scales, the superiority of this method is mainly reflected in the designed dynamic sampling strategy, which adjusts the sampling rate according to the rate of change (gradient) of the data. In other words, in areas where the data changes dramatically (high-gradient regions), the sampling rate is increased to capture more details and changes. In areas where the data changes more slowly (low-gradient regions), reducing the sampling rate and redundant data can improve data processing efficiency. This dynamic sampling can adaptively adjust the sampling strategy according to the data characteristics, without the need to pre-set a fixed sampling rate, and is suitable for data with a variety of different characteristics.

[0048] Furthermore, from the perspective of extracting the dynamics of nonlinear systems, cubic B-spline interpolation is particularly effective for fitting such systems due to its smoothness and high-order continuity, avoiding the jagged effects that may occur with general interpolation methods such as linear interpolation. The innovative construction of the Block-Hankel matrix preserves the interrelationships between different time series and captures the dynamic characteristics of multivariate systems. By applying kernel technology, the Block-Hankel matrix is ​​nonlinearly expanded, allowing for kernel singular value decomposition, which retains the main dynamic information of the system in high-dimensional space. Constructing the SINDy model based on the extracted features can reduce the order of nonlinear systems and, to a certain extent, reduce the impact of noise on the model, thereby improving the accuracy and robustness of the model, enabling it to more accurately guide system predictive control. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 Flow chart of the method of the present invention.

[0050] Figure 2 Schematic diagram of a jacketed continuous stirred tank reactor system.

[0051] Figure 3 A comparison chart of the real data of a jacketed non-isothermal CSTR system, the SINDy results, and the results of the method proposed in the present invention.

[0052] Figure 4 It is a comparison chart of the real data of the multi-reaction isothermal batch reactor system, the SINDy results and the results of the method proposed in the present invention.

[0053] Figure 5 This is a comparison chart of the SINDy results of the actual grinding-classification system and the mean square error results of the method proposed in this invention. DETAILED DESCRIPTION

[0054] To make the above-mentioned objects, features, and advantages of the present invention more readily apparent, the specific implementation methods of the present invention are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways than those described herein, and those skilled in the art may make similar modifications without departing from the scope of the invention. Therefore, the present invention is not limited to the specific implementation methods disclosed below.

[0055] Unless otherwise defined, all technical and scientific terms used herein have the same meanings as commonly understood by those skilled in the art to which this invention belongs. The terms used in this specification are for the purpose of describing specific embodiments only and are not intended to limit the present invention. The present invention will be further described in detail below with reference to the accompanying drawings and examples.

[0056] This paper presents a unified framework for system mechanism modeling based on the control requirements of industrial systems. Experiments were conducted on two simulated reactor systems and a real grinding and sorting process. The experimental results demonstrate that this paper can accurately model nonlinear, multi-scale complex systems, demonstrating strong robustness and universal applicability. Therefore, this paper can be extended to general industrial systems, establishing reduced-order mechanism models to describe these systems and guiding practical industrial prediction and control.

[0057] like Figure 1The figure shows a flow chart of an example method of the present invention. A method for inferring the control mechanism of a nonlinear multi-scale dynamic system based on the Koopman operator theory. First, the observed time series data is dynamically sampled based on the adaptive sampling strategy designed by the present invention, and the sampling rate is adjusted according to the data changes to effectively separate information of different time scales. Subsequently, the discrete sampling points are subjected to local data smoothing and numerical differentiation using cubic B-spline interpolation. On this basis, the nonlinear modeling capability of the Koopman operator theory is utilized to innovatively construct a Block-Hankel matrix to capture the multivariate dynamic characteristics of the system; and the kernel singular value decomposition technique is used to extract the main dynamic modes, extending the system identification to nonlinear dynamic systems. Finally, the SINDy algorithm is used to identify the reduced-order model to achieve accurate modeling of complex nonlinear multi-scale systems. The programming language used in the program execution steps of the present invention is not limited to MATLAB, Python, etc.

[0058] The specific steps of the present invention are as follows:

[0059] Step 1: Obtain system observation data.

[0060] Example: The continuous stirred tank reactor (CSTR) is a chemical system widely used in process industries. This paper investigates a non-isothermal CSTR in which a first-order exothermic irreversible reaction occurs, which is controlled by the rate constant k. The schematic diagram of the reaction system is shown in Figure 2 shown.

[0061] In this process, the meaning and corresponding values ​​of each parameter are listed in Table 1 below. Based on these parameters, the material balance of substance A and the energy balance of the reactor and jacket can be formulated as follows:

[0062]

[0063]

[0064] where the reaction rate r is given by:

[0065]

[0066] The sampling details of the system are summarized in Table 2 below. Inspection of the sampled datasets shows that C A The concentration changes slowly, while the temperature variable T r and T j Shows rapid dynamic changes.

[0067] Table 1 Parameter values ​​and meanings of jacketed continuous stirred tank reactor system

[0068]

[0069]

[0070] Table 2 Sampling details of the jacketed continuous stirred tank reactor system

[0071]

[0072] The method of the present invention can be used to determine the concentration C of the reactant A in the reaction system. A , reactor temperature T r and jacket temperature T j Make predictions.

[0073] Step 2: Perform adaptive sampling to dynamically adjust the sampling rate according to the dynamic changes of the data to ensure that multi-scale information can be effectively captured;

[0074] Step 2-1: For the data x(t), calculate the rate of change as follows:

[0075]

[0076] where x t and x t+1 are the data values ​​of two consecutive sampling points.

[0077] Step 2-2: Set the initial sampling rate r0 and define the gradient threshold θ to trigger the sampling rate adjustment. If g>θ, increase the sampling rate r. Otherwise, decrease the sampling rate r. The sampling rate adjustment function f(r,g) can be defined as:

[0078] r new =r old ·f(r old ,g)=r old ·(1+α·(g-θ)) (6)

[0079] Where α is the adjustment coefficient that controls the sensitivity of sampling rate changes. Record the index of the selected sampling point to form a set of sampling points t s .

[0080] Step 3: Apply cubic B-spline interpolation to the sampled data for local smoothing;

[0081] For the sampled data points {t i ,x i}(i=0,1,...,n), define the B-spline basis function N i (t):

[0082]

[0083] where t iare the elements of the knot vector. For cubic B-spline, using k=3, the above recursive process produces cubic polynomial B-spline basis functions. The interpolation function S(t) is a linear combination of basis functions and can be expressed as:

[0084]

[0085] where c i is the control point coefficient to be determined.

[0086] In order to ensure that the interpolation curve S(t) passes through the given data points {t i ,x i}, the following conditions must be met:

[0087] S(t i )=x i ,for i=0,1,...,n.(10)

[0088] By introducing natural boundary conditions (the second-order derivative of the curve at the boundary is zero), the control point coefficient c can be solved i .

[0089] The cubic B-spline interpolation has a significant advantage, namely that it provides C 2 Continuity, which means that the interpolation curve is analytically differentiable and has continuous first-order and second-order derivatives across the segment. Its first-order derivative can be expressed as:

[0090]

[0091] Step 4: Construct a Block-Hankel matrix based on the interpolated time series data and use its high-dimensional structure to extract the spatiotemporal characteristics of the system;

[0092] The traditional Hankel alternative view of Koopman analysis method (HAVOK) includes:

[0093] Combining Koopman operator theory with time series analysis techniques, the dynamic characteristics of the system are revealed by constructing the Hankel matrix. The core idea of ​​this method is to use the linear characteristics of the Koopman operator to analyze and predict the behavior of nonlinear dynamic systems. Specifically, the Koopman operator is an infinite-dimensional linear operator acting on the state space of the dynamic system, which transforms the nonlinear evolution of the system into a linear evolution in the function space. For the equation x, t+1 =f(x t ) describes a nonlinear dynamic system, Koopman operator The Koopman operator operates on the observable function g and provides the evolution of the function in the state space. The operation of the Koopman operator is as follows:

[0094]

[0095] For time series data For the generated dynamic system, constructing the Hankel matrix requires selecting an embedding dimension m such that The rows of the Hankel matrix consist of consecutive subsequences in the time series, and the columns are formed by delayed copies of the subsequences. This design effectively captures the temporal dependencies present in the time series. The standard Hankel matrix is ​​defined as:

[0096]

[0097] The traditional HAVOK algorithm uses singular value decomposition (SVD) to analyze the Hankel matrix and decompose it into three parts:

[0098] H=U∑V * (14)

[0099] Where U and V are orthogonal matrices, and ∑ is a diagonal matrix whose diagonal elements correspond to the singular values ​​of the Hankel matrix. SVD is used to select the largest singular value and its corresponding singular vector, which form the basis of the Koopman invariant subspace. By analyzing the columns of matrix V, the main dynamic modes of the system can be identified. When the Koopman operator is applied, these main modes are preserved, thus forming the Koopman invariant subspace. If r is the number of preserved singular values, the traditional standard embedding matrix HAVOK model can be approximated as:

[0100]

[0101] Here, U r ,∑ r and V r is a matrix consisting of the first r singular values ​​and their corresponding singular vectors.

[0102] The present invention constructs a new Block-Hankel matrix that is different from the traditional standard embedding matrix by introducing an appropriate window length L:

[0103]

[0104] Where K = N-L+1, N is the total length of the time series. Hankel matrix Connect vertically to form a Block-Hankel matrix This Block-Hankel matrix not only captures the dynamic characteristics of each time series, but also preserves the interdependencies between different time series.

[0105] Step 5: Decompose the constructed Block-Hankel matrix using kernel singular value decomposition (Kernel SVD) to extract the main dynamic modes and characteristic subspaces of the system;

[0106] Step 5-1: Use Gaussian kernel function Map the Block-Hankel matrix to a high-dimensional feature space:

[0107]

[0108] in and are any two columns of the Block-Hankel matrix.

[0109] Step 5-2: Kernel matrix K pq Perform SVD:

[0110] K=UΣV T (18)

[0111] After the KSVD decomposition, the eigenvector matrix V is obtained. This matrix consists of a set of principal singular vectors in the kernel space, corresponding to the main dynamic modes in the data. It is also considered to be the basis of the Koopman operator's approximate invariant subspace. By utilizing these principal singular vectors, the SINDy algorithm can be used to identify functions associated with these principal delay coordinates.

[0112] Step 6: Combine the extracted low-dimensional feature subspace with the enhanced SINDy method to construct a sparse dynamic model.

[0113] Step 6-1. Construct a function library Θ(V) containing various potential dynamic forms, such as polynomials, trigonometric functions, etc.:

[0114]

[0115] Where m is the number of time points and p is the number of functions in the library.

[0116] Step 6-2: Obtain the reduced-order model of the system through sparse regression:

[0117]

[0118] The obtained model constitutes an approximate representation of the Koopman operator and can be used to predict the future state of the system or analyze its dynamic characteristics.

[0119] Thus, the present invention obtains a reduced-order mechanism model that accurately describes the system. Figure 3 In order to compare the real data and SINDy results of a jacketed non-isothermal CSTR system with the results of the method proposed in the present invention, in addition to this example, the present invention also tests a simulated multi-reaction isothermal batch reactor system and a real grinding-classification system. Figure 4 It is a comparison chart of the real data of the multi-reaction isothermal batch reactor system, the SINDy results and the results of the method proposed in the present invention. Figure 5 The figure shows a comparison of the SINDy results of a real grinding-classification system and the mean square error results of the proposed method. It can be seen that the present invention facilitates an efficient and accurate modeling process and effectively captures the dynamic characteristics of the system.

[0120] Step 7: Analyze the system state changes based on the obtained reduced-order mechanism model to achieve precise control of the system.

[0121] The present invention establishes a reduced-order mechanism model that describes the dynamic characteristics of nonlinear, multi-scale complex systems from observed time series data. The model is universal and can be generalized to the field of actual engineering control.

[0122] In summary, in solving the complex and challenging problem of nonlinear multi-scale industrial system modeling, the present invention proposes a new method for inferring the control mechanism of the system. The present invention adopts dynamic sampling technology to adaptively adjust the sampling strategy according to the data characteristics, effectively isolating the dynamics of different time scales. In order to obtain continuous time series data and its corresponding derivatives from discrete sample points, the present invention uses cubic B-spline interpolation to achieve high-order smoothing, providing more accurate data support for subsequent model identification. A new Block-Hankel matrix is ​​constructed, and kernel SVD analysis is applied to effectively capture the dynamic characteristics of the system. This study adopts the SINDy method to identify reduced-order dynamic models, so that complex nonlinear multi-scale systems can be accurately modeled. The reduced-order model obtained by the present invention can guide actual industrial control.

[0123] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any way. Any simple modification, change and equivalent structural change made to the above embodiment based on the technical essence of the present invention shall still fall within the scope of protection of the technical solution of the present invention.

Claims

1. A mechanistic modeling method for nonlinear, multi-scale dynamic systems based on Koopman embedding theory, applied to a continuous stirred tank reactor system, characterized in that: This method effectively separates multi-timescale information through a designed adaptive sampling strategy and utilizes the nonlinear modeling capabilities of Koopman operator theory to generate a generalizable and interpretable reduced-order model from the observed time series data. The method includes the following steps: S1, obtaining system observation data; the system observation data includes input variables and output variables, and the input variables include the reactor liquid volume V r , inlet flow rate F, inlet concentration of reactant A C A 0. Inlet temperature of reactant A T A 0. Jacket volume V j , Jacket circulating fluid flow F j , jacket fluid inlet temperature T j 0. Density of the reaction liquid ρ m , heat capacity of the reaction liquid c p,m , the density of the heat transfer fluid ρ j , heat capacity of heat transfer fluid c p,j , reaction enthalpy change ΔH r , heat transfer coefficient U, heat transfer area A r , gas constant R, pre-exponential factor k0, activation energy E of the reaction, reactor flow F r , and the output variable is the concentration C of reactant A A , reactor temperature T r and jacket temperature T j ; S2, performs adaptive sampling according to the dynamic changes of data to effectively capture multi-scale information; S3, the sampled data {t i ,x i } Apply cubic B-spline interpolation for local smoothing; S4. Analyze the dynamic characteristics of the system, construct a Block-Hankel matrix for the interpolated time series data, and use its high-dimensional structure to extract the spatiotemporal characteristics of the system; including: by introducing a window length L, construct a new Block-Hankel matrix that is different from the traditional standard embedding matrix The Block-Hankel matrix is the Hankel matrix Vertical connection formation; the Block-Hankel matrix It is not only used to capture the dynamic characteristics of each time series, but also preserves the interdependence between different time series; S5. Decompose the constructed Block-Hankel matrix using kernel singular value decomposition (Kernel SVD) to extract the main dynamic modes and characteristic subspaces of the system; including: using Gaussian kernel function Map the Block-Hankel matrix to a high-dimensional feature space: in and are any two columns of the Block-Hankel matrix; Kernel matrix K pq Perform SVD decomposition: K=UΣV T Then, after KSVD decomposition, the eigenvector matrix V is obtained, which consists of a set of principal singular vectors in the kernel space and is used to represent the principal dynamic mode and the eigensubspace; S6. Combining the extracted low-dimensional feature subspace with the enhanced SINDy method to construct a sparse dynamic model, which is used to obtain an approximate representation of the Koopman operator that characterizes the incremental value of the next time point of the current attribute data by calculating the reduced-order model, and superimposing the value of the previous time point on it as the predicted value of the future state or dynamic characteristics of the current attribute data; the current attribute data is the concentration C of the reactant A. A , reactor temperature T r and jacket temperature T j ; Step 7: Use the obtained reduced-order mechanism model to analyze the future state changes of system data information, thereby achieving precise control.

2. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The observation data is actually collected long-term historical observation information, or time series information of the system evolving over time simulated in a simulation environment under a variety of different initial conditions.

3. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The adaptive sampling is to dynamically adjust the sampling rate within a range according to the change rate of the data values ​​of the continuous sampling points of the observed data, and obtain the long time series sampling point data of different time scales {t i ,x i }.

4. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The cubic B-spline interpolation includes: for a given data point {t i ,x i }(i=0,1,...,n), by recursively defining the B-spline basis function N i (t), we get a piecewise cubic B-spline polynomial Use its smooth interpolation data point t i , which is used to maintain the smoothness and stability of the curve while ensuring the interpolation accuracy.

5. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 4, characterized in that: The recursive definition of B-spline basis function N i (t) specifically include: where t i are the elements of the knot vector. For cubic B-spline k=3, the above recursive process produces cubic polynomial B-spline basis functions; The interpolation function S(t) is a linear combination of basis functions and is expressed as: where c i is the control point coefficient to be determined, which is solved by introducing boundary conditions; To ensure that the interpolation curve S(t) passes through the given data point x i , the following conditions must be met: S(t i )=x i ,for i=0,1,...,n. Therefore, the first-order derivative of the cubic B-spline basis function is expressed as:

6. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The Hankel matrix for: Where K = N-L+1, N is the total length of the time series.

7. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The SINDy algorithm specifically includes: Construct a function library Θ(V) that encapsulates the eigenvector matrix V using one of multiple potential dynamical function forms; Where m is the number of time points and p is the number of functions in the library; Obtain a reduced-order model of the system through sparse regression: Reduced-order model The calculation result of approximately represents the Koopman operator of the incremental value of the next time point of the current attribute data.

8. The mechanism modeling method for nonlinear, multi-scale dynamic systems according to claim 1, characterized in that: The dynamic function forms include polynomials and trigonometric functions.

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