An electric propulsion digital twin simulation data reduction method

Abstract

The application relates to the technical field of electric propulsion digital twin simulation, and discloses an electric propulsion digital twin simulation data reduction method, which comprises the following steps: acquiring simulation data of electric propulsion plasma; based on the simulation data, snapshot data is extracted in a grid unit of a calculation domain; singular value decomposition is performed on the snapshot data, a reduced orthogonal basis is established, and reduced data representation is obtained through projection; based on a plasma fluid control equation, a semi-discrete ordinary differential algebraic equation group is constructed, and a coefficient matrix of a reduced model is determined through least square fitting; real-time simulation of an electric propulsion digital twin system is performed by using the reduced model, and field data is output. According to the application, the calculation precision and the simulation result obtained by PIC / MCC simulation are in the same order of magnitude, the calculation time reaches the level of seconds, and the real-time simulation demand can be met.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of electric propulsion digital twin simulation, and particularly relates to an electric propulsion digital twin simulation data reduction method. BACKGROUND

[0002] In the electric propulsion digital twin fine modeling and simulation, the internal complex electromagnetic field simulation mainly applies high-precision digital simulation methods such as PIC / MCC, and the calculation is time-consuming and the data volume is large, which is difficult to be directly integrated into the electric propulsion real-time simulation and monitoring system. At present, there is no technical scheme to meet the real-time electromagnetic field online simulation requirement in the electric propulsion digital twin system. SUMMARY

[0003] In view of the above problems in the prior art, the present application provides an electric propulsion digital twin simulation data reduction method to replace high-precision digital simulation such as PIC / MCC, and to meet the time and precision requirements of the electric propulsion electromagnetic simulation in the electric propulsion digital twin system.

[0004] In order to achieve the above-mentioned application purposes, the technical scheme adopted by the present application is as follows:

[0005] An electric propulsion digital twin simulation data reduction method comprises the following steps:

[0006] Obtaining simulation data of electric propulsion plasma, the simulation data is generated by a particle cloud grid and Monte Carlo coupling method, including sample data at multiple time points, and the sample data at least includes electric field data on grid nodes, positions and speeds of plasma particles;

[0007] Based on the simulation data, snapshot data is extracted in the grid elements of the calculation domain, and the snapshot data includes the number of particles, average speed, equivalent temperature and electric field information in each grid element;

[0008] The snapshot data is singular value decomposed, an orthogonal basis is established, and a reduced data representation is obtained by projection;

[0009] Based on the plasma fluid control equation, a semi-discrete ordinary differential algebraic equation set is constructed, and the coefficient matrix of the reduced model is determined by least square fitting;

[0010] The reduced model is used for real-time simulation of the electric propulsion digital twin system, and field data is output.

[0011] Further, based on the simulation data, snapshot data is extracted in the grid elements of the calculation domain, including:

[0012] For each grid element, the electric field data on the grid nodes is interpolated to the grid center point by using a full linear interpolation method;

[0013] Use an alternating digital binary tree to quickly search for particles within a grid cell, count the number of particles, and calculate the average velocity and equivalent temperature.

[0014] Calculate auxiliary quantities, including momentum, energy, and collision-related parameters.

[0015] Furthermore, collision-related parameters include:

[0016]

[0017]

[0018]

[0019]

[0020]

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] in, The first collision parameter, For the second collision parameter, For the third collision parameter, For the fourth collision parameter, For the fifth collision parameter, The sixth collision parameter, The seventh collision parameter, The eighth collision parameter, This is the ninth collision parameter. This is the tenth collision parameter. Let i be the electronic equivalent temperature of grid cell i at time k. The first ionization energy, For electron-neutral atom collision cross section, Neutral atom number density Boltzmann's constant, For electronic quality, For electron charge, This represents the electron number density.

[0027] Furthermore, the snapshot data includes:

[0028]

[0029]

[0030]

[0031]

[0032]

[0033]

[0034] in, For snapshot data matrix, This is a snapshot data matrix at time k. Let i be the snapshot data matrix of grid cell i at time k. Positive particle number density The momentum density vector of a neutral atom. For positive particles, the momentum density vector is... Let K be the energy density of electrons, and K be the total number of time steps. This represents the total number of grid cells in the computational domain. To enhance the snapshot data matrix, For the enhanced snapshot data matrix at time k, This is the enhanced snapshot data matrix of grid cell i at time k. Let T be the electric field strength at the center point, and T be the transpose.

[0035] Furthermore, singular value decomposition is performed on the snapshot data to establish a reduced-order orthogonal basis, and the reduced-order data representation is obtained through projection, including:

[0036] Perform singular value decomposition on the snapshot data to obtain singular values ​​and orthogonal matrices;

[0037] Based on a preset cumulative energy threshold, the orthogonal basis corresponding to the singular value is selected as the reduced-order orthogonal basis;

[0038] The snapshot data is projected onto the reduced-order orthogonal basis to obtain the reduced-order data representation.

[0039] Furthermore, based on a preset cumulative energy threshold, the orthogonal basis corresponding to the singular values ​​is selected as a reduced-order orthogonal basis, including:

[0040] Based on a preset cumulative energy threshold, the reduced dimension is determined according to the singular value matrix, specifically as follows:

[0041]

[0042] in, To preset the cumulative energy threshold, It is a singular value. To reduce the dimension, The rank of the snapshot data;

[0043] Take the first left singular vector matrix The series yields a reduced-order orthogonal basis, specifically:

[0044]

[0045] in, It is a reduced-order orthogonal basis.

[0046] Further, the snapshot data is projected onto the reduced-order orthogonal basis to obtain a reduced-order data representation, including:

[0047] Projecting the snapshot data onto the reduced-order orthogonal basis specifically involves:

[0048]

[0049]

[0050]

[0051] in, It is a reduced-order coordinate matrix. To enhance the reduced-order coordinate matrix, To enhance the order reduction orthogonal basis;

[0052] Approximate calculation of reduced-order coordinate matrix derivative with respect to time .

[0053] Furthermore, based on the plasma fluid control equations, a system of semi-discrete ordinary differential algebraic equations is constructed, and the coefficient matrix of the reduced-order model is determined by least-squares fitting, including:

[0054] The plasma fluid control equations are discretized into a semi-discrete set of ordinary differential algebraic equations.

[0055] By projecting the field quantities onto the reduced-order orthogonal basis, the order of the semi-discrete ordinary differential algebraic equation system is reduced to obtain the reduced-order model.

[0056] By solving the regularized least squares problem, the coefficient matrix of the reduced-order model is determined to minimize the projection error.

[0057] Furthermore, the order reduction model is specifically as follows:

[0058]

[0059] in, , , , , This is the coefficient matrix of the reduced-order model. The derivative of the reduced-order state vector. For the reduced-order state vector, For the input vector, It is the Kronecker product.

[0060] Furthermore, by solving the regularized least squares problem, the coefficient matrix of the reduced-order model is determined to minimize the projection error, including:

[0061] By solving the regularized least squares problem:

[0062]

[0063] in, To find the minimum value function, For the characteristic matrix, This is the total parameter matrix. For regularization parameters, It is an L2 norm;

[0064] Obtain the coefficient matrix of the reduced-order model , , , , .

[0065] The present invention has the following beneficial effects:

[0066] This invention utilizes high-precision microscopic sample data of plasma obtained from PIC / MCC simulations, combined with macroscopic control equations given by a magnetohydrodynamic model, to present a data-driven plasma order reduction model. Its computational accuracy is on the same order of magnitude as the simulation results obtained from PIC / MCC simulations, and its computation time is in the second range, meeting the requirements for real-time simulations. Attached Figure Description

[0067] Figure 1 A schematic diagram of a method for reducing the order of digital twin simulation data for electric propulsion.

[0068] Figure 2 This is a schematic diagram of a local mesh.

[0069] Figure 3 Here is a flowchart of the PIC / MCC calculation process;

[0070] Figure 4 This is a schematic diagram of grid center interpolation. Detailed Implementation

[0071] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0072] like Figure 1 As shown, an embodiment of the present invention provides a method for reducing the order of electric propulsion digital twin simulation data, comprising the following steps S1 to S5:

[0073] S1. Acquire simulation data of electric propulsion plasma. The simulation data is generated by the particle cloud grid coupled with the Monte Carlo method and includes sample data at multiple time points. The sample data includes at least the electric field data at the grid nodes and the position and velocity of the plasma particles.

[0074] In an optional embodiment of the present invention, the coupling of particle cloud mesh (PIC) with Monte Carlo collision (MCC) is one of the important methods for plasma simulation. Based on first principles, it can capture some non-equilibrium phenomena and dynamic characteristics that are difficult to obtain by magnetohydrodynamic models. However, the computational cost is much larger than that of magnetohydrodynamic models and it cannot be directly used in digital twin electromagnetic field simulation.

[0075] like Figure 2 As shown, in this embodiment, the computational domain is meshed, and the number of mesh cells and mesh nodes are respectively... and .

[0076] like Figure 3 As shown, given the initial and boundary value conditions, simulation is performed using the PIC / MCC method to obtain... Time points Sample data:

[0077]

[0078] in, for The calculated data at each time point includes the electric field data at the grid nodes. The position of plasma particles With speed ,Right now

[0079]

[0080]

[0081]

[0082]

[0083] in, This represents the total number of particles in the plasma (including neutral atoms, positive ions, and electrons).

[0084] S2. Based on the simulation data, extract snapshot data within the grid cells of the computational domain. The snapshot data includes the number of particles, average velocity, equivalent temperature, and electric field information within each grid cell.

[0085] In an optional embodiment of the present invention, step S2, based on the simulation data, extracts snapshot data within the grid cells of the computational domain, including:

[0086] For each grid cell, the electric field data at the grid nodes is interpolated to the grid center point using a fully linear interpolation method.

[0087] Use an alternating digital binary tree to quickly search for particles within a grid cell, count the number of particles, and calculate the average velocity and equivalent temperature.

[0088] Calculate auxiliary quantities, including momentum, energy, and collision-related parameters.

[0089] like Figure 4 As shown, in this embodiment, at each sampling point At any given time, for each unit Perform the following operations, namely

[0090] Electric field at element node using fully linear interpolation method Interpolate to the grid center point to obtain ;

[0091] Using an Alternating Digital Tree (ADT) to quickly search for various particles located within a given grid cell (using superscripts) (Let represent neutral atoms, positive particles, and electrons respectively), count the number of particles in this unit. Calculate the average velocity of various particles Calculate the equivalent temperature using electron kinetic energy ;

[0092] calculate ;

[0093] calculate ;

[0094] Calculate auxiliary quantities ( (The first ionization energy of the working fluid)

[0095]

[0096]

[0097]

[0098]

[0099]

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] in, The first collision parameter, For the second collision parameter, For the third collision parameter, For the fourth collision parameter, For the fifth collision parameter, The sixth collision parameter, The seventh collision parameter, The eighth collision parameter, This is the ninth collision parameter. This is the tenth collision parameter. Let i be the electronic equivalent temperature of grid cell i at time k. The first ionization energy, For electron-neutral atom collision cross section, Neutral atom number density Boltzmann's constant, For electronic quality, For electron charge, This represents the electron number density.

[0106] After performing the above operations at each sampling time, snapshot data at the grid center is obtained, i.e.

[0107]

[0108]

[0109]

[0110]

[0111]

[0112]

[0113] in, For snapshot data matrix, This is a snapshot data matrix at time k. Let i be the snapshot data matrix of grid cell i at time k. Positive particle number density The momentum density vector of a neutral atom. For positive particles, the momentum density vector is... Let K be the energy density of electrons, and K be the total number of time steps. This represents the total number of grid cells in the computational domain. To enhance the snapshot data matrix, For the enhanced snapshot data matrix at time k, This is the enhanced snapshot data matrix of grid cell i at time k. Let T be the electric field strength at the center point, and T be the transpose.

[0114] S3. Perform singular value decomposition on the snapshot data, establish a reduced-order orthogonal basis, and obtain the reduced-order data representation through projection;

[0115] In an optional embodiment of the present invention, step S3 performs singular value decomposition on the snapshot data, establishes a reduced-order orthogonal basis, and obtains the reduced-order data representation through projection, including:

[0116] Perform singular value decomposition on the snapshot data to obtain singular values ​​and orthogonal matrices;

[0117] Based on a preset cumulative energy threshold, the orthogonal basis corresponding to the singular value is selected as the reduced-order orthogonal basis;

[0118] The snapshot data is projected onto the reduced-order orthogonal basis to obtain the reduced-order data representation.

[0119] This embodiment describes the snapshot data matrix. Perform SVD (Singular Value Decomposition), that is

[0120]

[0121] in

[0122]

[0123] The columns are orthogonal. It is an orthogonal array. Diagonal matrix:

[0124]

[0125] satisfy

[0126]

[0127] Given a cumulative energy threshold Find the minimum value that satisfies the following conditions. Obviously there is :

[0128]

[0129] in, To preset the cumulative energy threshold, It is a singular value. To reduce the dimension, The rank of the snapshot data;

[0130] Pick The former Column, i.e.

[0131]

[0132] The columns are the front Proper Orthogonal Decomposition (POD) orthogonal basis.

[0133] Snapshot data Projected onto the POD orthonormal basis, i.e.

[0134]

[0135]

[0136] Approximate calculation The derivative with respect to time, i.e.

[0137]

[0138] For snapshot data matrix Perform SVD decomposition as above, based on the threshold. Select Take the front The orthogonal basis of order POD is ,data exist The projection on is

[0139]

[0140] No calculation is needed here. The derivative with respect to time.

[0141] S4. Based on the plasma fluid control equations, a semi-discrete ordinary differential algebraic equation system is constructed, and the coefficient matrix of the reduced-order model is determined by least squares fitting.

[0142] In an optional embodiment of the present invention, step S4 constructs a semi-discrete ordinary differential algebraic equation system based on the plasma fluid control equations, and determines the coefficient matrix of the reduced-order model by least squares fitting, including:

[0143] The plasma fluid control equations are discretized into a semi-discrete set of ordinary differential algebraic equations.

[0144] By projecting the field quantities onto the reduced-order orthogonal basis, the order of the semi-discrete ordinary differential algebraic equation system is reduced to obtain the reduced-order model.

[0145] By solving the regularized least squares problem, the coefficient matrix of the reduced-order model is determined to minimize the projection error.

[0146] Under the aforementioned assumptions, this embodiment considers the conservation form of the plasma fluid control equations as follows.

[0147]

[0148]

[0149]

[0150]

[0151]

[0152]

[0153]

[0154] in

[0155]

[0156]

[0157]

[0158]

[0159]

[0160]

[0161]

[0162] These are the diffusion coefficient and mobility of electrons, respectively. The ground-state ionization coefficient;

[0163]

[0164]

[0165] Electron thermal conductivity; For the middle The constant; The temperatures of neutral atoms and positive ions are assumed to be the background temperatures.

[0166] Introduce the following notation (each component is defined in step 2).

[0167]

[0168]

[0169]

[0170]

[0171] Based on this notation, the semi-discrete form of the above plasma conservation equations after spatial discretization can be written in the form of the following system of ordinary differential algebraic equations, namely...

[0172]

[0173] in ; For input data (such as boundary conditions or experimental fusion data input, etc.). Represents the Kronecker product;

[0174] ,

[0175]

[0176] for The first unit matrix, i.e., the first unit matrix The equations are differential equations with conserved variables, and then... The system consists of a constrained algebraic equations (the constraints are shown in the auxiliary quantities in step 2).

[0177] field volume Project onto the following POD orthonormal basis (refer to step 3), i.e.

[0178]

[0179] The above system of ordinary differential algebraic equations can be reduced to the following order:

[0180]

[0181] in, , , , , This is the coefficient matrix of the reduced-order model. The derivative of the reduced-order state vector. For the reduced-order state vector, For the input vector, It is the Kronecker product.

[0182]

[0183]

[0184]

[0185]

[0186]

[0187] use The orthogonality of is obviously...

[0188]

[0189] for An identity matrix of order.

[0190] Solving for the coefficient matrix of the reduced-order model above requires knowing the specific form of the aforementioned semi-discrete ordinary differential algebraic equation system. Here, we do not consider the specific discretization format, but determine the coefficient matrix of the reduced-order model through the following optimization problem.

[0191] Search Minimization

[0192]

[0193] matrix ,in As given in step 3, that is

[0194]

[0195]

[0196] The above optimization problem can be rewritten as the following least squares problem.

[0197]

[0198] in

[0199]

[0200]

[0201] Typically, the number of samples satisfy The above least squares problem is overdetermined. To avoid overfitting, a regularization parameter is introduced. The least squares problem is modified as follows:

[0202]

[0203] in, To find the minimum value function, For the characteristic matrix, This is the total parameter matrix. For regularization parameters, It is an L2 norm;

[0204] Solve the introduced least squares problem to determine the matrix. A reduced-order model of the plasma can then be obtained. .

[0205] S5. Use the reduced-order model to perform real-time simulation of the electric propulsion digital twin system and output field data.

[0206] In an optional embodiment of the present invention, step S5 uses the obtained reduced-order model to calculate the value at any given time. Then by This allows us to obtain field data at any given time.

[0207] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0208] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0209] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0210] Specific embodiments have been used to illustrate the principles and implementation methods of this invention. The descriptions of the embodiments above are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

[0211] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for reducing the order of digital twin simulation data for electric propulsion, characterized in that, Includes the following steps: Simulation data of electrically propelled plasma is obtained. The simulation data is generated by a particle cloud grid coupled with the Monte Carlo method and includes sample data at multiple time points. The sample data includes at least the electric field data at the grid nodes and the position and velocity of the plasma particles. Based on the simulation data, snapshot data is extracted within the grid cells of the computational domain. The snapshot data includes the number of particles, average velocity, equivalent temperature, and electric field information within each grid cell. Singular value decomposition is performed on the snapshot data to establish a reduced-order orthogonal basis, and the reduced-order data representation is obtained through projection; wherein, the snapshot data Performing SVD is represented as ; in, , For each column to be orthogonal, It is an orthogonal array. It is a diagonal matrix; Pick The former Column, represented as ;in, The columns are the front orthogonal basis of order POD; Snapshot data Projected onto the POD orthonormal basis, it is represented as: , ; calculate The derivative with respect to time is expressed as ,in, It is a reduced-order coordinate matrix; For snapshot data matrix Perform SVD decomposition as above, based on the threshold. Select ,in Take the front The orthogonal basis of order POD is ; Snapshot Data Matrix exist The projection on is ; To enhance the reduced-order coordinate matrix, To enhance the order reduction orthogonal basis; Based on the plasma fluid control equations, a semi-discrete ordinary differential algebraic equation system is constructed, and the coefficient matrix of the reduced-order model is determined by least squares fitting. The reduced-order model is used for real-time simulation of the electric propulsion digital twin system, outputting field data; wherein, the reduced-order model is: in, , , , , This is the coefficient matrix of the reduced-order model. The derivative of the reduced-order state vector. For the reduced-order state vector, For the input vector, The product of Kronecker; Calculate the value at any time using the obtained reduced-order model. Then by This allows us to obtain field data at any given time. in, For field quantity, the stated Projecting onto the POD orthonormal basis is represented as .

2. The method for reducing the order of electric propulsion digital twin simulation data according to claim 1, characterized in that, Based on the simulation data, snapshot data is extracted within the grid cells of the computational domain, including: For each grid cell, the electric field data at the grid nodes is interpolated to the grid center point using a fully linear interpolation method. Use an alternating digital binary tree to quickly search for particles within a grid cell, count the number of particles, and calculate the average velocity and equivalent temperature. Calculate auxiliary quantities, including momentum, energy, and collision-related parameters.

3. The method for reducing the order of electric propulsion digital twin simulation data according to claim 2, characterized in that, Collision-related parameters include: 、 in, The first collision parameter, For the second collision parameter, For the third collision parameter, For the fourth collision parameter, For the fifth collision parameter, The sixth collision parameter, The seventh collision parameter, The eighth collision parameter, This is the ninth collision parameter. This is the tenth collision parameter. Let i be the electronic equivalent temperature of grid cell i at time k. The first ionization energy, For electron-neutral atom collision cross section, Neutral atom number density Boltzmann's constant, For electronic quality, For electron charge, This represents the electron number density.

4. The method for reducing the order of electric propulsion digital twin simulation data according to claim 3, characterized in that, Snapshot data includes: 、 in, For snapshot data matrix, This is a snapshot data matrix at time k. Let i be the snapshot data matrix of grid cell i at time k. Positive particle number density The momentum density vector of a neutral atom. For positive particles, the momentum density vector is... Let K be the energy density of electrons, and K be the total number of time steps. This represents the total number of grid cells in the computational domain. To enhance the snapshot data matrix, For the enhanced snapshot data matrix at time k, This is the enhanced snapshot data matrix of grid cell i at time k. Let T be the electric field strength at the center point, and T be the transpose.

5. The method for reducing the order of electric propulsion digital twin simulation data according to claim 4, characterized in that, Singular value decomposition is performed on the snapshot data to establish a reduced-order orthogonal basis, and the reduced-order data representation is obtained through projection, including: Perform singular value decomposition on the snapshot data to obtain singular values ​​and orthogonal matrices; Based on a preset cumulative energy threshold, the orthogonal basis corresponding to the singular value is selected as the reduced-order orthogonal basis; The snapshot data is projected onto the reduced-order orthogonal basis to obtain the reduced-order data representation.

6. The method for reducing the order of electric propulsion digital twin simulation data according to claim 5, characterized in that, Based on a preset cumulative energy threshold, orthogonal bases corresponding to singular values ​​are selected as reduced-order orthogonal bases, including: Based on a preset cumulative energy threshold, the reduced dimension is determined according to the singular value matrix, specifically as follows: in, To preset the cumulative energy threshold, It is a singular value. To reduce the dimension, The rank of the snapshot data.

7. The method for reducing the order of electric propulsion digital twin simulation data according to claim 1, characterized in that, Based on the plasma fluid control equations, a system of semi-discrete ordinary differential algebraic equations is constructed, and the coefficient matrix of the reduced-order model is determined by least-squares fitting, including: The plasma fluid control equations are discretized into a semi-discrete set of ordinary differential algebraic equations. By projecting the field quantities onto the reduced-order orthogonal basis, the order of the semi-discrete ordinary differential algebraic equation system is reduced to obtain the reduced-order model. By solving the regularized least squares problem, the coefficient matrix of the reduced-order model is determined to minimize the projection error.

8. The method for reducing the order of electric propulsion digital twin simulation data according to claim 1, characterized in that, By solving the regularized least squares problem, the coefficient matrix of the reduced-order model is determined to minimize the projection error, including: By solving the regularized least squares problem: in, To find the minimum value function, For the characteristic matrix, This is the total parameter matrix. For regularization parameters, It is an L2 norm; Obtain the coefficient matrix of the reduced-order model , , , , .

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