Unsteady aerodynamic reduced-order model construction method based on KAN network and application

Abstract

The invention belongs to the technical field of aeroelastic design and analysis of aircrafts, and particularly relates to an unsteady aerodynamic reduced-order model construction method and application based on a KAN network, and the method comprises the steps: firstly obtaining an aerodynamic response sample sequence of an aircraft in a limit cycle oscillation state; designing and filtering Gaussian white noise as an excitation signal, and obtaining high-precision aerodynamic data through fluid dynamics simulation; utilizing a linear subspace projection method to identify and obtain a linear state space model; a state estimation value of the linear model is used as input, a prediction residual error of the linear model is used as output, a KAN network is trained, and nonlinear mapping learned by the network is converted into an explicit mathematical expression through symbol regression; and finally, combining the linear state space model with an explicit nonlinear term to obtain a physically interpretable white-box aerodynamic force reduced-order model. According to the method, the defects that a traditional CFD method is high in calculation cost and has a neural network black box problem are overcome, and an effective tool is provided for aeroelastic mechanism analysis and optimization design.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aircraft aeroelastic design and analysis, and particularly relates to a non-steady aerodynamic force reduced-order model construction method based on a KAN network and application. BACKGROUND

[0002] In aircraft design and analysis, non-steady aerodynamic force modeling is a key link for aeroelastic stability analysis, flutter prediction and limit cycle oscillation (LCO) research. Although the traditional computational fluid dynamics (CFD) method can provide high-precision flow field simulation, the calculation cost is high, and it is difficult to meet the real-time simulation and multi-parameter optimization requirements. Therefore, developing an efficient and high-precision reduced-order model has become a research hotspot in the field of aeroelasticity, aiming to retain the main dynamic characteristics of the system through a simplified mathematical model, and significantly improve the calculation efficiency.

[0003] At present, the non-steady aerodynamic force reduced-order model is mainly divided into a feature extraction-based method and a system identification-based method. The feature extraction method such as proper orthogonal decomposition and dynamic mode decomposition has a complex model structure and a fuzzy physical meaning of parameters, and belongs to a typical "gray box model". The system identification method establishes a parameterized model using input and output data, including an autoregressive moving average model, a Volterra series model and a neural network model. Among them, the Volterra series model describes the nonlinear system through a kernel function, but the identification of high-order kernel functions is difficult, and the physical meaning of the kernel function is not clear, resulting in poor model interpretability; the neural network model such as a radial basis function neural network and a recursive RBF neural network has strong nonlinear fitting ability, but due to its "black box" characteristics, the model decision mechanism is difficult to understand. For example, the center, width and weight parameters of the RBF neural network lack direct physical correlation, and the training process relies on data driving, and users cannot intuitively judge the specific influence of parameter changes on the aerodynamic response. In the prior art, even if an optimization algorithm or a verification signal mechanism is introduced, the model accuracy can only be improved, and the essential problem of poor interpretability cannot be solved. SUMMARY

[0004] The purpose of the present application is to provide a non-steady aerodynamic force reduced-order model construction method based on a KAN network and application, so as to solve the technical problem that the traditional non-steady aerodynamic force reduced-order model cannot balance the low calculation efficiency and poor model interpretability.

[0005] The present application achieves the above-mentioned purpose through the following technical solutions: In a first aspect, the present application provides a non-steady aerodynamic force reduced-order model construction method based on a KAN network, which comprises the following steps: S1, obtaining a sample sequence of initial generalized displacement and initial generalized aerodynamic force of an aircraft in a limit cycle oscillation state; S2, constructing a filtered white Gaussian noise signal as an excitation input based on the sample sequence, and applying the excitation input to the fluid dynamics model to obtain a corresponding simulated generalized aerodynamic force time sequence; S3, taking the sample sequence of the excitation input and the simulated generalized aerodynamic force time sequence as input-output data pairs, performing system identification by a linear subspace projection method, and establishing a linear state space model of the unsteady aerodynamic force system; S4, taking the state variable estimate value of the linear state space model as input and the residual between the simulated generalized aerodynamic force time sequence and the model prediction value as output, training a KAN network, performing symbolic regression on the trained KAN network, and obtaining an explicit nonlinear term representing nonlinear dynamics; S5, combining the linear state space model and the explicit nonlinear term to obtain an unsteady aerodynamic force reduced-order model.

[0006] Further, the sample sequence is specifically obtained based on fluid dynamics / structure dynamics coupling simulation or physical experiment, and the aircraft is in a limit cycle oscillation state when the sample sequence is obtained; the method further comprises: analyzing the sample sequence to determine a change relationship between the amplitude of the limit cycle oscillation and the reduction speed, and a change relationship between the reduction frequency of the limit cycle oscillation and the reduction speed; based on the change relationship, determining a signal amplitude and a frequency band range for constructing the excitation input.

[0007] Further, step S3 specifically comprises: S31, taking the sample sequence of the excitation input as input data , organizing it into a block Hankel matrix, and dividing it into a past window and a future window , as follows: ; taking the simulated generalized aerodynamic force time sequence as output data , organizing it into a block Hankel matrix, and dividing it into a past window and a future window , as follows: ; wherein subscript p represents the past window, f represents the future window, is a prediction window length, is a data block column number; combining the past and future input-output data into a matrix and a matrix , as follows: , ; Among them, matrix Includes initial condition information, matrix Includes information on future responses; S32, Window to the Future Projection to matrix and the window to the future The projection result is obtained on the spanned space. As shown in the following formula: ; Let the weight matrix Calculate the pseudo-inverse The projection of the future output onto the specified space is calculated using the following formula. : ; S33, Projection results Perform singular value decomposition to determine the effective order and extract the state sequence. As shown in the following formula: ; , ; in, To extend the observability matrix, U is the left singular vector matrix. This is the transpose of the right singular vector matrix. For the front A submatrix composed of left singular vectors The effective order of the system is determined based on the magnitude of the singular values. for The false reversal; S34, Based on the state sequence Input data and output data The system matrices A, B, C, and D are determined by constructing and solving the least squares problem, thus obtaining a linear state-space model.

[0008] Furthermore, in step S34, based on the state sequence... Input data and output data Construct and solve the least squares problem to determine the system matrices A, B, C, D, where: The state sequence matrix Represented as a discrete time series ,in For time indexing, ; The least squares problem is established based on the following state-space model relationship: ; in, for Input data at any time, for Output data at any given time; Apply the above relationship to the time index The matrix equation is constructed as follows: ; in, , This represents the state at the next moment; for The former The first column of the row to the first column A matrix composed of columns To output the dimension; for The former The first column of the row to A matrix composed of columns Input dimension; Solving the matrix equations, we determine the system matrices A, B, C, and D of the linear state-space model, as shown below: ; The linear state-space model is obtained as follows: ; in, for Structural displacement at time t, for aerodynamic force at all times State sequence of Time element.

[0009] Furthermore, step S4 includes: S41. The state sequence of the linear state-space model. As input feature vector The simulation of generalized aerodynamic time series The predicted output of the linear state-space model The residuals between As output target Together they constitute the training set. ; S42. Initialize the KAN network structure and train the KAN network using the training set. S43. Perform symbolic regression on the trained KAN network to obtain the explicit nonlinear term expression.

[0010] Furthermore, in step S42, the activation function of the KAN network is a learnable B-spline basis function, expressed as: Where b(x) is the basis function and spline(x) is the B-spline approximation. and These are learnable parameters.

[0011] Furthermore, in step S43, the symbolic regression specifically involves converting the activation function learned in the KAN network into an explicit nonlinear expression containing polynomial, trigonometric, or piecewise function terms.

[0012] Furthermore, in step S5, the unsteady aerodynamic order reduction model Represented as: ; in, It is a linear state-space model. This is an explicit nonlinear term.

[0013] Secondly, this invention proposes a method for aeroelastic analysis using a reduced-order unsteady aerodynamic model constructed as described above, comprising: The unsteady aerodynamic reduced-order model is coupled with the aircraft structural dynamics equations, and the response of the coupled system is solved by numerical integration to perform aeroelastic stability analysis and limit cycle oscillation characteristic prediction.

[0014] The beneficial effects of this invention are as follows: This invention employs a technical approach combining linear subspace projection and nonlinear fitting using KAN networks. In terms of modeling accuracy, the adaptive basis functions of the KAN network can accurately approximate the nonlinear aerodynamic characteristics under complex flow conditions such as transonic and high angle-of-attack, ensuring high model fidelity. Regarding model interpretability, symbolic regression transforms nonlinear terms into explicit mathematical expressions, overcoming the black-box limitations of traditional neural networks and giving the model white-box characteristics, facilitating the analysis of aeroelastic mechanisms. In terms of computational efficiency, the established reduced-order model shortens the time for a single aeroelastic analysis from hours in traditional CFD methods to minutes, significantly improving computational efficiency. Furthermore, the state-space model established by this method preserves the dynamic characteristics of the system, reliably predicts bifurcation points and limit cycle oscillations, and possesses good generalization ability, making it applicable to highly nonlinear scenarios such as large deformations and flow separation. Attached Figure Description

[0015] Fig. 1A flowchart illustrating a method for constructing a reduced-order unsteady aerodynamic model based on a KAN network, as provided in an embodiment of the present invention. Fig. 2 This is a flowchart illustrating the method for constructing a reduced-order unsteady aerodynamic model proposed in this invention and its application. Fig. 3 This is another flowchart illustrating the method for constructing and applying the unsteady aerodynamic order reduction model proposed in this invention. Detailed Implementation

[0016] The present application will now be described in further detail with reference to the accompanying drawings. It should be noted that the following specific embodiments are only used to further illustrate the present application and should not be construed as limiting the scope of protection of the present application. Those skilled in the art can make some non-essential improvements and adjustments to the present application based on the above application content.

[0017] Example 1 A specific embodiment of the present invention proposes a method for constructing a reduced-order unsteady aerodynamic model based on KAN network, which is applied to aeroelastic problems with strong nonlinear factors such as transonic speed, large deformation, and flow separation. It has good physical interpretability and computational efficiency, and can especially serve the aeroelastic design of transonic flight.

[0018] Please see Figs. 1-3 The method includes the following steps: S1. Data Preparation and Preprocessing: Through coupled computational fluid dynamics (CFD) / computational structural dynamics (CSD) simulations or physical wind tunnel experiments, generalized displacement and generalized aerodynamic response data of the aircraft under limiting cycle oscillation (LCO) conditions are acquired to form an initial sample sequence. This sequence is time-series data, recording the motion and force conditions of the aircraft structure under nonlinear vibration conditions.

[0019] In one implementation, after step S1, the method further includes: analyzing the sample sequence to determine the relationship between the amplitude of the limiting cycle oscillation and the reduction rate, and the relationship between the reduction frequency of the limiting cycle oscillation and the reduction rate; and based on the relationship, determining the signal amplitude and frequency band range used to construct the excitation input.

[0020] S2. CFD Aerodynamic Calculation: Based on the system dynamic characteristics (including the amplitude and frequency characteristics of the limiting cycle oscillation) reflected in the sample sequence obtained in S1, a filtered Gaussian white noise signal with specific frequency band and amplitude parameters is designed as the excitation input. Specifically, by analyzing the relationship between the amplitude of the limiting cycle oscillation (LCO) and the reduction speed in the sample sequence, the average amplitude of the random signal is determined; a Fourier transform is performed on the LCO response to determine its dominant frequency range, thereby setting the frequency band of the filtered Gaussian white noise to ensure that the excitation signal can cover the key dynamics of the system; simultaneously, by analyzing the eigenvectors at the flutter points through linear flutter analysis, the inherent amplitude ratio between different degrees of freedom is obtained, which can be used as the basis for determining the amplitude relationship between multiple input signals. This excitation signal is applied to the fluid dynamics model for unsteady calculations to obtain the corresponding high-precision generalized aerodynamic output data, forming a simulated generalized aerodynamic time series.

[0021] S3. Linear Term Identification Using Subspace Projection: Using the excitation input time series and the simulated generalized aerodynamic force output time series obtained in S2 as input-output data pairs, the system is identified using the linear subspace projection method. This method directly extracts the system's state sequence and dynamic characteristics from the input-output data, establishes a linear state-space model of the unsteady aerodynamic system, and captures the system's main linear dynamics.

[0022] Preferably, step S3 specifically includes: S31. Data Organization and Hankel Matrix Construction The sample sequence of the stimulus input is used as the input data. The organization is a block Hankel matrix, divided into past windows. and the window to the future As shown in the following formula: ; The simulated generalized aerodynamic time series is used as the output data. The organization is a block Hankel matrix, divided into past windows. and the window to the future As shown in the following formula: ; In this context, the subscript 'p' represents the past window, and 'f' represents the future window. To predict the window length, Number of columns in the data block; Combine past and future input and output data into a matrix sum matrix As shown in the following formula: , ; Among them, matrix Includes initial condition information, matrix Includes information on future responses; S32, Orthogonal Projection and State Sequence Extraction Window to the Future Projection to matrix and the window to the future The projection result is obtained on the spanned space. As shown in the following formula: ; Let the weight matrix Calculate the pseudo-inverse The projection of the future output onto the specified space is calculated using the following formula. : ; S33. Singular Value Decomposition (SVD) and Determination of System Order For projection results Perform singular value decomposition to determine the effective order and extract the state sequence. As shown in the following formula: SVD decomposition: ; Extracting the extended observability matrix and state sequence: , ; in, To extend the observability matrix, It is a left singular vector matrix. This is the transpose of the right singular vector matrix. For the front A submatrix composed of left singular vectors The effective order of the system is determined based on the magnitude of the singular values. for The false reversal; S34, Solving the System Matrix Based on state sequences Input data and output data By constructing and solving the least squares problem, the system matrices A, B, C, and D are determined, resulting in a linear state-space model. The state sequence matrix is ​​then... Represented as a discrete time series ,in For time indexing, ; Preferably, in step S34, the least squares problem is established based on the following state-space model relationship: ; in, for Input data at any time, for Output data at any given time; Apply the above relationship to the time index The matrix equation is constructed as follows: ; in, , This represents the state at the next moment; for The former The first column of the row to the first column A matrix composed of columns To output the dimension; for The former The first column of the row to A matrix composed of columns The input dimension.

[0023] By solving the matrix equations, the system matrices A, B, C, and D of the linear state-space model are determined as follows: ; The linear state-space model is obtained as follows: ; in, for Structural displacement at time t, for aerodynamic force at all times State sequence of Time element.

[0024] S4, KAN network nonlinear term fitting: Based on the linear state-space model established in S3, the residual between its predicted value and high-precision CFD data is used as the learning target, and the nonlinear dynamics represented by the residual are fitted using the KAN network.

[0025] This step aims to leverage the superior nonlinear fitting and symbolic regression capabilities of the KAN network by using the dynamic residuals that linear models fail to capture as the learning objective, ultimately obtaining an interpretable white-box nonlinear term model with an explicit mathematical expression. To achieve this goal, the core process is as follows: Fig. 3 As shown, this covers the complete process from data preprocessing to model output: Preferably, step S4 includes: S41. Data Preparation and Normalization The state sequence of the linear state-space model As input feature vector The simulation of generalized aerodynamic time series Predicted output of linear state-space model The residuals between As output target Together they constitute the training set. .

[0026] To improve training stability and convergence speed, the input and output data are standardized to conform to a standard normal distribution with a mean of 0 and a standard deviation of 1. The normalization formula is: ; in, The mean of all samples. This represents the standard deviation of all sample data. The KAN network will then learn this normalized mapping.

[0027] S42 and KAN network structure design, initialization, and network training Initialize the KAN network structure and train the KAN network using the training set.

[0028] Network width and depth: Number of neurons in the network input layer Equals the dimension of the state variables and the number of neurons in the output layer. This is equal to the dimension of the nonlinear residual. The recommended width of the hidden layer is... or When starting the initialization, it is recommended to begin with 1 or 2 layers. The width and depth can be dynamically increased later based on the training results (such as underfitting or overfitting).

[0029] Activation Functions and Basis Functions: The core of the KAN network lies in replacing the fixed activation functions in traditional neural networks with learnable, univariate functions parameterized by B-spline basis functions. The activation function on each connection edge... Represented as: ; Where b(x) is the basis function, and spline(x) is the B-spline approximation. and These are learnable parameters. The order of the B-spline and the number of grid points are key initialization hyperparameters.

[0030] The training objective is to minimize the predictions made by the KAN network. With the true residual target The difference between them, loss function Mean squared error is used, and L1 regularization is added to induce sparsity, thereby improving the simplicity and interpretability of the model: ; in is the regularization coefficient, used to control the degree of sparsity.

[0031] Gradient descent and its variants (such as Adam and L-BFGS) are used to optimize network parameters. During training, the loss curves on both the training and validation sets must be monitored to prevent overfitting and ensure the model has good generalization ability.

[0032] S43, Symbolic Regression and Explicit Expression Extraction Basis function simplification and symbolization: When the training loss converges to a predetermined threshold, such as... Next, the symbolic regression stage begins. This stage aims to transform the complex functions learned in the KAN network, represented by combinations of B-spline basis functions, into concise, human-readable explicit mathematical expressions.

[0033] The first method is manual analysis: you can observe the visualization of the activation function on each edge (such as sine, exponential, polynomial, etc.), manually specify its function form, and determine its coefficients by fitting using the least squares method.

[0034] The second approach is automatic search: it can integrate symbolic mathematics libraries or genetic programming algorithms to automatically search for mathematical expression templates that best match the learned function shape, such as... , and optimize its parameters through training.

[0035] Expression integration and explicit output: The symbolic expressions of all activation functions in the network are combined and simplified according to their network connectivity to obtain the explicit mathematical expression of the entire nonlinear term. For example, a simple possible output form is: ; in, , , For state variables, , , , These are the constant coefficients obtained from the fitting.

[0036] Model saving: Two results are saved in the end: the trained KAN network parameter model file, which is used for high-precision prediction; and the explicit mathematical expression obtained by symbolic regression, which is used for physical analysis, theoretical derivation and engineering design optimization.

[0037] Through the above four sub-steps, the S4 stage completes the core transformation from data to a white-box model, ensuring the dual advantages of this invention in terms of accuracy and interpretability.

[0038] S5. By combining the linear state-space model with explicit nonlinear terms, a reduced-order unsteady aerodynamic model is constructed.

[0039] Preferably, in step S5, the unsteady aerodynamic order reduction model Represented as: ; in, It is a linear state-space model. It is an explicit nonlinear function obtained through symbolic regression using the KAN network, demonstrating the model's interpretability and nonlinear fitting capability.

[0040] Example 2 Another embodiment of the present invention proposes a method for aeroelastic analysis using a reduced-order unsteady aerodynamic model constructed as in Embodiment 1, comprising the following step S6: The unsteady aerodynamic reduced-order model is coupled with the aircraft structural dynamics equations, and the response of the coupled system is solved by numerical integration to perform aeroelastic stability analysis and limit cycle oscillation characteristics prediction.

[0041] For example, consider structural equations (such as a two-dimensional wing model): ; in, For the quality matrix, Here is the stiffness matrix. It is a generalized displacement vector. This represents the generalized aerodynamic force vector. A nonlinear aerodynamic force reduction model based on a KAN network is used to handle the aerodynamic forces on the right-hand side of the equation. The system response is solved by numerical integration (such as the Newmark method) to predict the bifurcation point and LCO characteristics.

[0042] In the above embodiments, implementation can be achieved, in whole or in part, through software, hardware, firmware, or any combination thereof. When implemented in software, it can be implemented, in whole or in part, as a computer program product. The computer program product includes one or more computer instructions. When the computer program instructions are loaded and executed on a computer, all or part of the processes or functions described in the embodiments of the present invention are generated.

[0043] The computer may be a general-purpose computer, a special-purpose computer, a computer network, or other programmable device. The computer instructions may be stored in a computer-readable storage medium or transmitted from one computer-readable storage medium to another. For example, the computer instructions may be transmitted from one website, computer, server, or data center to another via wired (e.g., coaxial cable, fiber optic, digital subscriber line (DSL)) or wireless (e.g., infrared, wireless, microwave, etc.) means. The computer-readable storage medium may be any available medium that a computer can access, or a data storage device such as a server or data center that integrates one or more available media. The available medium may be a magnetic medium (e.g., floppy disk, hard disk, magnetic tape), an optical medium (e.g., DVD), or a semiconductor medium (e.g., a solid-state drive (SSD)).

[0044] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0045] In addition, the functional modules in the various embodiments of this application can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit.

[0046] The above embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application.

Claims

1. A method for constructing a reduced-order unsteady aerodynamic model based on KAN networks, characterized in that, The method includes the following steps: S1. Obtain the sample sequence of the initial generalized displacement and initial generalized aerodynamic force of the aircraft in the limiting cycle oscillation state; S2. Based on the sample sequence, construct a filtered Gaussian white noise signal as an excitation input and apply it to the fluid dynamics model to obtain the corresponding simulated generalized aerodynamic time series. S3. Using the sample sequence of the excitation input and the simulated generalized aerodynamic time series as input-output data pairs, the system is identified by the linear subspace projection method, and a linear state-space model of the unsteady aerodynamic system is established. S4. Using the estimated state variables of the linear state-space model as input, and the residual between the simulated generalized aerodynamic time series and the model prediction as output, train the KAN network, and perform symbolic regression on the trained KAN network to obtain the explicit nonlinear term representing the nonlinear dynamics. S5. Combine the linear state-space model with the explicit nonlinear term to construct a reduced-order unsteady aerodynamic model.

2. The method for constructing a reduced-order unsteady aerodynamic model based on KAN networks according to claim 1, characterized in that, The sample sequence is specifically obtained based on coupled fluid dynamics / structural dynamics simulation or physical experiments, and the spacecraft is in a limiting cycle oscillation state at the time of acquisition; the method also includes: Analyze the sample sequence to determine the relationship between the amplitude of the limiting cycle oscillation and the reduction rate, as well as the relationship between the reduction frequency of the limiting cycle oscillation and the reduction rate. Based on the aforementioned relationship, the signal amplitude and frequency band range used to construct the excitation input are determined.

3. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 1, characterized in that, Step S3 specifically includes: S31. Use the sample sequence of the excitation input as input data. The organization is a block Hankel matrix, divided into past windows. and the window to the future As shown in the following formula: ； The simulated generalized aerodynamic time series is used as output data. The organization is a block Hankel matrix, divided into past windows. and the window to the future As shown in the following formula: ； In this context, the subscript 'p' represents the past window, and 'f' represents the future window. To predict the window length, Number of columns in the data block; Combine past and future input and output data into a matrix sum matrix As shown in the following formula: ， ； Among them, matrix Includes initial condition information, matrix Includes information on future responses; S32, Window to the Future Projection to matrix and the window to the future The projection result is obtained on the spanned space. As shown in the following formula: ； Let the weight matrix Calculate the pseudo-inverse The projection of the future output onto the specified space is calculated using the following formula. : ； S33, Projection results Perform singular value decomposition to determine the effective order and extract the state sequence. As shown in the following formula: ； ， ； in, To extend the observability matrix, U is the left singular vector matrix. This is the transpose of the right singular vector matrix. For the front A submatrix composed of left singular vectors The effective order of the system is determined based on the magnitude of the singular values. for The false reversal; S34, Based on the state sequence Input data and output data The system matrices A, B, C, and D are determined by constructing and solving the least squares problem, thus obtaining a linear state-space model.

4. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 3, characterized in that, In step S34, based on the state sequence... Input data and output data Construct and solve the least squares problem to determine the system matrices A, B, C, D, where: The state sequence matrix Represented as a discrete time series ,in For time indexing, ; The least squares problem is established based on the following state-space model relationship: ； in, for Input data at any time, for Output data at any given time; Apply the above relationship to the time index The matrix equation is constructed as follows: ； in, , This represents the state at the next moment; for The former The first column of the row to the first column A matrix composed of columns To output the dimension; for The former The first column of the row to A matrix composed of columns Input dimension; Solving the matrix equations, we determine the system matrices A, B, C, and D of the linear state-space model, as shown below: ； The linear state-space model is obtained as follows: ； in, for Structural displacement at time t, for aerodynamic force at all times State sequence of Time element.

5. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 4, characterized in that, Step S4 includes: S41. The state sequence of the linear state-space model. As input feature vector The simulation of generalized aerodynamic time series The predicted output of the linear state-space model The residuals between As output target Together they constitute the training set ; S42. Initialize the KAN network structure and train the KAN network using the training set. S43. Perform symbolic regression on the trained KAN network to obtain the explicit nonlinear term expression.

6. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 5, characterized in that, In step S42, the activation function of the KAN network is a learnable B-spline basis function, expressed as: Where b(x) is the basis function and spline(x) is the B-spline approximation. and These are learnable parameters.

7. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 6, characterized in that, In step S43, the symbolic regression specifically involves converting the activation function learned in the KAN network into an explicit nonlinear expression containing polynomial, trigonometric, or piecewise function terms.

8. The method for constructing a reduced-order unsteady aerodynamic model based on a KAN network according to claim 5, characterized in that, In step S5, the unsteady aerodynamic order reduction model Represented as: ； in, It is a linear state-space model. This is an explicit nonlinear term.

9. A method for performing aeroelastic analysis using a reduced-order unsteady aerodynamic model constructed according to any one of claims 1-8, characterized in that, include: The unsteady aerodynamic reduced-order model is coupled with the aircraft structural dynamics equations, and the response of the coupled system is solved by numerical integration to perform aeroelastic stability analysis and limit cycle oscillation characteristic prediction.

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