A symbolic regression-based aerodynamic force modeling transfer learning method
By employing symbolic regression and transfer learning methods, an aerodynamic modeling model is constructed, which solves the problems of high cost of aerodynamic data acquisition and unreliable extrapolation, and achieves high-precision aerodynamic prediction under sparse samples.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- NORTHWESTERN POLYTECHNICAL UNIV
- Filing Date
- 2026-04-20
- Publication Date
- 2026-06-23
AI Technical Summary
Existing aerodynamic data acquisition is costly, requires a large number of samples, and is unreliable in extrapolation, making it difficult to achieve high-precision aerodynamic modeling, especially under sparse sample conditions.
A transfer learning method based on symbolic regression for aerodynamic modeling is adopted. By pre-training the correlation function in the source domain aircraft, multinomial fitting and transfer learning are used, combined with sparse sample fine-tuning, to construct the aerodynamic model.
High-precision prediction of aerodynamic coefficients was achieved under sparse sample conditions, reducing the amount of aerodynamic data required and improving the extrapolation capability of the model.
Smart Images
Figure CN122087963B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of aerodynamic modeling technology in aerodynamics, specifically involving a transfer learning method for aerodynamic modeling based on symbolic regression. Background Technology
[0002] With the continuous advancement of aerospace technology, accurately understanding the aerodynamic characteristics of aircraft has become a crucial foundation for the overall design of advanced aircraft. Currently, commonly used methods for acquiring aerodynamic data mainly include numerical simulation, wind tunnel experiments, and flight tests. Although all three methods play a key role in engineering applications, they all face significant challenges such as high cost and long development cycles when it is necessary to cover a wide range of flight envelopes and obtain a large number of aerodynamic samples.
[0003] Numerical simulation, wind tunnel testing, and flight testing are the three main methods for obtaining the aerodynamic characteristics of aircraft. Numerical simulation, capable of large-scale parallel computation under different states and flight conditions, is one of the most widely used methods. However, high-precision numerical simulation places extremely high demands on the performance of the computing platform; as the complexity of flight conditions increases, the computational load also grows exponentially, resulting in enormous computing power consumption. Wind tunnel testing can obtain aerodynamic data with a high confidence level, but high Mach number and high Reynolds number conditions place stringent requirements on wind tunnel capabilities, leading to huge construction and operating costs and increased difficulty in organizing the tests. In contrast, flight testing yields results closest to the real flight environment, but requires the manufacture of a flyable prototype and the assumption of test risks, resulting in a much higher economic investment than other methods. Therefore, in engineering practice, aerodynamic data often faces a significant contradiction between the large sample size requirement and the high acquisition cost.
[0004] Furthermore, aerodynamic data is inherently high-dimensional and incomplete. The flight state space often consists of multiple variables, such as Mach number, Reynolds number, angle of attack, sideslip angle, and control surface deflection angle. The increase in dimensionality leads to a sharp increase in the amount of data required to cover the complete flight envelope. However, actual flight tests can usually only sample on a limited trajectory, making it difficult to achieve comprehensive coverage of the high-dimensional space. This results in the aerodynamic data often being sparsely distributed, and the prediction process inevitably needs to rely on extrapolation, which further increases the difficulty of modeling.
[0005] In data-driven aerodynamic modeling, methods can be broadly categorized into black-box and white-box models. Currently, achieving high-precision aerodynamic modeling using sparse samples (e.g., less than 20) remains a challenging engineering problem. Black-box models, such as those based on deep neural networks, primarily learn the mapping relationship between state variables and aerodynamic coefficients, and can be combined with strategies like transfer learning to improve predictive performance. They offer advantages such as fast inference speed and convenient computation, but they are highly dependent on training samples and have insufficient extrapolation capabilities. Under sparse sample conditions, they often struggle to guarantee prediction accuracy, typically requiring hundreds or even more samples to obtain reliable results. Conversely, white-box modeling methods, such as symbolic regression and sparse identification, can generate well-defined analytical expressions, possessing good interpretability and extrapolation performance. They have relatively low sample size requirements, typically requiring only a few dozen samples for modeling. However, when dealing with complex aerodynamic problems, these models often require optimization of numerous parameters, making it difficult to achieve the required engineering accuracy even under extremely sparse sample conditions. Summary of the Invention
[0006] To address the aforementioned shortcomings in existing technologies, this invention provides a transfer learning method for aerodynamic modeling based on symbolic regression, which solves the problems of large aerodynamic data requirements and unreliable aerodynamic model extrapolation in existing aerodynamic database construction methods.
[0007] To achieve the aforementioned objectives, the present invention employs the following technical solution: a transfer learning method for aerodynamic modeling based on symbolic regression, comprising the following steps:
[0008] S1: Collect aerodynamic data of the source domain aircraft under various flow conditions, including flow state variables and aerodynamic coefficients;
[0009] S2: In the pre-training phase, based on the flow state variables, combined with symbolic regression and shape parameter optimization methods, the correlation function expression of the aerodynamic coefficients of the source domain aircraft is constructed;
[0010] S3: Acquire sparse sample aerodynamic data of the target domain aircraft;
[0011] S4: In the fine-tuning stage, the correlation function parameters are fine-tuned and aerodynamic modeling is performed based on sparse samples of the target domain. The optimal parameters of the correlation function are evaluated, and the optimal correlation function is used to perform polynomial fitting on the sparse sample aerodynamic data to achieve aerodynamic modeling transfer learning based on symbolic regression.
[0012] Furthermore, the flow state variables include the incoming Mach number, Reynolds number, angle of attack, sideslip angle, and rudder deflection angle;
[0013] The aerodynamic coefficients are obtained based on numerical simulation or experiments.
[0014] Furthermore, step S2 includes the following sub-steps:
[0015] S21: Using the flow state variables as inputs to symbolic regression and the aerodynamic coefficients as the correlated physical quantities, symbolic regression is used to generate candidate expressions for the correlation function. ,in, For flow state variables, Represents the coefficients in the correlation function expression;
[0016] S22: For the candidate expressions of the association function generated by symbolic regression, optimize their constant parameters individually on the dataset of each shape. So that the polynomial of the correlation function can be passed. The aerodynamic force of each configuration is fitted with the smallest error;
[0017] In fixed Under the condition, coefficient Calculated using least squares:
[0018]
[0019] in, For the outer function, For inner function, For polynomial coefficients, Let be the degree of the terms in the polynomial. Let be the highest degree of the polynomial. for The vector formed For the first The configuration number Aerodynamic coefficients of a single aerodynamic sample For the first The configuration number The flow state variables of an aerodynamic sample These are the adjustable coefficients in the correlation function. For configuration The number of samples;
[0020] S23: Optimization This makes the aerodynamic model in the first The normalized mean absolute error is minimized for each configuration. The optimization algorithm can be either gradient optimization or non-gradient optimization:
[0021]
[0022]
[0023] in, These are the optimized constant parameters. For the first Normalized mean absolute error on each configuration;
[0024] S24: Use the average error of each configuration as the total loss function of the association function expression, use the total loss function value to evaluate the fitness of each candidate expression, and treat each candidate expression as an individual to perform genetic, mutation, crossover and recombination operations to update the individual ranking;
[0025] S25: Repeat steps S21 to S24 until the termination condition set by the algorithm is met, and select the correlation function expression of the aerodynamic coefficient based on the fitness of each candidate expression.
[0026] Furthermore, in S3, sparse sample aerodynamic data acquisition of the target domain aircraft specifically involves: acquiring multiple incoming flow state samples within the flight envelope of the target domain aircraft to obtain a sparse sample aerodynamic dataset.
[0027] Furthermore, step S4 includes the following sub-steps:
[0028] S41: Perform multiple random repeated samplings on the sparse sample aerodynamic dataset, and each time split the dataset to generate a training set and a test set;
[0029] S42: Optimize the coefficients of the correlation function for the training set. So that the polynomial of the correlation function can be passed. The aerodynamic force fitting this configuration has the smallest mean square error, thus obtaining the first... Each member model and its mean squared error on the test set:
[0030] The membership model is represented as:
[0031]
[0032] in, For the first Member model, For the first Parameters of each member model;
[0033] S43: Select the member model parameters that minimize the mean square error of the test set. As the final parameters of the aerodynamic model, the model is refitted using a sparse sample aerodynamic dataset to obtain the final aerodynamic model:
[0034]
[0035] in, For the final aerodynamic model, It is a polynomial of order K.
[0036] The beneficial effects of this invention are:
[0037] (1) This invention integrates the idea of transfer learning into the learning of correlation functions. By reserving a fine-tuning interface (adjustable coefficients in the correlation function expression) when training the model, a correlation function framework for shape generalization is obtained by pre-training multiple source domain shape aerodynamic data, so that the extracted aerodynamic correlation function has shape generalization ability.
[0038] (2) The correlation function proposed in this invention can perform nonlinear dimensionality reduction on the incoming flow state variables so that the aerodynamic coefficients can be expressed in polynomial form of the correlation function. Correlating the force coefficients of the new shape within the dimensionality-reduced correlation function space can effectively reduce the aerodynamic samples required for aerodynamic modeling of the new shape.
[0039] (3) Since the correlation function obtained by pre-training already contains global aerodynamic distribution information, when the present invention uses sparse samples of new shapes to fine-tune the parameters of the correlation function and model aerodynamic forces, it can reliably extrapolate aerodynamic data to other states outside the sample area by using data within a smaller range of incoming flow states. Attached Figure Description
[0040] Figure 1 The flowchart of the aerodynamic modeling transfer learning method based on symbolic regression provided by this invention is shown.
[0041] Figure 2 The normal force coefficients of HB-2 and the cone before association provided by this invention vary with Ma and Distribution map.
[0042] Figure 3 The correlation diagram of the correlation function of HB-2 and the cone provided by the present invention on the normal force coefficient is shown.
[0043] Figure 4 The image shows the prediction effect of the polynomial model of the normal force coefficient with respect to the correlation function established based on the sparse sample data of the double ellipsoid shape provided by this invention. Detailed Implementation
[0044] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.
[0045] like Figure 1 As shown, a transfer learning method for aerodynamic modeling based on symbolic regression includes the following steps:
[0046] S1: Collect aerodynamic data of the source domain aircraft under various flow conditions, including flow state variables and aerodynamic coefficients;
[0047] Numerical simulations or experiments are conducted on the source-domain aircraft under various flow conditions, and corresponding aerodynamic data are collected; wherein, the flow state variables include the incoming Mach number. Reynolds number Angle of attack Sideslip angle and rudder deflection The aerodynamic coefficients are obtained based on numerical simulation or experiments. The aerodynamic database obtained from sampling is used... Indicates subscript Indicates the first Each configuration, subscript Indicates the first The first pneumatic sample, the... Each configuration has One sample.
[0048] S2: In the pre-training phase, based on the flow state variables, combined with symbolic regression and shape parameter optimization methods, the correlation function expression of the aerodynamic coefficients of the source domain aircraft is constructed;
[0049] The definition of an association function is:
[0050] Given output y and input X, if the output can be expressed as a composite function of the input, that is... Then the inner function It is the correlation function of y, and its outer function g(S) can be fitted by a polynomial function. The coefficient c of the correlation function can vary with different aerodynamic shapes, thus enabling the expression to adaptively transfer and characterize the aerodynamic characteristics of different shapes.
[0051] S2 includes the following steps:
[0052] S21: Transfer flow state variables As input to the symbolic regression, the aerodynamic coefficients As the associated physical quantity, candidate expressions for the correlation function are generated using symbolic regression. ,in, For flow state variables, Represents the coefficients in the correlation function expression;
[0053] S22: For the candidate expressions of the association function generated by symbolic regression, in each shape of the dataset... Optimize its constant parameters separately. So that the polynomial of the correlation function can be passed. The aerodynamic force of each configuration is fitted with the smallest error;
[0054] In fixed Under the condition, coefficient Calculated using least squares:
[0055]
[0056] in, For the outer function, For inner function, For polynomial coefficients, Let be the degree of the terms in the polynomial. Let be the highest degree of the polynomial. for The vector formed For the first The configuration number Aerodynamic coefficients of a single aerodynamic sample For the first The configuration number The flow state variables of an aerodynamic sample These are the adjustable coefficients in the correlation function. For configuration The number of samples;
[0057] S23: Optimization This makes the aerodynamic model in the first The normalized mean absolute error is minimized for each configuration. The optimization algorithm can be either gradient optimization or non-gradient optimization:
[0058]
[0059] The loss for a given shape is calculated using the Normalized Mean Absolute Error (NMAE):
[0060]
[0061] in, These are the optimized constant parameters. For the first Normalized mean absolute error on each configuration;
[0062] S24: The average error of each configuration (N configurations in total) As the total loss function of the association function expression, the fitness of each candidate expression is evaluated using the total loss function value, and each candidate expression is treated as an individual for genetic, mutation, crossover and recombination operations to update the individual ranking;
[0063] S25: Repeat steps S21 to S24 until the termination condition set by the algorithm is met, and select the correlation function expression of the aerodynamic coefficient based on the fitness of each candidate expression.
[0064] S3: Acquisition of sparse sample aerodynamic data of target domain aircraft ,in, For the first The flow state variables of an aerodynamic sample Let be the aerodynamic coefficient of the j-th sample of the new shape;
[0065] In step S3, sparse sample aerodynamic data acquisition of the target domain aircraft is specifically performed by acquiring multiple incoming flow state samples in the flight envelope of the target domain aircraft to obtain a sparse sample aerodynamic dataset.
[0066] S4: In the fine-tuning stage, the correlation function parameters are fine-tuned and aerodynamic modeling is performed based on sparse samples of the target domain. The optimal parameters of the correlation function are evaluated, and the optimal correlation function is used to perform polynomial fitting on the sparse sample aerodynamic data to achieve aerodynamic modeling transfer learning based on symbolic regression.
[0067] S4 includes the following steps:
[0068] S41: Perform multiple random repeated samplings on the sparse sample aerodynamic dataset, and each time split the dataset to generate a training set and a test set;
[0069] S42: Optimize the coefficients of the correlation function for the training set. So that the polynomial of the correlation function can be passed. The aerodynamic force fitting this configuration has the smallest mean square error, thus obtaining the first... Each member model and its mean squared error on the test set:
[0070] The membership model is represented as:
[0071]
[0072] in, For the first Member model, For the first Parameters of each member model;
[0073] S43: Select the member model parameters that minimize the mean square error of the test set. As the final parameters of the aerodynamic model, the model is refitted using a sparse sample aerodynamic dataset to obtain the final aerodynamic model:
[0074]
[0075] in, For the final aerodynamic model, It is a polynomial of order K.
[0076] In one embodiment of the present invention, a correlation function is extracted for the normal force coefficients of HB-2 and the conical shape, and this correlation function is transferred to the double ellipsoid shape. The parameters of the correlation function are then fine-tuned using sparse aerodynamic samples of the double ellipsoid to establish a normal force coefficient model for the double ellipsoid. Specifically, the method is as follows:
[0077] Step 1: Numerical calculations of the aerodynamic forces of HB-2 and the pointed cone shape are performed using the CFD method. The calculation ranges for each state variable are as follows: Mach number range is... Angle of attack range is The Reynolds number range is The sideslip angle and rudder deflection angle are both 0. The normal force coefficients of HB-2 and the cone are... Follow and The distribution pattern is as follows Figure 2 As shown. Let the incoming flow state variable be... .
[0078] Step 2: Based on the symbolic regression method and shape parameter optimization method, the normal force coefficient is extracted using the aerodynamic data of the source domain shape HB-2 and the cone. correlation function Association function The expression is , where the coefficient It changes in response to changes in shape. The extracted correlation function is used. For normal force coefficient Perform third-order polynomial fitting modeling, and the modeling results are as follows: Figure 3 As shown.
[0079] Step 3: Select 16 different state samples The sampling state range is: Mach number range is Angle of attack range is The Reynolds number range is The aerodynamic forces of the double ellipsoidal shape were numerically calculated using CFD methods to obtain the normal force coefficients corresponding to these samples. .
[0080] Step 4: Randomly sample 13 data points from the 16 data points 30 times each time, using the 13 samples in each sample as the training set and the remaining 3 samples as the test set. Use the training set to analyze the normal force correlation function. coefficients in Fine-tuning was performed to minimize the mean square error of the correlation function fitting the force coefficients. The BFGS algorithm was used for optimization. The result was obtained... The member model is ,in It is a third-order polynomial. Obtain the member model coefficients that minimize the mean squared error of the test set. , as the coefficient of the correlation function. Using A third-order polynomial fit is performed on the force coefficients of all sparse sample data to obtain the expression for the corresponding normal force coefficient. The correlation function is fine-tuned using sparse samples of a double ellipsoidal shape, and the resulting aerodynamic model for the normal force coefficient is as follows: Figure 4As shown in the figure, the prediction results are displayed on the training set and the test set.
[0081] Using the established double-ellipsoidal normal force coefficient model expression, the relative prediction error of the normal force coefficients for the test set is 3.60%, while the Mach number range of the test set is... Angle of attack range is The Reynolds number range is The range of Mach numbers in the test set is much wider than that in the training samples, demonstrating that this method has a strong ability to extrapolate aerodynamic characteristics.
[0082] Based on the correlation function obtained through pre-training in the above embodiments, and by fine-tuning the correlation function coefficients and modeling aerodynamic coefficients using sparse samples of the new shape, sparse reconstruction of the aerodynamic model of the new shape can be achieved. Since the correlation function already contains the aerodynamic physical distribution characteristics obtained during pre-training, it has excellent state extrapolation capability on the new shape.
[0083] 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 the invention.
Claims
1. A transfer learning method for aerodynamic modeling based on symbolic regression, characterized in that, Includes the following steps: S1: Collect aerodynamic data of the source domain aircraft under various flow conditions, including flow state variables and aerodynamic coefficients; S2: In the pre-training phase, based on the flow state variables, combined with symbolic regression and shape parameter optimization methods, the correlation function expression of the aerodynamic coefficients of the source domain aircraft is constructed; S2 includes the following steps: S21: Using the flow state variables as inputs to symbolic regression and the aerodynamic coefficients as the correlated physical quantities, symbolic regression is used to generate candidate expressions for the correlation function. ,in, For flow state variables, Represents the coefficients in the correlation function expression; S22: For the candidate expressions of the association function generated by symbolic regression, optimize their constant parameters individually on the dataset of each shape. So that the polynomial of the correlation function can be passed. The aerodynamic force of each configuration is fitted with the smallest error; In fixed Under the condition, coefficient Calculated using least squares: ； in, For the outer function, For inner function, For polynomial coefficients, Let be the degree of the terms in the polynomial. Let be the highest degree of the polynomial. for The vector formed For the first The configuration number Aerodynamic coefficients of a single aerodynamic sample For the first The configuration number The flow state variables of an aerodynamic sample These are the adjustable coefficients in the correlation function. For configuration The number of samples; S23: Optimization This makes the aerodynamic model in the first The normalized mean absolute error is minimized for each configuration. The optimization algorithm can be either gradient optimization or non-gradient optimization: ； ； in, These are the optimized constant parameters. For the first Normalized mean absolute error on each configuration; S24: Use the average error of each configuration as the total loss function of the association function expression, use the total loss function value to evaluate the fitness of each candidate expression, and treat each candidate expression as an individual to perform genetic, mutation, crossover and recombination operations to update the individual ranking; S25: Repeat steps S21 to S24 until the termination condition set by the algorithm is met, and select the correlation function expression of the aerodynamic coefficient based on the fitness of each candidate expression. S3: Acquire sparse sample aerodynamic data of the target domain aircraft; S4: In the fine-tuning stage, the correlation function parameters are fine-tuned and aerodynamic modeling is performed based on sparse samples of the target domain. The optimal parameters of the correlation function are evaluated, and the optimal correlation function is used to perform polynomial fitting on the sparse sample aerodynamic data to achieve aerodynamic modeling transfer learning based on symbolic regression.
2. The aerodynamic modeling transfer learning method based on symbolic regression according to claim 1, characterized in that, The flow state variables include the incoming Mach number, Reynolds number, angle of attack, sideslip angle, and rudder deflection angle; The aerodynamic coefficients are obtained based on numerical simulation or experiments.
3. The aerodynamic modeling transfer learning method based on symbolic regression according to claim 1, characterized in that, In step S3, sparse sample aerodynamic data acquisition of the target domain aircraft is specifically performed by acquiring multiple incoming flow state samples in the flight envelope of the target domain aircraft to obtain a sparse sample aerodynamic dataset.
4. The aerodynamic modeling transfer learning method based on symbolic regression according to claim 3, characterized in that, S4 includes the following steps: S41: Perform multiple random repeated samplings on the sparse sample aerodynamic dataset, and each time split the dataset to generate a training set and a test set; S42: Optimize the coefficients of the correlation function for the training set. So that the polynomial of the correlation function can be passed. The aerodynamic force fitting this configuration has the smallest mean square error, thus obtaining the first... Each member model and its mean squared error on the test set; The membership model is represented as: ； in, For the first Member model, For the first Parameters of each member model These are the polynomial coefficients; S43: Select the member model parameters that minimize the mean square error of the test set. As the final parameters of the aerodynamic model, the model is refitted using a sparse sample aerodynamic dataset to obtain the final aerodynamic model: ； in, For the final aerodynamic model, It is a polynomial of order K.