An interpretable aerodynamic coefficient prediction method and system based on dynamic weighting

Abstract

The application provides an interpretable aerodynamic coefficient prediction method and system based on dynamic weighting. The method comprises: obtaining a flight state parameter feature dataset composed of a plurality of sample points and a corresponding aerodynamic coefficient label dataset, performing normalization processing on the feature dataset to obtain standardized sample features; training a group of heterogeneous machine learning models in parallel based on the standardized sample features and the label dataset, and obtaining unbiased prediction values of each model for each sample point through cross-validation; for each sample point, calculating and fusing three types of sample-level evaluation weights based on the unbiased prediction values of each model corresponding to the sample point to generate dynamic fusion weights of each model for the sample point; for each sample point, performing weighted summation on the unbiased prediction values of each model for the sample point by using the dynamic fusion weights corresponding to the sample point to obtain a final fusion prediction value of the sample point; and summarizing the final fusion prediction values of all sample points to output an aerodynamic coefficient prediction result and a corresponding interpretability analysis report.

Description

Technical Field

[0001] This invention relates to the field of interdisciplinary technology of machine learning and computational fluid dynamics, and in particular to a method and system for predicting interpretable aerodynamic coefficients based on dynamic weighting. Background Technology

[0002] Aerodynamic modeling is a core technology in aerospace, aircraft design, and control system development, primarily addressing the accurate prediction of high-dimensional nonlinear aerodynamic parameters (such as force and moment coefficients) under complex flight conditions. Traditional modeling methods rely on wind tunnel experiments or computational fluid dynamics (CFD) simulations, which, while offering high physical fidelity, suffer from limitations such as high cost, long development cycles, and difficulty in achieving real-time response. In recent years, data-driven machine learning methods have gradually become an important means of constructing efficient surrogate models.

[0003] Existing machine learning modeling solutions often employ a single model architecture, such as Multilayer Perceptron (MLP), Support Vector Regression (SVR), or Random Forest. These methods often suffer from insufficient generalization ability, poor interpretability, and weak adherence to physical laws when processing aerodynamic data with strong nonlinearity and multi-scale coupling characteristics. Especially when modeling multiple output variables (such as CA / CN / CZ / CMx / CMy / CMz) jointly, a single model struggles to balance accuracy, stability, and engineering reliability. Furthermore, most deep learning models are considered "black boxes," lacking transparency in their internal decision-making mechanisms, which limits their deployment and application in safety-critical systems.

[0004] With the development of explainable artificial intelligence (XAI) technology, researchers have begun to explore fusion strategies that integrate multiple modeling paradigms to enhance the understandability and physical consistency of models while maintaining high predictive performance. Symbolic Regression (SR) can generate explicit mathematical expressions, facilitating derivative analysis and verification of physical laws; Decision Tree (DT) provides a clear logical path through piecewise rules; attention mechanisms can implement importance weighting of input features in neural networks; and the emerging Kolmogorov-Arnold Network (KAN) is based on function approximation theory, using spline basis functions to achieve stronger nonlinear fitting and support fine-grained visualization analysis. However, existing fusion methods typically employ static weight combinations or simple voting mechanisms, failing to fully consider the differences in local adaptability, predictive stability, and physical plausibility of each sub-model in different sample regions. This results in fusion results that may still exhibit non-physical incomprehensibility or sensitive fluctuations under certain conditions.

[0005] Therefore, there is an urgent need for a dynamic adaptive fusion framework for multi-physical quantity prediction tasks. This framework should be able to introduce a sample-level adaptive weighting mechanism while preserving the interpretability advantages of each model, and explicitly integrate domain knowledge (such as physical constraints like monotonicity, symmetry, and symbol consistency) into the fusion process, thereby improving the robustness, reliability, and practicality of the overall model in complex engineering scenarios. Summary of the Invention

[0006] The purpose of this invention is to provide a dynamically weighted interpretable aerodynamic coefficient prediction method and system, which aims to solve the technical problems of insufficient model fusion flexibility, weak physical consistency guarantee, and ineffective utilization of interpretable information in the prior art.

[0007] This invention provides a dynamically weighted, interpretable aerodynamic coefficient prediction method, comprising: A flight state parameter feature dataset consisting of multiple sample points and a corresponding aerodynamic coefficient label dataset are obtained, and the feature dataset is normalized to obtain standardized sample features. A set of heterogeneous machine learning models are trained in parallel based on the standardized sample features and the labeled dataset, and the unbiased prediction value of each model for each sample point is obtained through cross-validation. For each sample point, the three types of sample-level evaluation weights are calculated and fused based on the unbiased prediction values ​​of each model corresponding to that sample point, thereby generating the dynamic fusion weights of each model at that sample point. For each sample point, the unbiased prediction values ​​of each model for that sample point are weighted and summed using the corresponding dynamic fusion weights to obtain the final fusion prediction value for that sample point. The final fusion prediction values ​​of all sample points are summarized, and the final aerodynamic coefficient prediction results and corresponding interpretability analysis report are output.

[0008] This invention provides a dynamically weighted interpretable aerodynamic coefficient prediction system, comprising: The data preprocessing module is used to acquire a flight state parameter feature dataset consisting of multiple sample points and a corresponding aerodynamic coefficient label dataset, and to normalize the feature dataset to obtain standardized sample features. The model training and prediction module is used to train a set of heterogeneous machine learning models in parallel based on the standardized sample features and the labeled dataset, and to obtain the unbiased prediction value of each model for each sample point through cross-validation. The dynamic weight generation module is used to calculate and fuse the three types of sample-level evaluation weights for each sample point based on the unbiased prediction values ​​of each model, and generate the dynamic fusion weights of each model at that sample point. The weighted fusion module is used to sum the unbiased predictions of each model for each sample point using its corresponding dynamic fusion weights, so as to obtain the final fusion prediction value of the sample point. The results output and visualization module is used to summarize the final fusion prediction values ​​of all sample points, and output the final aerodynamic coefficient prediction results and the corresponding interpretability analysis report.

[0009] This invention also provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the computer program is executed by the processor, it implements the steps of the above-described dynamically weighted interpretable aerodynamic coefficient prediction method.

[0010] This invention also provides a computer-readable storage medium storing an information transmission implementation program, which, when executed by a processor, implements the steps of the above-described dynamically weighted interpretable aerodynamic coefficient prediction method.

[0011] The embodiments of this invention can include the following beneficial effects: These embodiments provide a comprehensive solution for modeling, simulating, and optimizing complex engineering systems, combining high precision, strong robustness, and deep interpretability. This solution addresses the modeling challenges of achieving high precision, strong interpretability, and physical consistency in complex engineering scenarios (such as aerodynamic coefficient prediction). It is particularly suitable for constructing alternative models and discovering knowledge for nonlinear multi-output problems in fields such as aerodynamics, structural mechanics, and thermal control systems. Attached Figure Description

[0012] To more clearly illustrate the technical solutions in one or more embodiments of this specification or in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in this specification. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0013] Figure 1 This is a flowchart of the interpretable aerodynamic coefficient prediction method based on dynamic weighting according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the network architecture according to an embodiment of the present invention; Figure 3 This is an embodiment of the present invention. Schematic diagram of the evaluation results of the predictive network; Figure 4 This is an embodiment of the present invention. Schematic diagram of the evaluation results of the predictive network; Figure 5This is a schematic diagram of an interpretable aerodynamic coefficient prediction system based on dynamic weighting, according to an embodiment of the present invention. Detailed Implementation

[0014] To enable those skilled in the art to better understand the technical solutions in one or more embodiments of this specification, the technical solutions in one or more embodiments of this specification will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this specification, and not all of the embodiments. Based on one or more embodiments of this specification, all other embodiments obtained by those skilled in the art without creative effort should fall within the protection scope of this document.

[0015] Method Implementation Examples According to embodiments of the present invention, a method for predicting interpretable aerodynamic coefficients based on dynamic weighting is provided. Figure 1 This is a flowchart of the dynamically weighted interpretable aerodynamic coefficient prediction method according to an embodiment of the present invention, as shown below. Figure 1 As shown, the dynamically weighted interpretable aerodynamic coefficient prediction method according to an embodiment of the present invention specifically includes: Step S101: Obtain the flight state parameter feature dataset and the corresponding aerodynamic coefficient label dataset composed of multiple sample points, and normalize the feature dataset to obtain standardized sample features. Step S102: Based on the standardized sample features and the labeled dataset, a set of heterogeneous machine learning models are trained in parallel, and the unbiased prediction value of each model for each sample point is obtained through cross-validation; the set of heterogeneous machine learning models includes a symbolic regression model for generating explicit mathematical expressions, a decision tree model that provides interpretable decision paths based on feature splitting rules, a multilayer perceptron model that adaptively weights input features using an attention mechanism, and a Kolmogorov-Arnold network model; specifically including: The model parameters and training hyperparameters of the symbolic regression model, decision tree model, multilayer perceptron model, and Kolmogorov-Arnold network model are configured independently respectively. The standardized sample features and the corresponding label dataset are divided into K non-overlapping folds using the same K-fold cross-validation partitioning strategy; For each model type, K training and prediction loops are executed independently in sequence. The predicted values ​​obtained on each validation set in each of the K loops are then concatenated in the original sample order to form an unbiased prediction vector for that model type on the entire dataset. Each element in the unbiased prediction vector is the unbiased prediction value output by that model for the corresponding sample point. Specifically, this includes: In the k-th iteration, the k-th fold of data is used as the validation set, and the remaining K-1 folds of data are used as the training set; The model is trained using the current training set, and predictions are made on the corresponding validation set to obtain the predicted values ​​of all sample points on the validation set. The unbiased prediction vectors of all models are organized into an unbiased prediction matrix by rows; wherein the rows of the unbiased prediction matrix correspond to different models and the columns correspond to different sample points.

[0016] Step S103: For each sample point, calculate and fuse the three types of sample-level evaluation weights based on the unbiased prediction values ​​of each model, generating the dynamic fusion weights of each model at that sample point; specifically including: For each sample point, three types of weight scores are calculated based on the model prediction values ​​of each corresponding column in the unbiased prediction matrix, and each type of weight score is normalized; wherein, the three types of weight scores include consistency score, stability score and physical rationality score. For each model, the normalized three-category weighted scores are weighted and summed according to preset weight coefficients to obtain the comprehensive score of the model at that sample point. The comprehensive score of all models at the same sample point is normalized to obtain the dynamic fusion weight of each model at that sample point.

[0017] Step S104: For each sample point, the unbiased prediction values ​​of each model are weighted and summed using its corresponding dynamic fusion weights to obtain the final fusion prediction value for that sample point; specifically including: S61: For each sample point, extract the corresponding column from the unbiased prediction matrix to obtain the unbiased prediction value vector of each model at that sample point; S62: For the same sample point, extract the dynamic fusion weights calculated by each model at that sample point, and form the fusion weight vector of that sample point. S63: Multiply the unbiased prediction vector element by element with the corresponding fusion weight vector to obtain the weighted prediction value of each model at the sample point. S64: Sum the weighted predicted values ​​to obtain the final fused predicted value for the sample point; S65: Traverse all sample points and repeat steps S61 to S64 to generate the final fused prediction set of all sample points. Each predicted value in the final fused predicted value set corresponds to a predicted result of an aerodynamic coefficient label, which includes at least one of axial force coefficient, normal force coefficient, vertical force coefficient, roll moment coefficient, pitch moment coefficient, and yaw moment coefficient.

[0018] Step S105: Summarize the final fused prediction values ​​of all sample points, and output the final aerodynamic coefficient prediction results and the corresponding interpretability analysis report, specifically including: The final fused prediction values ​​of each sample point are arranged and integrated according to the original sample order to generate the final aerodynamic coefficient prediction result; wherein the aerodynamic coefficient prediction result corresponds to the dimension and order of the label dataset; Based on the aerodynamic coefficient prediction results, the unbiased prediction values ​​of each model, the dynamic fusion weights, and the intermediate interpretable data generated during training, a multi-dimensional interpretable analysis report is generated; wherein, the interpretable analysis report includes at least one or more of the following analysis contents: overall performance evaluation, model contribution analysis, feature importance analysis, symbolic regression analytical expression, key decision rule path, spline basis function curve, sensitivity analysis curve, and dynamic changes of model weights at different sample points; The predicted aerodynamic coefficients and the interpretability analysis report are then stored in a structured manner and visualized.

[0019] The following describes the specific details of the dynamically weighted interpretable aerodynamic coefficient prediction method according to embodiments of the present invention, such as... Figure 2 As shown, the above technical solutions of the embodiments of the present invention will be described in detail.

[0020] This invention proposes an interpretable fusion modeling framework based on multi-model collaboration and dynamic weighting mechanisms for high-precision prediction of multivariate responses (such as aerodynamic coefficients and torques) in complex physical fields. This framework integrates various heterogeneous learning models, including Symbolic Regression (SR), Decision Tree (DT), Attention-MLP, and Kolmogorov-Arnold Network (KAN), and combines a triple adaptive weighting strategy—consistency weight, stability weight, and physical plausibility weight—to significantly enhance the interpretability and engineering credibility of the model output while improving prediction accuracy.

[0021] This invention first employs K-fold cross-validation to generate Out-of-Fold (OOF) prediction results for each base model, avoiding data leakage during training and achieving unbiased prediction using OOF prediction. Subsequently, in the fusion stage, fusion weights are dynamically calculated for each sample point based on the prediction behavior on the validation set: consistency weights measure the consistency of predictions from different models on the same sample point, which can be calculated based on the statistical dispersion of predicted values ​​(such as standard deviation) or correlation measures (such as correlation coefficient); stability weights measure the robustness of the model under noise disturbances; and physical rationality weights introduce domain prior knowledge (such as monotonicity, symmetry, symbol constraints, etc.) to penalize predictions that violate physical laws. Finally, the weighted fusion output not only possesses statistical optimality but also satisfies key physical conservation and behavioral constraints, making it suitable for applications with stringent safety requirements such as aircraft modeling and control system design.

[0022] The embodiments of the present invention specifically include the following steps: 1. Input the original feature dataset With label dataset Where n represents the number of samples, and "3" refers to the three input variables: Mach number (Ma), sideslip angle (Ma), and sideslip angle (Ma). ) and angle of attack ( ), m represents the number of output targets, corresponding to six dimensionless aerodynamic coefficients: axial force coefficient Normal force coefficient Vertical force coefficient Rolling torque coefficient Pitch moment coefficient With yaw moment coefficient Data can come from structured data files, such as CSV files, and serve as the basic input for the entire modeling process. 2. For the input feature matrix Perform normalization and calculate the minimum value for each column. With the maximum value Then, each column of features is mapped to the interval [-1, 1] using the following linear transformation formula: ; in To prevent division by zero of small constants, this normalization process unifies the features of different physical dimensions to the same numerical range, which is beneficial to improving the stability of subsequent neural network training and facilitates the effective fitting of spline functions in the normalized space; the final output is the normalized feature matrix. ; 3. Construct four types of base learners and independently perform K-fold cross-validation training to generate Out-of-Fold (OOF) prediction results for each model on the validation fold, specifically including: Attention Multilayer Perceptron (Attention MLP): Employs a self-attention mechanism to weight input features. Its structure includes a linear attention module and a fully connected network. During training, it optimizes the mean squared error loss and outputs the OOF prediction value for each sample. and attention weight matrix ; Kolmogorov-Arnold Network (KAN): Based on the Kolmogorov-Arnold representation theorem, it uses B-spline functions to approximate univariate functions and implements nonlinear transformations at each layer. During training, the spline grid positions are updated in the first few steps to adapt to the data distribution, and subsequent steps fix the grid for parameter optimization. It outputs an Out-of-Flight (OOF) prediction. With the final model state dictionary list; Decision Tree: Constructs a regression tree based on the CART algorithm, achieving piecewise linear fitting through feature splitting rules; sets a maximum depth to control complexity and prevent overfitting; outputs an OOF (Out of Frame) prediction. A list of tree model objects obtained from each fold of training; Symbolic Regression: This method uses genetic programming to search for the optimal mathematical expression, attempting to discover explicit analytical relationships in the data. If the environment supports it, it runs the PySR library for symbolic learning; otherwise, it automatically reverts to a linear regression model to ensure the integrity of the process. It outputs an Out-of-Flight (OOF) prediction. A list of symbolic expression strings generated for each fold; 4. Load the OOF prediction results generated by each base model during cross-validation, and denote the prediction of the i-th model as... Where n is the number of samples. The predictions of all k models (usually 4) are stacked row-wise into a matrix to form the prediction matrix. This step is used for subsequent fusion weight calculation; it ensures that the fusion stage relies solely on cross-validation extrapolation prediction, avoiding the leakage of training information, thereby guaranteeing the reliability and generalization ability of the evaluation. 5. Calculate the triple dynamic sample-level fusion weights, namely consistency weight, stability weight, and physical rationality weight: Consistency weight: For each sample point, calculate the standard deviation of the predicted values ​​of each model. This reflects the prediction discrepancy on this sample. The standard deviation is then normalized. ; The inverted value is used as the basic consistency score. The higher the score, the stronger the consensus among the models on this sample. Stability weights: Apply a noise perturbation stability test to each base model, i.e., add small-amplitude Gaussian noise to the input X. Repeat the process multiple times to obtain the predicted volatility variance; calculate the average noise response stab and normalize it. Its inverse ratio is used as the stability score; Physical rationality weighting: Prior physical constraints are defined for different aerodynamic coefficients and substituted into the predictions of each model for verification; for example, for... Requirements Monotonically non-decreasing, for Requirements and Consistent signs, for Requirements Monotonically non-increasing; outputs a score indicating the degree to which physical laws are satisfied for each sample. ∈[0,1]; 6. Construct the final fusion weight by combining the three weights. A weighted index normalization strategy is adopted: set up =0.4、 =0.3、 =0.3 is the preset hyperparameter, then the overall score is: ; The final weights are then obtained by normalization using the softmax function. ; make sure =1 and >0; This mechanism ensures that the fusion results simultaneously take into account statistical consistency, anti-interference stability, and engineering physical reliability; 7. Perform weighted fusion prediction and calculate the fusion output: ; That is, for each sample, the predictions of each base model are combined according to its dynamic weights; the fused point estimation results are output. This serves as the final, highly accurate, and interpretable prediction of aerodynamic coefficients. 8. Output multi-dimensional visualization analysis reports to aid in understanding model behavior and fusion mechanisms. Model contribution bar chart: Shows the average absolute contribution of each base model in the fusion, reflecting the overall weight distribution; KAN path contribution heatmap: via Jacobian matrix Quantify the gradient propagation path strength from the original features to the hidden channels and then to the output to reveal the internal nonlinear mechanism; The spline function curve of the last layer: plot the spline activation function shape of each channel of the last layer of KAN to intuitively show the local nonlinear shaping effect; Univariate prediction curve: With other features fixed as the mean, the output response curve is scanned when a single input changes, for sensitivity analysis. Comparison of Path Contribution and SHAP Value: The results of the above path contribution and SHAP interpretation methods are normalized and compared, and the correlation coefficient is calculated to verify the consistency of different interpretation methods. Comprehensive Summary Diagram: Integrating performance indicator radar charts, symbolic regression equations, decision tree rule fragments, attention importance, fusion contribution, and KAN visualization, it provides a comprehensive diagnostic view.

[0023] Taking a hypersonic vehicle aerodynamic database as an example, input feature data and output label data are loaded, where the input includes Mach number (Ma), sideslip angle (Ma), and other parameters. ) and angle of attack ( The output is a six-dimensional dimensionless aerodynamic coefficient: axial force coefficient ( ), normal force coefficient ( Vertical force coefficient () ), rolling moment coefficient ( ), pitching moment coefficient ( ) and yaw moment coefficient ( ).

[0024] 1. Perform independent K-fold cross-validation training on each base model and generate Out-of-Fold (OOF) prediction results: Attention-MLP model: Training begins with the hyperparameters defined in the configuration file (hidden layer structure [64, 32], learning rate 1e-2, training steps 200). The Attention-MLP network is trained on each fold and the attention weights are output. Finally, the validation set predictions of each fold are concatenated into a complete OOF vector, and the attention matrix and model parameters are cached. KAN model: Based on the configuration (64 hidden units, spline grid size 7, 10 grid update steps, 250 total training steps), it is trained fold by fold using an efficient KAN architecture to capture the nonlinear function relationship between input and output; after training, it outputs OOF predictions and saves the model state dictionary for each fold; Decision tree model: Set the maximum depth to 5, complete five-fold training, generate piecewise linear fitting results, its rule path has strong interpretability, and save OOF predictions; Symbolic regression model: Using configured genetic programming parameters (50 iterations, binary operators ["+", ""], unary operators ["sin", "cos", "exp"]), an explicit mathematical expression is searched; the resulting symbolic equations are serialized and saved. It is worth noting that the above parameters are merely examples and do not constitute a limitation on the scope of protection of this invention.

[0025] 2. The dynamic weighted fusion script loads the corresponding label prediction arrays from the OOF directories of each model to construct a multi-model prediction set. The fusion weight strategy is set as follows: Consistency weight: Calculate the Pearson correlation coefficient matrix among the OOF predictions of each model to reflect the degree of group consensus; Stability weight: The sensitivity of each model to small changes in input is evaluated through noise perturbation test. The smaller the predicted fluctuation index (such as variance or standard deviation), the higher the score. Physical rationality weight: Prior physical constraints are set for different labels: Requirements Monotonically non-decreasing; for and Requirements and Same sign; opposite Requirements | |follow| |The enhancement shows a positive correlation trend; for Requirements Increases monotonically but decreases (statically stable configuration); points will be deducted for violating the above rules, and points will be added for conforming to them.

[0026] Combining the three types of weights, the sample-level adaptive fusion weights are calculated using a weighted exponential normalization method: ; in, Let represent the consistency, stability, and physical plausibility scores of the i-th model, respectively. =0.4、 =0.3、 =0.3 is the task customization adjustment coefficient, and j represents all models traversed in the summation, i.e., i, j K is the total number of base models.

[0027] 3. Apply fusion weights to perform a weighted sum of the OOF predictions from each model to obtain the fused composite prediction value. For example, after dynamic weighted fusion, the validation set yielded MSE=0.002872, RMSE=0.0536, and MAE=0.0407. =0.9981; The performance indicators of the remaining tags are summarized as follows: Table 1 Results of performance indicators for each label

[0028] 4. Output integrated analysis reports and visualization charts, such as Figure 3 , 4 The figures shown are embodiments of the present invention. , Schematic diagram of the evaluation results of the prediction network: Generate a bar chart of the fusion contribution of each label to show the average absolute contribution of each base model in the fusion process; We use SHAP and path gradient methods to analyze the internal mechanism of KAN, revealing feature importance and local nonlinear response; Generate a comprehensive summary diagram that integrates multi-dimensional information such as performance index comparison, symbolic regression formula, decision tree rule fragments, attention distribution and fusion contribution; Finally, the fusion verification metrics of all labels are summarized and used as the basis for evaluating the overall performance of the model.

[0029] In summary, this invention provides a physically constrained, dynamically weighted, interpretable aerodynamic coefficient prediction method. This method constructs a heterogeneous model fusion framework integrating symbolic regression (SR), decision tree (DT), attention-based multilayer perceptron (Attention-MLP), and Kolmogorov-Arnold network (KAN). Unbiased prediction results from each base model are generated through out-of-fold (OOF) cross-validation. A sample-level adaptive dynamic weighting mechanism is designed, comprehensively considering the consistency, stability, and physical rationality of the models to achieve optimal fusion output. This method not only significantly improves prediction accuracy but also enhances model transparency through multi-source interpretability methods (such as symbolic expressions, rule paths, attention weights, spline function visualization, and SHAP value analysis). It is applicable to engineering scenarios such as aerodynamic / torque modeling of hypersonic vehicles, control system design, and sensitivity analysis.

[0030] Compared to single-model prediction, the embodiments of this invention exhibit stronger generalization ability and robustness; compared to traditional weighted averaging or stacking fusion, the embodiments of this invention introduce physical rationality constraints as explicit weight adjustment factors to ensure that the fusion result conforms to aerodynamic prior laws (such as...). Monotonically increasing with angle of attack (Consistent with sideslip angle sign, etc.) effectively suppresses the generation of non-physical interpretations; compared with black-box deep learning models, the embodiments of this invention provide a complete interpretable system from global to local, from structure to behavior through a multi-dimensional interpretation toolchain. Therefore, the embodiments of this invention have good scalability and can be extended to other multiphysics modeling tasks, with advantages of high accuracy, strong interpretability, physical consistency, ease of diagnosis, and engineering reliability.

[0031] System Implementation Examples According to embodiments of the present invention, a dynamically weighted interpretable aerodynamic coefficient prediction system is provided. Figure 5 This is a schematic diagram of a dynamically weighted interpretable aerodynamic coefficient prediction system according to an embodiment of the present invention, as shown below. Figure 5 As shown, the dynamically weighted interpretable aerodynamic coefficient prediction system according to an embodiment of the present invention specifically includes: The data preprocessing module 50 is used to acquire a flight state parameter feature dataset consisting of multiple sample points and a corresponding aerodynamic coefficient label dataset, and to normalize the feature dataset to obtain standardized sample features. The model training and prediction module 52 is used to train a set of heterogeneous machine learning models in parallel based on the standardized sample features and the labeled dataset, and to obtain the unbiased prediction value of each model for each sample point through cross-validation. The dynamic weight generation module 54 is used to calculate and fuse the three types of sample-level evaluation weights for each sample point based on the unbiased prediction values ​​of each model, and generate the dynamic fusion weights of each model at that sample point. The weighted fusion module 56 is used to perform a weighted summation of the unbiased prediction values ​​of each model for each sample point using its corresponding dynamic fusion weights, so as to obtain the final fusion prediction value of the sample point. The results output and visualization module 58 is used to summarize the final fusion prediction values ​​of all sample points and output the final aerodynamic coefficient prediction results and the corresponding interpretability analysis report.

[0032] The embodiments of the present invention are system embodiments corresponding to the above method embodiments. The specific operation of each module can be understood by referring to the description of the method embodiments, and will not be repeated here.

[0033] In summary, this invention proposes a physically constrained, dynamically weighted, interpretable aerodynamic coefficient prediction method. This method integrates various heterogeneous learners, including symbolic regression, decision trees, attention-based multilayer perceptrons, and Kolmogorov-Arnold networks, and innovatively combines a triple dynamic weighting mechanism based on consistency, stability, and physical plausibility. This successfully constructs a robust, accurate, and engineering-credible fusion prediction model for high-dimensional nonlinear systems (such as aerodynamic coefficient prediction). This method not only achieves high-fidelity approximation of complex physical fields but also significantly improves the understandability and auditability of data-driven models in key engineering fields through a multi-layered visualization and interpretation toolchain.

[0034] The technical solutions adopted in the embodiments of this invention are universal and not limited to specific aerodynamic data or tasks. They can be extended to any scenario involving multivariate joint modeling and multi-source prediction result fusion, and are applicable to many engineering fields such as aerospace, fluid mechanics, structural analysis, and intelligent manufacturing that require simultaneous consideration of prediction accuracy and model interpretability. Compared with the prior art, the beneficial effects of the embodiments of this invention specifically include: 1. Compared with traditional ensemble learning methods that rely solely on prediction error or cross-validation performance for static weighting, this invention introduces a sample-level dynamic triple weighting mechanism (consistency + stability + physical rationality), which can adaptively adjust the contribution of each base model according to the input conditions, significantly improving the robustness and engineering credibility of the fusion results in complex nonlinear regions. 2. Compared to single interpretable models (such as overly simplified symbolic regression expressions, weak extrapolation capabilities of decision trees, and lack of structural transparency in attention mechanisms), this invention's embodiments utilize a multi-paradigm model collaborative architecture (symbolic regression provides explicit formulas, decision trees preserve segmented logic, attention networks capture dynamic feature weights, and KAN achieves high-precision spline approximation), which balances accuracy, generalization ability, and multi-level interpretability, meeting the dual requirements of industrial-grade modeling for "accuracy and understanding." 3. Compared to black-box neural networks, which struggle to verify whether their outputs conform to physical laws, this invention incorporates physical rationality as an explicit constraint into the fusion process, defining corresponding criteria for different aerodynamic coefficients (such as...). Monotonic non-decreasing with angle of attack (Consistent with the sideslip angle sign, etc.), and quantitatively evaluated on a sample-by-sample basis within the OOF framework, effectively filtering out abnormal predictions that violate the mechanism and enhancing the safety and reliability of the model under unknown operating conditions.

[0035] Device Example 1 This invention provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the computer program is executed by the processor, it performs the steps described in the method embodiment.

[0036] Device Example 2 This invention provides a computer-readable storage medium storing an information transmission implementation program, which, when executed by a processor, performs the steps described in the method embodiment.

[0037] The computer-readable storage media described in this embodiment include, but are not limited to, ROM, RAM, disk, or optical disk.

[0038] 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 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 or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for predicting interpretable aerodynamic coefficients based on dynamic weighting, characterized in that, include: A flight state parameter feature dataset consisting of multiple sample points and a corresponding aerodynamic coefficient label dataset are obtained, and the feature dataset is normalized to obtain standardized sample features. A set of heterogeneous machine learning models are trained in parallel based on the standardized sample features and the labeled dataset, and the unbiased prediction value of each model for each sample point is obtained through cross-validation. For each sample point, the three types of sample-level evaluation weights are calculated and fused based on the unbiased prediction values ​​of each model corresponding to that sample point, thereby generating the dynamic fusion weights of each model at that sample point. For each sample point, the unbiased prediction values ​​of each model for that sample point are weighted and summed using the corresponding dynamic fusion weights to obtain the final fusion prediction value for that sample point. The final fusion prediction values ​​of all sample points are summarized, and the final aerodynamic coefficient prediction results and corresponding interpretability analysis report are output.

2. The method according to claim 1, characterized in that, The set of heterogeneous machine learning models includes a symbolic regression model for generating explicit mathematical expressions, a decision tree model that provides interpretable decision paths based on feature splitting rules, a multilayer perceptron model that uses an attention mechanism to adaptively weight input features, and a Kolmogorov-Arnold network model.

3. The method according to claim 2, characterized in that, A set of heterogeneous machine learning models are trained in parallel based on the standardized sample features and the labeled dataset, and the unbiased prediction value of each model for each sample point is obtained through cross-validation. Specifically, this includes: The model parameters and training hyperparameters of the symbolic regression model, decision tree model, multilayer perceptron model, and Kolmogorov-Arnold network model are configured independently respectively. The standardized sample features and the corresponding label dataset are divided into K non-overlapping folds using the same K-fold cross-validation partitioning strategy; For each type of model, K training and prediction cycles are executed independently in sequence. The predicted values ​​obtained by the model on each validation set in each of the K cycles are concatenated in the original sample order to form the unbiased prediction vector of the model on the entire dataset. Each element in the unbiased prediction vector is the unbiased prediction value output by the model for the corresponding sample point. The unbiased prediction vectors of all models are organized into an unbiased prediction matrix by rows; wherein the rows of the unbiased prediction matrix correspond to different models and the columns correspond to different sample points.

4. The method according to claim 3, characterized in that, For each type of model, K training and prediction cycles are performed independently in sequence, specifically including: In the k-th iteration, the k-th fold of data is used as the validation set, and the remaining K-1 folds of data are used as the training set; The model is trained using the current training set, and predictions are made on the corresponding validation set to obtain the predicted values ​​for all sample points on that validation set.

5. The method according to claim 3, characterized in that, For each sample point, the dynamic fusion weights of the three sample-level evaluations are calculated and fused based on the unbiased predictions of each model, generating the dynamic fusion weights of each model at that sample point. Specifically, this includes: For each sample point, three types of weight scores are calculated based on the model prediction values ​​of each corresponding column in the unbiased prediction matrix, and each type of weight score is normalized; wherein, the three types of weight scores include consistency score, stability score and physical rationality score. For each model, the normalized three-category weighted scores are weighted and summed according to preset weight coefficients to obtain the comprehensive score of the model at that sample point. The comprehensive score of all models at the same sample point is normalized to obtain the dynamic fusion weight of each model at that sample point.

6. The method according to claim 5, characterized in that, For each sample point, the unbiased predictions of each model for that sample point are weighted and summed using its corresponding dynamic fusion weights to obtain the final fusion prediction value for that sample point. Specifically, this includes: S61: For each sample point, extract the corresponding column from the unbiased prediction matrix to obtain the unbiased prediction value vector of each model at that sample point; S62: For the same sample point, extract the dynamic fusion weights calculated by each model at that sample point, and form the fusion weight vector of that sample point. S63: Multiply the unbiased prediction vector element by element with the corresponding fusion weight vector to obtain the weighted prediction value of each model at the sample point. S64: Sum the weighted predicted values ​​to obtain the final fused predicted value for the sample point; S65: Traverse all sample points and repeat steps S61 to S64 to generate the final fused prediction set of all sample points. Each predicted value in the final fused predicted value set corresponds to a predicted result of an aerodynamic coefficient label, which includes at least one of axial force coefficient, normal force coefficient, vertical force coefficient, roll moment coefficient, pitch moment coefficient, and yaw moment coefficient.

7. The method according to claim 6, characterized in that, The final fused prediction values ​​of all sample points are summarized, and the final aerodynamic coefficient prediction results and corresponding interpretability analysis report are output, including: The final fused prediction values ​​of each sample point are arranged and integrated according to the original sample order to generate the final aerodynamic coefficient prediction result; wherein the aerodynamic coefficient prediction result corresponds to the dimension and order of the label dataset; Based on the aerodynamic coefficient prediction results, the unbiased prediction values ​​of each model, the dynamic fusion weights, and the intermediate interpretable data generated during training, a multi-dimensional interpretable analysis report is generated; wherein, the interpretable analysis report includes at least one or more of the following analysis contents: overall performance evaluation, model contribution analysis, feature importance analysis, symbolic regression analytical expression, key decision rule path, spline basis function curve, sensitivity analysis curve, and dynamic changes of model weights at different sample points; The predicted aerodynamic coefficients and the interpretability analysis report are then stored in a structured manner and visualized.

8. A predictive system for interpretable aerodynamic coefficients based on dynamic weighting, characterized in that, include: The data preprocessing module is used to acquire a flight state parameter feature dataset consisting of multiple sample points and a corresponding aerodynamic coefficient label dataset, and to normalize the feature dataset to obtain standardized sample features. The model training and prediction module is used to train a set of heterogeneous machine learning models in parallel based on the standardized sample features and the labeled dataset, and to obtain the unbiased prediction value of each model for each sample point through cross-validation. The dynamic weight generation module is used to calculate and fuse the three types of sample-level evaluation weights for each sample point based on the unbiased prediction values ​​of each model, and generate the dynamic fusion weights of each model at that sample point. The weighted fusion module is used to sum the unbiased predictions of each model for each sample point using its corresponding dynamic fusion weights, so as to obtain the final fusion prediction value of the sample point. The results output and visualization module is used to summarize the final fusion prediction values ​​of all sample points, and output the final aerodynamic coefficient prediction results and the corresponding interpretability analysis report.

9. An electronic device, characterized in that, include: A memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the computer program, when executed by the processor, implements the steps of the dynamically weighted interpretable aerodynamic coefficient prediction method as described in any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores an implementation program for information transmission, which, when executed by a processor, implements the steps of the dynamically weighted interpretable aerodynamic coefficient prediction method as described in any one of claims 1-7.

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