A high-speed aerodynamic derivative analytical modeling method and system based on symbolic regression

Abstract

This invention discloses a high-speed aerodynamic derivative analytical modeling method and system based on symbolic regression, belonging to the interdisciplinary field of aircraft aerodynamic modeling and artificial intelligence. The method involves acquiring and preprocessing the aerodynamic dataset of a high-speed aircraft; defining a syntax for generating candidate symbolic expressions and constructing an expression tree; using a hierarchical symbolic regression framework based on the training set to obtain an analytical model; validating the prediction accuracy of the analytical model on a test set and performing physical consistency analysis; quantifying the uncertainty of the analytical model to obtain an uncertainty estimate of its predictions; and finally outputting an analytical model and its uncertainty estimate in the form of a mathematical expression describing the relationship between aerodynamic derivatives and flight state variables. This invention overcomes the shortcomings of traditional methods in terms of accuracy, generalization, and interpretability, achieving fully automated and efficient modeling from data to analytical models, and is applicable to the control design, stability analysis, and real-time simulation of high-speed aircraft.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of aircraft aerodynamic modeling and artificial intelligence, and in particular to a high-speed aerodynamic derivative analytical modeling method and system based on symbolic regression. Background Technology

[0002] The aerodynamic characteristics of high-speed aircraft exhibit strong nonlinearity, multivariable coupling, and shock wave / viscous disturbance effects, such as the lift coefficient. drag coefficient Pitch moment coefficient Accurate modeling of parameters is a key prerequisite for the design of flight control systems.

[0003] Currently, high-speed aerodynamic modeling mainly relies on the following three methods:

[0004] 1. Physics-based engineering estimation / empirical formula method: Simplified formulas derived using Newtonian flow theory, shock wave expansion wave theory, etc. These methods have clear physical meaning but low accuracy, especially in complex flow regimes and over a wide parameter range where errors are significant, failing to meet the requirements for high-precision control.

[0005] 2. Data-driven polynomial fitting and table lookup: This is the mainstream method in current engineering practice. Data points are obtained through CFD calculations or experiments at discrete state points, and then polynomial surface fitting or direct multidimensional interpolation is used. The main drawbacks of this method are: ① Poor generalization ability: predictions are unreliable in sparse data points or extrapolated regions; ② Lack of physical insight: polynomial coefficients have no clear physical meaning, the model is complex, and it is difficult to use for mechanistic analysis; ③ Curse of dimensionality: as the number of state variables increases, the required data volume grows exponentially, and the burden of table lookup storage and retrieval becomes heavy.

[0006] 3. Deep Learning-Based Black-Box Surrogate Model: This model utilizes the powerful nonlinear fitting capabilities of deep neural networks to establish a mapping from state variables to aerodynamic coefficients. Although DNNs can achieve high accuracy, they are essentially "black-box" models: tens of millions or even hundreds of millions of parameters and their complex nonlinear combinations cannot provide analytical expressions that humans can understand, making it difficult to integrate with flight mechanics theory and hindering frequency domain analysis, stability verification, and online fault diagnosis of control systems.

[0007] Therefore, existing technologies face a "trilemma" in the field of high-speed aerodynamic modeling: it is difficult to simultaneously meet the three core requirements of high accuracy, strong generalization and interpretability. Summary of the Invention

[0008] The purpose of this invention is to provide a high-speed aerodynamic derivative analytical modeling method and system based on symbolic regression to solve the problems mentioned in the background art.

[0009] To achieve the above objectives, this invention provides a method and system for analytical modeling of high-speed aerodynamic derivatives based on symbolic regression, comprising the following steps:

[0010] S1. Obtain the aerodynamic dataset of the high-speed aircraft under a specific configuration. The aerodynamic dataset includes flight state variables and corresponding aerodynamic derivatives. Preprocess the aerodynamic dataset and divide the preprocessed aerodynamic dataset into training set and test set.

[0011] S2. Define the syntax for generating candidate symbolic expressions, and construct an expression tree based on the syntax for generating candidate symbolic expressions. The expression tree includes leaf nodes and internal nodes. Leaf nodes include state variables and trainable constants, and internal nodes contain unary functions and binary functions.

[0012] S3. Based on the training set, a hierarchical symbolic regression framework is used for modeling, including:

[0013] In the first stage, a global structural search of the expression tree is performed based on genetic programming to obtain a set of candidate symbolic expression structures.

[0014] In the second stage, the candidate symbolic expression structures in the candidate symbolic expression structure set are converted into differentiable symbolic neural networks, and the parameters of the symbolic neural networks are optimized based on the training set to obtain the analytical model.

[0015] S4. Perform numerical verification of the analytical model on the test set, obtain the prediction accuracy and conduct comparative analysis; perform physical consistency analysis on the analytical model to verify whether the analytical model conforms to physical laws.

[0016] S5. Quantify the uncertainty of the analytical model to obtain the uncertainty estimate of the analytical model's prediction. The uncertainty estimate is used to indicate the reliability of the analytical model's prediction in different input regions.

[0017] S6. Output the analytical model and the uncertainty estimate of the analytical model. The analytical model is a mathematical expression describing the relationship between the aerodynamic derivative and the flight state variables.

[0018] Preferably, the univariate functions in S2 include sine, cosine, exponential, logarithmic, absolute value, square, and cubic functions; the bivariate functions include addition, subtraction, multiplication, and division.

[0019] Preferably, the specific steps of the first stage of genetic programming in S3 are as follows:

[0020] Generate an initial population containing multiple expression tree individuals;

[0021] The fitness of each individual is evaluated using a fitness function, which includes the initial fitness and a physical constraint penalty term.

[0022] The selection operation is performed based on fitness, and the selected individuals are subjected to crossover and mutation operations to generate the next generation population;

[0023] Iterate until the termination condition is met, and output a set of candidate symbolic expression structures.

[0024] Preferably, the fitness function is calculated as follows:

[0025] The constant leaf nodes in the expression tree are replaced with trainable parameters, and the trainable parameters are optimized on the training set by gradient descent to obtain the data fitting error as the original fitness.

[0026] Construct physical constraint penalty terms based on prior physical knowledge;

[0027] The overall fitness is obtained by weighted summation of the original fitness and the physical constraint penalty term.

[0028] Preferably, the physical constraint penalty terms include symmetry constraints, antisymmetry constraints, derivative sign constraints, and asymptotic behavior constraints.

[0029] Preferably, in the second stage of S3, the network structure of the symbolic neural network is defined by the candidate symbolic expression structure, and the weights of the symbolic neural network are constant terms in the candidate symbolic expression structure;

[0030] Based on the training set, the parameters of the symbolic neural network are optimized using gradient descent. The loss function includes data fitting loss, physical constraint loss, and L2 regularization term.

[0031] Preferably, in S5, uncertainty quantification employs a Bayesian method or a Bootstrap ensemble method to obtain the mean and variance of the analytical model predictions.

[0032] A high-speed aerodynamic derivative analytical modeling system based on symbolic regression, comprising:

[0033] The data acquisition and preprocessing module is used to acquire and preprocess the aerodynamic dataset of the high-speed aircraft.

[0034] The syntax definition and expression tree building module is used to define the syntax for generating candidate symbol expressions and build expression trees;

[0035] The hierarchical symbolic regression module is used to model based on the training set using a hierarchical symbolic regression framework to obtain an analytical model.

[0036] The uncertainty quantification module is used to quantify the uncertainty of the analytical model and obtain an estimate of the uncertainty of the model prediction.

[0037] The output module is used to output the analytical model and uncertainty estimates.

[0038] Preferably, the hierarchical symbolic regression module includes:

[0039] The global structure search unit is used to perform a global structure search on the expression tree based on genetic programming to obtain a set of candidate symbolic expression structures.

[0040] The local parameter optimization unit is used to convert the candidate symbolic expression structures in the candidate symbolic expression structure set into a symbolic neural network, and optimize the parameters of the symbolic neural network based on the training set to obtain an analytical model.

[0041] Therefore, the present invention employs the above-mentioned analytical modeling method and system for high-speed aerodynamic derivatives based on symbolic regression, which has the following beneficial effects:

[0042] (1) Through two-stage optimization, genetic programming is used to explore the global structure and avoid getting trapped in local optima, while gradient descent is used to fine-tune the local parameters. The final analytical model has significantly higher prediction accuracy in the test set and extrapolation region than traditional empirical formulas and polynomial fitting, and the calculation speed is much faster than table lookup interpolation.

[0043] (2) The output is a mathematical expression, rather than a large number of incomprehensible network parameters. Each item has a potential physical meaning, which makes it easier for engineers to understand and analyze the source of aerodynamic characteristics;

[0044] (3) By introducing physical constraints into the loss function, we ensure that the discovered model follows known physical laws, avoid the physical absurdity that may be generated by pure data-driven approaches, and improve the reliability and extrapolation ability of the model.

[0045] (4) Through the Bayesian framework or the Bootstrap method, the model can be aware of its uncertainty and give a low confidence warning when the data is sparse or the operating conditions are extreme, which is crucial for safety-critical high-speed flight control.

[0046] (5) It realizes a fully automated process from raw data to usable analytical model, greatly reducing manual trial and error and reliance on expert experience, and shortening the aerodynamic modeling cycle from several weeks to several hours.

[0047] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0048] Figure 1 This is a flowchart of an analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression, according to the present invention.

[0049] Figure 2 This is a schematic diagram of the hierarchical symbolic regression framework of a high-speed aerodynamic derivative analytical modeling method based on symbolic regression according to the present invention;

[0050] Figure 3 This is a schematic diagram illustrating the conversion between expression trees and symbolic neural networks in a high-speed aerodynamic derivative analytical modeling method based on symbolic regression according to the present invention.

[0051] Figure 4 This is a schematic diagram illustrating how the physical constraint penalty term is introduced into the fitness function in the high-speed aerodynamic derivative analytical modeling method based on symbolic regression according to the present invention.

[0052] Figure 5 This is a comparison diagram of the predicted surface and data points of the high-speed aerodynamic derivative analytical modeling method based on symbolic regression of the present invention, wherein (a) is the distribution diagram of training data points, and (b) is the distribution diagram of the predicted surface and test data points;

[0053] Figure 6 The figure shows the test set prediction accuracy verification results of the high-speed aerodynamic derivative analytical modeling method based on symbolic regression of the present invention, where (a) is a scatter plot comparing the true value and the predicted value, and (b) is a residual distribution plot.

[0054] Figure 7 This is a schematic diagram illustrating the quantification of uncertainty in the analytical model of a high-speed aerodynamic derivative analytical modeling method based on symbolic regression according to the present invention. Detailed Implementation

[0055] The following detailed description of embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely illustrates selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0056] like Figure 1 As shown, this invention provides an analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression, including the following steps S1-S6. In this embodiment, the lift coefficient of a high-speed blunt-nosed body (HB2 model) is used. Regarding the angle of attack and Mach number Taking the modeling as an example, the implementation process of the present invention will be described in detail.

[0057] S1. Obtain the aerodynamic dataset of the high-speed aircraft under a specific configuration. The aerodynamic dataset includes flight state variables and corresponding aerodynamic derivatives. According to high-speed aerodynamic theory, the true physical law of the lift coefficient of the HB2 model can be expressed as:

[0058] ;

[0059] in, The lift coefficient, Angle of attack (radians). Let be the Mach number, take , Within the angle of attack range (Convert to radians), Mach number range Uniform random sampling within Each point is used, and Gaussian noise with a standard deviation of 0.005 is added to obtain noisy observation data. The input feature is then used. Output .

[0060] Analyze the physical meaning of the above expression: First term Corresponding to linear lift at small angles of attack and nonlinear correction at medium angles of attack; the second term This demonstrates the effect of Mach number on lift, when The contribution is positive and increases nonlinearly with increasing angle of attack, which is consistent with the compressibility effect of high-speed flow. The entire expression is concise and each term has a clear physical meaning.

[0061] The aerodynamic dataset is preprocessed, including standardization, to ensure that the input features and output target have zero mean and unit variance. , .

[0062] in, and Input features The mean and standard deviation, and Output target The mean and standard deviation;

[0063] Standardization helps improve the search efficiency and numerical stability of genetic programming. The preprocessed aerodynamic dataset is divided into a training set (200 samples) and a test set (50 samples).

[0064] S2. Define the syntax for generating candidate symbolic expressions, and construct an expression tree based on the syntax. The expression tree includes leaf nodes and internal nodes. Leaf nodes include state variables (such as...). ) and trainable constants (such as The internal nodes contain univariate and bivariate functions; univariate functions include sine, cosine, exponential, logarithmic, absolute value, square, and cubic functions, such as... Etc.; Bivariate functions include addition, subtraction, multiplication, and division, for example... (The division is performed using protected division to avoid division by zero).

[0065] Expression trees are constructed in the form of binary trees, for example This can be represented as the root node being The left and right subtrees are both multiplicative binary trees.

[0066] S3, such as Figure 2 As shown, the hierarchical symbolic regression framework includes a first-stage global structure search and a second-stage local parameter optimization, demonstrating the synergy between genetic programming and symbolic neural networks. Based on the training set, the hierarchical symbolic regression framework is used for modeling, including:

[0067] In the first stage, a global structural search of the expression tree is performed based on genetic programming to obtain a set of candidate symbolic expression structures.

[0068] The specific steps of genetic planning are as follows:

[0069] An initial population containing multiple expression tree individuals is generated; the population size in this embodiment is... Evolutionary algebra .

[0070] The fitness of each individual is evaluated using a fitness function, which includes the initial fitness and a physical constraint penalty term.

[0071] The fitness function is calculated as follows:

[0072] The constant leaf nodes in the expression tree are replaced with trainable parameters, and the trainable parameters are optimized on the training set by gradient descent to obtain the data fitting error as the original fitness.

[0073] The data fitting error is expressed as mean square error. calculate:

[0074] ;

[0075] in, The data fitting error is the mean square error. The number of samples in the training set. For the first The true aerodynamic derivative values ​​of each sample. For the expression tree, the first The predicted value for each sample, These are the trainable constant parameters in the expression tree.

[0076] Physical constraint penalty terms are constructed based on prior physical knowledge; these terms include symmetry constraints, antisymmetry constraints, derivative sign constraints, and asymptotic behavior constraints.

[0077] Symmetry / antisymmetry constraints: For example, for the longitudinal aerodynamic derivative, there is usually... (Opposing view) (Symmetrical). Define constraint violation degree: .

[0078] in, This is a symmetry penalty term used to quantify the degree to which the model violates the symmetry of odd functions. and The expression tree at the angle of attack and The predicted value at that location, It is a set of symmetrical sampling points. The sign coefficient is +1 when the constraint is symmetric and -1 when the constraint is antisymmetric.

[0079] Derivative sign constraint: at the equilibrium point (e.g.) Near a certain distance, the aerodynamic derivative should satisfy a specific sign, for example... This is a necessary condition for static stability. Calculate the numerical derivative and compare it with the expected sign.

[0080] Asymptotic behavior constraint: At large angles of attack, some aerodynamic derivatives may tend to saturate.

[0081] In this embodiment, based on the physical properties of the lift coefficient ( It should be an odd function of the angle of attack: Using symmetry constraints, a symmetry penalty function is defined:

[0082] ;

[0083] in, The number of symmetrical sampling points selected. For the model at the angle of attack The predicted value at that location, For the model at the angle of attack The predicted value at that location, These are positive angle-of-attack samples selected from the training set. This penalty term ensures that the model's predicted values ​​satisfy the odd function property, i.e. .

[0084] Overall fitness calculation: ,

[0085] in, For expression trees Overall adaptability The original fitness (i.e., the data fitting error) ), This is a tradeoff coefficient used to balance data fitting accuracy and physical consistency. This is a physical constraint penalty term. A lower fitness value is better. (Pass / Follow) During the structure search phase, expressions that do not conform to basic physical laws are eliminated.

[0086] In this embodiment, based on the physical characteristics of the lift coefficient, a symmetry constraint is used as a physical constraint penalty term. The original fitness is weighted and summed with the physical constraint penalty term to obtain the comprehensive fitness.

[0087] ;

[0088] in, As a weighting factor, this embodiment takes... fitness The smaller the value, the better the individual. For example... Figure 4 As shown, the physical constraint penalty term is introduced into the fitness function through symmetry constraints.

[0089] The selection operation is performed based on fitness, followed by crossover and mutation operations on the selected individuals to generate the next generation population. The selection operation uses a tournament selection method, randomly selecting several individuals each time, and choosing the individual with the lowest fitness as the parent. The crossover operation randomly swaps the subtrees of the expression trees of two parent individuals to generate a new individual; the mutation operation randomly changes the operator of a node or replaces a subtree. In this embodiment, the crossover probability... Subtree mutation probability Point mutation probability Increase mutation probability .

[0090] Selection: Select parent individuals using roulette or tournament selection based on fitness.

[0091] Crossover: Randomly swap the subtrees of the expression trees of two parent individuals to generate a new individual.

[0092] Mutation: Randomly changes the operator of a node or replaces a subtree.

[0093] Iterative evolution continues until the termination condition is met (the preset number of generations is reached). (or fitness convergence), outputting a set of candidate symbolic expression structures. After the genetic programming run is complete, candidate expressions are collected from all generations, and duplicates are removed using a dictionary to obtain the set of candidate symbolic expression structures.

[0094] In the second stage, the candidate symbolic expression structures in the candidate symbolic expression structure set are converted into differentiable symbolic neural networks. The parameters of the symbolic neural network are then optimized based on the training set to obtain an analytical model, such as... Figure 3 As shown, the network structure of the symbolic neural network is defined by the candidate symbolic expression structure, and the weights of the symbolic neural network are the constant terms in the candidate symbolic expression structure.

[0095] This symbolic neural network is implemented using PyTorch and trained end-to-end using the Adam optimizer. The total loss function includes data fitting loss, physical constraint loss, and a regularization term:

[0096] ;

[0097] in, This is the total loss function value. The mean squared error of the training set. As a symmetry penalty term, symmetric pairs are constructed by randomly selecting positive angle-of-attack samples from the training set during each forward pass. This is an L2 regularization term (which helps keep the coefficients small). The physical constraint loss weighting coefficient is taken in this embodiment. , The regularization weight coefficient is taken in this implementation. .

[0098] The training parameters are a learning rate of 0.01, 1000 iterations, and all training data are used in batches. The training loss and validation loss are output every 200 iterations to monitor convergence. After training, an analytical model is obtained.

[0099] S4. Numerical verification of the analytical model is performed on the test set to obtain the prediction accuracy and conduct comparative analysis. In this embodiment, the root mean square error (RMSE) of the model on the test set is 0.0048, while the RMSE of the traditional fifth-order polynomial fitting is 0.012, indicating that the method in this embodiment has higher accuracy.

[0100] like Figure 5 As shown, where, Figure 5 (a) is a distribution map of training data points, with blue dots representing training samples, showing the training data at angles of attack. and Mach number Distribution within the parameter space; Figure 5 (b) is a distribution diagram of the predicted surface and test data points. The colored surface represents the predicted surface generated by the analytical model discovered by the method of this embodiment, and the red triangles represent test data points. From Figure 5 (b) It can be seen that the predicted surface is smooth and continuous, and the test data points closely fit the predicted surface, indicating that the model accurately captures the lift coefficient as a function of angle of attack. and Mach number The nonlinear variation law.

[0101] like Figure 6 As shown, where, Figure 6 (a) is a scatter plot comparing the actual and predicted values. The horizontal axis represents the actual values ​​and the vertical axis represents the model's predicted values. All points are closely distributed around the diagonal, indicating that the model's predicted values ​​are highly consistent with the actual values. Figure 6 (b) is a residual distribution plot, with the horizontal axis representing the predicted value and the vertical axis representing the residual (actual value minus predicted value). The residuals are randomly scattered around the zero value and have no trend structure, which verifies the high accuracy and error independence of the model.

[0102] Physical consistency analysis is performed on the analytical model to verify whether it conforms to physical laws. This analysis includes sign differentiation and limit analysis to verify whether its monotonicity, convexity, or parity conforms to aerodynamic theory. (Substitution) , The expression for the slope of the lifting line is obtained by sign differentiation:

[0103] ;

[0104] in, Let be a sign function. Analyze its monotonicity; calculate the limiting behavior to verify whether the model conforms to physical expectations at high angles of attack. The analysis results show that the model is physically reasonable, and the behavior of its derivatives is consistent with aerodynamic theory.

[0105] S5. Quantify the uncertainty of the analytical model to obtain the uncertainty estimate of the analytical model's prediction. The uncertainty estimate is used to indicate the reliability of the analytical model's prediction in different input regions.

[0106] This embodiment uses the Bootstrap ensemble method for uncertainty quantification: sampling with replacement from the training set to generate... A Bootstrap sample set The number of Bootstrap samples is [number], and the size of each sample set is the same as the original training set. For each Bootstrap sample set, its parameters are optimized using the same expression structure as in the first stage, resulting in a set of parameters. and the corresponding prediction function , The index of the Bootstrap sample set, for any input Integrated predicted mean for:

[0107] ;

[0108] The standard deviation of the forecast is .

[0109] Standard deviation This reflects the uncertainty in the model's prediction at that point. In areas with dense training data, the standard deviation is smaller; in areas with sparse data or extrapolation, the standard deviation increases, suggesting that the prediction results should be used with caution.

[0110] like Figure 7 As shown, Figure 7This is a schematic diagram illustrating the uncertainty quantification of the analytical model of this invention. The diagram shows a fixed Mach number. The horizontal axis represents the angle of attack. The vertical axis represents the lift coefficient. The blue solid line represents the prediction curves of the analytical model at different angles of attack. The light blue area represents Confidence band (i.e.) The red dots represent test data points with Mach numbers close to 7.5. As can be seen from the graph, the confidence band is located in areas of dense data (such as...). The narrower the area (nearby), the higher the model's predictive reliability in that region; in the boundary region (e.g., ... and The slight widening of the curve (near the input region) indicates increased uncertainty in the model's predictions in that area, suggesting that the predictions should be used with caution. This diagram accurately reflects the model's prediction confidence in different input regions, providing a reliability indicator for engineering applications.

[0111] As another implementation method, uncertainty quantification can also employ Bayesian methods: for a selected optimal symbolic expression structure, variational inference methods are used to infer the posterior distribution of its parameters. ,in For the training dataset; for new input The model's prediction is no longer a point estimate. It is not a distribution Its predicted mean can be calculated. With variance The magnitude of variance directly reflects the uncertainty of the model's prediction at that point. In regions with sparse training data or under extreme conditions, variance increases, thus issuing a warning to the user.

[0112] S6. Output the analytical model and the uncertainty estimate of the analytical model. The analytical model is a mathematical expression describing the relationship between the aerodynamic derivative and the flight state variables.

[0113] The analytical model output in this embodiment is:

[0114] ;

[0115] Simultaneously outputs the prediction standard deviation of each input region. This information is used to guide downstream applications. The final output includes analytical expressions for aerodynamic derivatives, model uncertainty information, physical consistency verification reports, and control design recommendations.

[0116] It can be directly used in the design of control laws for high-speed aircraft. For example, the slope of the lift curve can be obtained through sign differentiation:

[0117] ;

[0118] This expression can be used to calculate the trim angle of attack and dynamic derivative, and then to design a gain scheduling controller. The analytical model has extremely high computational efficiency and is suitable for real-time simulation and hardware-in-the-loop testing.

[0119] A high-speed aerodynamic derivative analytical modeling system based on symbolic regression, comprising:

[0120] The data acquisition and preprocessing module is used to acquire and preprocess the aerodynamic dataset of the high-speed aircraft.

[0121] The syntax definition and expression tree building module is used to define the syntax for generating candidate symbol expressions and build expression trees;

[0122] The hierarchical symbolic regression module is used to model based on the training set using a hierarchical symbolic regression framework to obtain an analytical model.

[0123] The hierarchical symbolic regression module includes:

[0124] The global structure search unit is used to perform a global structure search on the expression tree based on genetic programming to obtain a set of candidate symbolic expression structures.

[0125] The local parameter optimization unit is used to convert the candidate symbolic expression structures in the candidate symbolic expression structure set into a symbolic neural network, and optimize the parameters of the symbolic neural network based on the training set to obtain an analytical model.

[0126] The uncertainty quantification module is used to quantify the uncertainty of the analytical model and obtain an estimate of the uncertainty of the model prediction.

[0127] The output module is used to output the analytical model and uncertainty estimates.

[0128] Therefore, the present invention adopts the above-mentioned analytical modeling method and system for high-speed aerodynamic derivatives based on symbolic regression. By combining the global search of symbolic regression, the local optimization of neural networks and the strong constraints of physical priors, it successfully realizes the automatic mining of concise, accurate and interpretable analytical models from high-speed aerodynamic data, providing a revolutionary tool for aircraft design, control and evaluation.

[0129] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A high-speed aerodynamic derivative analytical modeling method based on symbolic regression, characterized in that, Includes the following steps: S1. Obtain the aerodynamic dataset of the high-speed aircraft under a specific configuration. The aerodynamic dataset includes flight state variables and corresponding aerodynamic derivatives. Preprocess the aerodynamic dataset and divide the preprocessed aerodynamic dataset into training set and test set. S2. Define the syntax for generating candidate symbolic expressions, and construct an expression tree based on the syntax for generating candidate symbolic expressions. The expression tree includes leaf nodes and internal nodes. Leaf nodes include state variables and trainable constants, and internal nodes contain unary functions and binary functions. S3. Based on the training set, a hierarchical symbolic regression framework is used for modeling, including: In the first stage, a global structural search of the expression tree is performed based on genetic programming to obtain a set of candidate symbolic expression structures. In the second stage, the candidate symbolic expression structures in the candidate symbolic expression structure set are converted into differentiable symbolic neural networks, and the parameters of the symbolic neural networks are optimized based on the training set to obtain the analytical model. S4. Perform numerical verification of the analytical model on the test set, obtain the prediction accuracy and conduct comparative analysis; perform physical consistency analysis on the analytical model to verify whether the analytical model conforms to physical laws. S5. Quantify the uncertainty of the analytical model to obtain the uncertainty estimate of the analytical model's prediction. The uncertainty estimate is used to indicate the reliability of the analytical model's prediction in different input regions. S6. Output the analytical model and the uncertainty estimate of the analytical model. The analytical model is a mathematical expression describing the relationship between the aerodynamic derivative and the flight state variables.

2. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 1, characterized in that: The univariate functions in S2 include sine, cosine, exponential, logarithmic, absolute value, square, and cubic functions; the bivariate functions include addition, subtraction, multiplication, and division.

3. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 1, characterized in that, The specific steps of the first stage of genetic programming in S3 are as follows: Generate an initial population containing multiple expression tree individuals; The fitness of each individual is evaluated using a fitness function, which includes the initial fitness and a physical constraint penalty term. The selection operation is performed based on fitness, and the selected individuals are subjected to crossover and mutation operations to generate the next generation population; Iterate until the termination condition is met, and output a set of candidate symbolic expression structures.

4. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 3, characterized in that, The fitness function is calculated as follows: The constant leaf nodes in the expression tree are replaced with trainable parameters, and the trainable parameters are optimized on the training set by gradient descent to obtain the data fitting error as the original fitness. Construct physical constraint penalty terms based on prior physical knowledge; The overall fitness is obtained by weighted summation of the original fitness and the physical constraint penalty term.

5. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 4, characterized in that: Physical constraint penalties include symmetry constraints, antisymmetry constraints, derivative sign constraints, and asymptotic behavior constraints.

6. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 3, characterized in that: In the second stage of S3, the network structure of the symbolic neural network is defined by the candidate symbolic expression structure, and the weights of the symbolic neural network are the constant terms in the candidate symbolic expression structure. Based on the training set, the parameters of the symbolic neural network are optimized using gradient descent. The loss function includes data fitting loss, physical constraint loss, and L2 regularization term.

7. The analytical modeling method for high-speed aerodynamic derivatives based on symbolic regression according to claim 1, characterized in that: In S5, uncertainty quantification employs either the Bayesian method or the Bootstrap ensemble method to obtain the mean and variance of the analytical model predictions.

8. A high-speed aerodynamic derivative analytical modeling system based on symbolic regression, used to implement the high-speed aerodynamic derivative analytical modeling method based on symbolic regression as described in any one of claims 1-7, characterized in that, include: The data acquisition and preprocessing module is used to acquire and preprocess the aerodynamic dataset of the high-speed aircraft. The syntax definition and expression tree building module is used to define the syntax for generating candidate symbol expressions and build expression trees; The hierarchical symbolic regression module is used to model based on the training set using a hierarchical symbolic regression framework to obtain an analytical model. The uncertainty quantification module is used to quantify the uncertainty of the analytical model and obtain an estimate of the uncertainty of the model prediction. The output module is used to output the analytical model and uncertainty estimates.

9. The high-speed aerodynamic derivative analytical modeling system based on symbolic regression according to claim 8, characterized in that, The hierarchical symbolic regression module includes: The global structure search unit is used to perform a global structure search on the expression tree based on genetic programming to obtain a set of candidate symbolic expression structures. The local parameter optimization unit is used to convert the candidate symbolic expression structures in the candidate symbolic expression structure set into a symbolic neural network, and optimize the parameters of the symbolic neural network based on the training set to obtain an analytical model.

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