Symmetrical airfoil unsteady flow field prediction method and system based on deep neural operator network

Abstract

The invention relates to a symmetric airfoil unsteady flow field prediction method and system based on a deep neural operator network. The method comprises the following steps: collecting flow field data sets of different symmetric airfoils; constructing an input and output data set; constructing a neural network model, wherein the neural network model comprises a static branch network, a dynamic backbone network and a fusion network; taking the static aerodynamic parameters as input of a static branch network, and outputting nonlinear features reflecting mapping of the static aerodynamic parameters to a potential space; the dynamic space-time coordinate features subjected to Fourier position coding processing serve as input of a dynamic backbone network, and potential dynamic features representing space-time coordinates in a flow field are output; splicing to obtain a high-dimensional feature vector; outputting an output predicted value representing the pressure field and the velocity field; training the neural network model; and carrying out neural network model lightweight processing, and carrying out flow field prediction. According to the method, flow field characteristic prediction can be rapidly and accurately carried out, and the aerodynamic performance in the aerodynamic design process of the aircraft is optimized.

Description

Technical Field

[0001] This invention relates to the fields of aircraft aerodynamic design and computational fluid dynamics, and in particular to a method and system for predicting unsteady flow fields of symmetric airfoils based on deep neural operator networks. Background Technology

[0002] Currently, research on airfoil flow field characteristics mainly relies on wind tunnel testing and computational fluid dynamics (CFD) methods. Wind tunnel testing can provide high-precision experimental data, but it is costly and time-consuming, making it difficult to meet the needs of rapid iterative design. CFD technology can obtain high-precision flow field predictions by solving the Navier-Stokes (NS) equations, and it is less expensive than wind tunnel testing. However, when dealing with large-scale parameter scans and long-term unsteady evolution, it still requires a lot of computational resources and time. A single simulation often takes several hours or even days, becoming a major bottleneck restricting the efficiency of airfoil optimization design.

[0003] With the rapid development of artificial intelligence, data-driven methods based on deep learning have become an effective alternative to traditional CFD methods. Compared with traditional neural networks, Deep Operator Networks (DeepONet) exhibit stronger expressive power and higher prediction efficiency when dealing with high-dimensional and nonlinear problems. In recent years, DeepONet has shown promising application prospects in airfoil aerodynamic characteristic prediction, turbulence modeling, and flow field reconstruction. Deep Operator Networks learn linear or nonlinear operators based on data, possessing advantages such as fast convergence speed and strong generalization ability, and have been used to solve problems such as differential equations and boundary layer unstable waves. Using DeepONet to model the dynamic stall aerodynamic forces under large pitch motion of an airfoil and coupling it with structural dynamic equations, the bifurcation velocity and limit cycle oscillation prediction of stall flutter were successfully achieved. This method more accurately characterizes hysteresis characteristics than traditional recurrent neural networks, reducing aerodynamic prediction errors by approximately 2%. Furthermore, data-driven modeling of flow around a cylinder based on DeepONet can accurately reconstruct unsteady flow fields evolving over time and maintain stability during long-term predictions. This method outperforms traditional order reduction methods in capturing nonlinear vortex structures, demonstrating the significant advantage of operator learning in the rapid prediction of unsteady flow fields.

[0004] However, existing deep learning methods still have some shortcomings in handling flow field prediction: (1) Limitations of single-task prediction: Existing methods mostly use a single prediction task, which makes it difficult to accurately predict multiple physical quantities (such as pressure field and velocity field) at the same time, resulting in insufficient physical consistency of prediction results.

[0005] (2) Limited ability to capture high-frequency features: There are multi-scale, high-frequency dynamic features in the flow field (such as vortex shedding, etc.). Traditional position coding methods are difficult to effectively express these high-frequency information, which affects the prediction accuracy.

[0006] (3) Simple feature fusion mechanism: The existing DeepONet architecture mostly adopts a simple linear combination method to fuse the features of the branch network and the backbone network, which fails to fully explore the nonlinear interaction relationship between aerodynamic parameters and spatiotemporal coordinates.

[0007] (4) Insufficient generalization ability: The generalization performance of the model needs to be improved under different airfoils and different operating conditions (Mach number, angle of attack).

[0008] Therefore, it is necessary to develop a new method for predicting unsteady flow fields, which can significantly improve computational efficiency while ensuring prediction accuracy, and enhance the ability to predict multiple physical quantities, multiple scales, and high-frequency dynamic characteristics. Summary of the Invention

[0009] To address these issues, this invention provides a method and system for predicting unsteady flow fields in symmetrical airfoils based on deep neural operator networks. This aims to solve the problems of high computational cost, long prediction cycle, insufficient joint prediction capability of multiple physical quantities, and limited high-frequency feature capture capability in traditional computational fluid dynamics (CFD) methods for predicting unsteady flow fields at high angles of attack. This invention is applicable to the prediction and analysis of unsteady flow fields in symmetrical airfoils under high angle-of-attack conditions, enabling rapid and accurate prediction of flow field characteristics and optimizing aerodynamic performance during aircraft aerodynamic design.

[0010] To address the aforementioned technical problems, this invention provides a method for predicting unsteady flow fields in symmetric airfoils based on deep neural operator networks, comprising the following steps: Collect flow field datasets for different symmetrical airfoils; wherein, the flow field datasets include CFD simulation data of different airfoils under different operating conditions; After preprocessing the flow field dataset, an input-output dataset is constructed; wherein, the input data includes static aerodynamic parameters and corresponding dynamic spatiotemporal coordinate features, and the output data includes pressure field and velocity field data; A neural network model is constructed, comprising a static branch network, a dynamic backbone network, and a fusion network. The static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space. The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of the spatiotemporal coordinates in the flow field. The nonlinear features and the latent dynamic features are fused to obtain a high-dimensional feature vector. The high-dimensional feature vector is input into the fusion network for nonlinear mapping, and the output predicted values ​​representing the pressure field and velocity field are output. The neural network model is trained based on the input and output datasets, using the mean squared error between the predicted output value and the target value as the loss function. The trained neural network model is lightweighted to predict the flow field under different operating conditions and airfoil conditions.

[0011] In one embodiment of the present invention, the flow field datasets for different symmetrical airfoils are collected, including: Flow field data of different aircraft airfoils are collected as the flow field dataset, and flow field data of at least one of the aircraft airfoils are used as the validation set; wherein, the flow field dataset includes flow field simulation results with Mach number range of 0.2-0.5 and angle of attack range of 25°-55°, and the flow field simulation results include pressure field and velocity field variables; Based on the flow field data of different airfoils, after meshing, flow field simulation was performed under different operating conditions, including changes in Mach number, angle of attack, and airfoil parameters. Based on the simulation results, the pressure and velocity field data near the airfoil were obtained.

[0012] In one embodiment of the present invention, after preprocessing the flow field dataset, an input-output dataset is constructed, including: The flow field dataset is cleaned and anomaly handled, and missing values ​​are filled in using linear interpolation to ensure data integrity and continuity. The preprocessed flow field dataset is normalized to map the values ​​of each physical quantity to the interval [-1, 1]. Based on aerodynamic principles and data-driven analysis methods, the original static parameters in the flow field dataset are screened; the importance of each original static parameter is evaluated using a random forest model, and finally the static aerodynamic parameters are selected, including Mach number, angle of attack, and airfoil geometry parameters including maximum thickness and the location of maximum thickness. The input and output are aligned with the coordinate axes and time index. The static aerodynamic parameters and dynamic spatiotemporal coordinate features are used as inputs, and the pressure field and velocity field components of the corresponding flow field are used as outputs to form an input-output dataset for the neural network model.

[0013] In one embodiment of the present invention, the static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space, including: The static aerodynamic parameters are processed by the static branch network, and the output feature dimension is (B, 256), where B is the batch size. This output contains the nonlinear feature representation of the static aerodynamic parameters mapped to the latent space. The neural network model adopts the Flow-Deeponet neural network framework, and the static branch network includes multiple ResMLP modules; Each of the ResMLP modules includes: a first fully connected layer, a GELU activation function, a Dropout layer, a second fully connected layer, and residual connections; The input dimension of the static branch network is (B, 4), where B is the batch size and 4 is the dimension of the static aerodynamic parameters.

[0014] In one embodiment of the present invention, the dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as the input of the dynamic backbone network, and the output is a latent dynamic feature characterizing the spatiotemporal coordinates in the flow field, including: Fourier position encoding is performed on the dynamic spatiotemporal coordinate features; The location-encoded dynamic spatiotemporal coordinate features are used as input to the dynamic backbone network to enhance the neural network model's ability to perceive temporal features. The backbone network adopts an MLP architecture and performs deep feature extraction through multiple fully connected layers; the dynamic spatiotemporal coordinate feature input has a dimension of (B, 19), including two coordinate axes, a time index (x, y, t), and 16 position-encoded frequency features; B is the batch size. After processing by the backbone network, the output feature dimension is (B, 256), which represents the potential dynamic features of the spatiotemporal coordinates in the flow field. The Fourier position encoding of the dynamic spatiotemporal coordinate features includes: The Fourier position encoding uses sine and cosine transforms to encode spatiotemporal coordinates based on multiple frequencies, as shown in the following formula: encoded_coord(x) = [x, sin(2 i πx), cos(2 i πx), …] for I∈{0,1,…,N}; Where x represents the input coordinates; encoded_coord(x) represents Fourier position encoding; i represents different frequency levels; and N represents the maximum frequency index used.

[0015] In one embodiment of the present invention, feature fusion is performed on the nonlinear features and the latent dynamic features to obtain a high-dimensional feature vector, including: The outputs of the static branch network and the dynamic backbone network are fused by concatenation and element-wise feature product to generate a high-dimensional feature vector with dimension (B, 512). The high-dimensional feature vector is input into the fusion network for nonlinear mapping processing, and the final output dimension is (B,3), which is used to characterize the components of the pressure field and velocity field in the x and y directions of the flow field; The fusion network consists of three fully connected layers, which compress and map the high-dimensional feature vector layer by layer. The final output is constrained to the range of [-1,1] by the tanh activation function to ensure the normalization consistency of each physical quantity.

[0016] In one embodiment of the present invention, after nonlinearly mapping the high-dimensional feature vector through the fusion network, output predicted values ​​representing the pressure field and velocity field are generated, including: The concatenated high-dimensional feature vector is input into the fusion network for nonlinear mapping processing and compressed to the output predicted value; The fusion network comprises three fully connected layers. The first layer compresses the high-dimensional features from 512 dimensions to 256 dimensions. The second layer further compresses the 256 dimensions to 128 dimensions. The third layer outputs three physical quantities, corresponding to the pressure field, the velocity field in the x-direction, and the velocity field in the y-direction of the flow field, respectively. The final output dimension is (B,3), which is used to represent the pressure and velocity of the flow field, where B is the batch size.

[0017] In one embodiment of the present invention, the loss function is expressed as: MSE = ;

[0018] in, For the target value, The output is the predicted value, where N is the number of samples.

[0019] In one embodiment of the present invention, the neural network model is trained based on the input-output dataset, and the trained neural network model is then subjected to lightweight processing, including: The neural network model is trained in a distributed parallel manner based on the PyTorch deep learning framework. During training, mixed precision training is applied to reduce memory usage, and dropout layers with a dropout rate of 0.2 are set in each network layer to prevent overfitting. The AdamW optimizer is used to update the network parameters, with an initial learning rate of 1×10⁻⁶. -4 And L1 regularization is introduced to improve the robustness of the model; An early stopping strategy was set during training, and model checkpoints were saved every 5 epochs. After training for 100 epochs, convergence was achieved on the validation set. The trained neural network model is lightweighted using a structured pruning strategy. This includes determining the importance of neurons based on the absolute value of the weights, removing low-importance neuron connections according to a preset pruning rate, fine-tuning the pruned network, and restoring model accuracy through a learning rate decay strategy. This achieves the structured lightweighting of the neural network model.

[0020] This invention also provides a system for predicting unsteady flow fields of symmetric airfoils based on deep neural operator networks, comprising: The flow field dataset acquisition module is used to collect flow field datasets for different symmetrical airfoils; wherein, the flow field dataset includes CFD simulation data of different airfoils under different operating conditions; The input / output dataset acquisition module is used to preprocess the flow field dataset and then construct the input / output dataset; wherein the input data includes static aerodynamic parameters and corresponding dynamic spatiotemporal coordinate features, and the output data includes pressure field and velocity field data; A neural network model construction module is used to construct a neural network model, which includes a static branch network, a dynamic backbone network, and a fusion network. The static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space. The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of the spatiotemporal coordinates in the flow field. A high-dimensional feature vector acquisition module is used to fuse the nonlinear features and the potential dynamic features to obtain a high-dimensional feature vector. The output prediction value acquisition module is used to input the high-dimensional feature vector into the fusion network for nonlinear mapping and output the output prediction value representing the pressure field and velocity field. The training module is used to train the neural network model based on the input and output dataset, using the mean squared error between the output predicted value and the target value as the loss function. The flow field prediction module is used to perform lightweight processing on the trained neural network model in order to predict the flow field under different operating conditions and different airfoil conditions.

[0021] The technical solution of the present invention has the following advantages compared with the prior art: This invention presents a method and system for predicting unsteady flow fields in symmetric airfoils based on deep neural operator networks, addressing the high computational cost and long computation cycle issues inherent in traditional computational fluid dynamics (CFD) methods for predicting unsteady flow fields at high angles of attack. The method employs a dual-path parallel network architecture, inputting aerodynamic parameters and spatiotemporal coordinates into separate branch and backbone networks for independent feature extraction. Multi-frequency Fourier position coding enhances the feature representation of spatiotemporal coordinates, effectively capturing multi-scale features and high-frequency dynamic behavior of the flow field. An improved feature fusion mechanism is used to concatenate and fuse the outputs of the branch and backbone networks and their element-wise products, achieving nonlinear interaction between static aerodynamic parameters and dynamic spatiotemporal features. A multi-task learning framework simultaneously predicts pressure and velocity field components, improving prediction accuracy and physical consistency. Experimental results show that the inference time of this invention reaches the second level, significantly improving computational efficiency compared to traditional CFD methods. It exhibits good generalization ability under different airfoils and operating conditions, providing an efficient and accurate flow field prediction tool for aircraft aerodynamic design. Attached Figure Description

[0022] To make the content of this invention easier to understand, the invention will be further described in detail below with reference to specific embodiments and accompanying drawings.

[0023] Figure 1 This is a flowchart of a method and system for predicting unsteady flow fields in airfoils based on deep neural operator networks, as proposed in this invention.

[0024] Figure 2 This is a diagram of the flow field prediction framework of the Flow-Deeponet neural network model of this invention.

[0025] Figure 3 This is a diagram of the Flow-Deeponet neural network model architecture of the present invention.

[0026] Figure 4a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Pressure contour map of the first airfoil at 0.45s, unit: Pa).

[0027] Figure 4b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Pressure contour map of the first airfoil at 0.6s, unit: Pa).

[0028] Figure 4c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Pressure contour map of the first airfoil at 0.7s, unit: Pa).

[0029] Figure 5aFor the Flow-Deeponet flow field prediction of this invention Figure 1 (Pressure contour plot of the second airfoil at 0.45s, unit: Pa).

[0030] Figure 5b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Pressure contour map of the second airfoil at 0.6s, unit: Pa).

[0031] Figure 5c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Pressure contour map of the second airfoil at 0.7s, unit: Pa).

[0032] Figure 6a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Pressure contour plot of the third airfoil at 0.45s, unit: Pa).

[0033] Figure 6b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Pressure contour plot of the third airfoil at 0.6s, unit: Pa).

[0034] Figure 6c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Pressure contour plot of the third airfoil at 0.7s, unit: Pa).

[0035] Figure 7a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the first airfoil in the X direction at 0.45s, unit: m / s).

[0036] Figure 7b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the first airfoil in the X direction at 0.6s, unit: m / s).

[0037] Figure 7c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the first airfoil in the X direction at 0.7s, unit: m / s).

[0038] Figure 8a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the second airfoil in the X direction at 0.45s, unit: m / s).

[0039] Figure 8b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the second airfoil in the X direction at 0.6s, unit: m / s).

[0040] Figure 8c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the second airfoil in the X direction at 0.7s, unit: m / s).

[0041] Figure 9a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the second airfoil in the X direction at 0.45s, unit: m / s).

[0042] Figure 9b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the second airfoil in the X direction at 0.6s, unit: m / s).

[0043] Figure 9c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the second airfoil in the X direction at 0.7s, unit: m / s).

[0044] Figure 10a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the first airfoil in the Y direction at 0.45s, unit: m / s).

[0045] Figure 10b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the first airfoil in the Y direction at 0.6s, unit: m / s).

[0046] Figure 10c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the first airfoil in the Y direction at 0.7s, unit: m / s).

[0047] Figure 11a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the second airfoil in the Y direction at 0.45s, unit: m / s).

[0048] Figure 11b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the second airfoil in the Y direction at 0.6s, unit: m / s).

[0049] Figure 11c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the second airfoil in the Y direction at 0.7s, unit: m / s).

[0050] Figure 12a For the Flow-Deeponet flow field prediction of this invention Figure 1 (Velocity contour map of the second airfoil in the Y direction at 0.45s, unit: m / s).

[0051] Figure 12b For the Flow-Deeponet flow field prediction of this invention Figure 2 (Velocity contour map of the second airfoil in the Y direction at 0.6s, unit: m / s).

[0052] Figure 12c For the Flow-Deeponet flow field prediction of this invention Figure 3 (Velocity contour map of the second airfoil in the Y direction at 0.7s, unit: m / s).

[0053] Figure 13 Error analysis of Flow-Deeponet and CAE results in this invention Figure 1 (First airfoil).

[0054] Figure 14 Error analysis of Flow-Deeponet and CAE results in this invention Figure 2 (Second airfoil).

[0055] Figure 15 Error analysis of Flow-Deeponet and CAE results in this invention Figure 3 (Third airfoil). Detailed Implementation

[0056] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0057] In this invention, when directions (up, down, left, right, front, and back) are described, it is only for the convenience of describing the technical solution of this invention, and does not indicate or imply that the technical features referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, it should not be construed as a limitation of this invention.

[0058] In this invention, "several" means one or more, "multiple" means two or more, "greater than," "less than," "exceeding," etc., are understood to exclude the stated number; "above," "below," "within," etc., are understood to include the stated number. In the description of this invention, the terms "first" and "second" are used only to distinguish technical features and should not be construed as indicating or implying relative importance, or implicitly indicating the number of indicated technical features, or implicitly indicating the order of the indicated technical features.

[0059] In this invention, unless otherwise explicitly defined, the terms "setting," "installing," and "connecting" should be interpreted broadly. For example, they can refer to a direct connection or an indirect connection through an intermediate medium; a fixed connection, a detachable connection, or an integrally formed connection; a mechanical connection, an electrical connection, or a connection capable of mutual communication; or the internal connection of two components or the interaction between two components. Those skilled in the art can reasonably determine the specific meaning of the above terms in this invention based on the specific content of the technical solution.

[0060] Example 1 Reference Figure 1 , Figure 2 As shown, the present invention provides a method for predicting unsteady flow fields of symmetric airfoils based on deep neural operator networks, comprising the following steps:

[0061] S1. Collect flow field datasets for different symmetrical airfoils; wherein, the flow field datasets include CFD simulation data of different airfoils under different operating conditions.

[0062] Specifically, in step S1, flow field datasets for different symmetrical airfoils are collected, including:

[0063] S11. Collect flow field data of different aircraft airfoils as the flow field dataset, and use the flow field data of at least one of the aircraft airfoils as a validation set (test set); In this embodiment, the flow field data of different symmetrical aircraft airfoils include NACA0006, NACA0012, NACA0018 and NACA0024, of which NACA0024 is used as the validation set;

[0064] The flow field dataset includes flow field simulation results with Mach numbers ranging from 0.2 to 0.5 and angles of attack ranging from 25° to 55°. The flow field simulation results include pressure field and velocity field variables; all data are sourced from the UIUC airfoil database.

[0065] S12. Based on the flow field data of different airfoils, after mesh generation, CFD flow field simulations were performed under different operating conditions, including variations in Mach number, angle of attack, and airfoil parameters. Specifically, high-precision mesh generation was performed for the flow field simulation data of different airfoils, with each airfoil containing approximately 615,200 mesh elements and 200 mesh nodes on the airfoil surface. The far-field boundary was positioned 10 times the airfoil chord length to minimize the influence of the far-field boundary conditions on local flow characteristics. During data generation, flow field simulations were performed for different operating parameters (including Mach number, angle of attack, and airfoil geometric parameters) with a simulation time step of 0.002 s and a total simulation duration of 1.2 s.

[0066] S13. Pressure and velocity field data in the vicinity of the airfoil were obtained through the above simulation, and flow field datasets for training deep neural operator network models were constructed based on these data.

[0067] S2. After preprocessing the flow field dataset, construct the input and output datasets; wherein, the input data includes static aerodynamic parameters and corresponding dynamic spatiotemporal coordinate features, and the output data includes pressure field and velocity field data (X-direction velocity and Y-direction velocity).

[0068] Specifically, step S2 includes:

[0069] S21. Perform data cleaning and anomaly handling on the flow field dataset (remove outliers using the 3σ principle) and use linear interpolation to fill in missing values ​​to ensure data integrity and continuity.

[0070] S22. Normalize the preprocessed flow field dataset, mapping the value range of each physical quantity to the interval [-1, 1].

[0071] S23. Based on aerodynamic principles and data-driven analysis methods, the original static parameters in the flow field dataset are screened; the importance of each original static parameter is evaluated using a random forest model, and finally the static aerodynamic parameters are selected. The static aerodynamic parameters include four items: Mach number, angle of attack, and airfoil geometric parameters including the maximum thickness and the location of the maximum thickness.

[0072] S24. Align the input and output with the coordinate axes and time index (x, y, t), take the static aerodynamic parameters and dynamic spatiotemporal coordinate features as input, and the pressure field and velocity field components of the corresponding flow field as output, to form an input-output dataset for the neural network model.

[0073] S3. Reference Figure 3As shown, a (Flow-Deeponet) neural network model is constructed, which includes a static branch network, a dynamic backbone network, and a fusion network. The static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space. The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of the spatiotemporal coordinates in the flow field.

[0074] The nonlinear features and the latent dynamic features are fused to obtain a high-dimensional feature vector.

[0075] The high-dimensional feature vector is input into the fusion network for nonlinear mapping, and the output predicted values ​​representing the pressure field and velocity field are output.

[0076] It should be noted that in the input feature separation stage, the input data includes static aerodynamic parameters and dynamic spatiotemporal coordinate features. Static aerodynamic parameters include Mach number, angle of attack, and airfoil geometry, which directly reflect the inherent properties of the aircraft and the incoming flow conditions. Dynamic spatiotemporal coordinate features include the spatial coordinates and time index of the flow field, reflecting the time-varying characteristics of the flow field. To avoid mutual interference between the two types of features, the input data is first separated, with static and dynamic features modeled through two independent network branches.

[0077] Specifically, the static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space, including:

[0078] The static aerodynamic parameters are processed by the static branch network, and the output feature dimension is (B, 256), where B is the batch size. This output contains the nonlinear feature representation of the static aerodynamic parameters mapped to the latent space.

[0079] The neural network model adopts the Flow-Deeponet neural network framework, and the static branch network includes multiple ResMLP modules;

[0080] Each ResMLP module includes: a first fully connected layer, a GELU activation function, a Dropout layer, a second fully connected layer, and residual connections; it can effectively learn the complex nonlinear mapping relationship between aerodynamic parameters and flow field solutions, improving the model's expressive power and training stability.

[0081] The input dimension of the static branch network is (B, 4), where B is the batch size and 4 is the dimension of the static aerodynamic parameters.

[0082] Specifically, the dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features characterizing the spatiotemporal coordinates in the flow field, including:

[0083] Fourier position encoding is performed on the dynamic spatiotemporal coordinate features to capture multi-scale and high-frequency dynamic features in the flow field.

[0084] The location-encoded dynamic spatiotemporal coordinate features are used as input to the dynamic backbone network to enhance the neural network model's ability to perceive temporal features.

[0085] The backbone network adopts an MLP architecture and performs deep feature extraction through multiple fully connected layers; the dynamic spatiotemporal coordinate feature input has a dimension of (B, 19), including two coordinate axes, a time index (x, y, t), and 16 position-encoded frequency features; B is the batch size.

[0086] After processing by the backbone network, the output feature dimension is (B, 256), which represents the potential dynamic features of the spatiotemporal coordinates in the flow field.

[0087] The Fourier position encoding of the dynamic spatiotemporal coordinate features includes:

[0088] The Fourier position encoding uses sine and cosine transforms to encode spatiotemporal coordinates based on multiple frequencies, as shown in the following formula:

[0089] encoded_coord(x) = [x, sin(2 i πx), cos(2 i πx), …] for I∈{0,1,…,N};

[0090] Where x represents the input coordinates; encoded_coord(x) represents Fourier position encoding; i represents different frequency levels; and N represents the maximum frequency index used.

[0091] Specifically, the nonlinear features and the latent dynamic features are fused to obtain a high-dimensional feature vector, including:

[0092] The outputs of the static branch network and the dynamic backbone network are fused by concatenation and element-wise feature product to generate a high-dimensional feature vector with dimension (B, 512).

[0093] The high-dimensional feature vector is input into the fusion network for nonlinear mapping processing, and the final output dimension is (B,3), which is used to characterize the components of the pressure field and velocity field in the x and y directions of the flow field;

[0094] The fusion network consists of three fully connected layers, which compress and map the high-dimensional feature vector layer by layer. The final output is constrained to the range of [-1,1] by the tanh activation function to ensure the normalization consistency of each physical quantity.

[0095] Specifically, after nonlinearly mapping the high-dimensional feature vectors through the fusion network, the output predicted values ​​representing the pressure field and velocity field are generated, including:

[0096] The concatenated high-dimensional feature vector is input into the fusion network for nonlinear mapping processing and compressed to the output predicted value;

[0097] The fusion network comprises three fully connected layers. The first layer compresses the high-dimensional features from 512 dimensions to 256 dimensions. The second layer further compresses the 256 dimensions to 128 dimensions. The third layer outputs three physical quantities, corresponding to the pressure field, the velocity field in the x-direction, and the velocity field in the y-direction of the flow field, respectively. The final output dimension is (B,3), which is used to represent the pressure and velocity of the flow field, where B is the batch size.

[0098] It should be noted that a three-way parallel fusion strategy is adopted in feature fusion: the branch features, the main features, and the element-wise product of the two are concatenated. The element-wise product term explicitly introduces the nonlinear interaction information between aerodynamic parameters and spatiotemporal position. Compared with the traditional linear combination method, it can more fully explore the coupling relationship between input features.

[0099] S4. Using the mean square error between the predicted output value and the target value as the loss function, train the neural network model based on the input and output dataset.

[0100] Specifically, the loss function is expressed as:

[0101] MSE = ;

[0102] in, For the target value, To output the predicted values, N represents the number of samples. For each sample, the model outputs three physical quantities: pressure P and velocity components U and V. The errors between the predicted and actual values ​​of each physical quantity are calculated using the mean square error and then weighted to obtain the total loss value. The model continuously updates the network weights by optimizing this loss function, so that the predicted flow field variables gradually approach the actual physical values.

[0103] S5. The trained neural network model is lightweighted to predict the flow field under different operating conditions and airfoil conditions.

[0104] The optimized Flow-Deeponet model can be deployed to computing devices and interface with database systems to perform rapid predictive inference for different airfoils under different flow conditions.

[0105] It should be noted that step S4 includes training hyperparameters through the Flow-Deeponet neural network model, minimizing the MSE prediction error, training the model constructed in step S3 to obtain the optimal model, and performing lightweight processing in step S5 refers to optimizing the number of model parameters through pruning.

[0106] Among them, the training hyperparameters specifically refer to the model training loss function using mean squared error (MSE), the optimizer being AdamW, the learning rate being 0.0001, and the use of early stopping during training to prevent overfitting.

[0107] Model pruning optimization specifically refers to the lightweighting of the Flow-Deeponet model after initial training using a structured pruning strategy. The specific steps include: determining neuron importance based on weight absolute values, removing low-importance neuron connections according to a preset pruning rate, fine-tuning the pruned network, and restoring model accuracy using a learning rate decay strategy.

[0108] Furthermore, during model training, the neural network model is trained in a distributed parallel manner based on the PyTorch deep learning framework, using a hardware configuration of four GPUs with 32GB of video memory and a 12-core Intel Xeon Platinum 8352V CPU. During training, mixed precision training is applied to reduce video memory usage, and dropout layers with a dropout rate of 0.2 are set in each layer of the network to prevent overfitting.

[0109] The AdamW optimizer is used to update the network parameters, with an initial learning rate of 1×10⁻⁶. -4 L1 regularization is introduced to improve the robustness of the model. The mean squared error (MSE) is used as the loss function to measure the difference between the model's predicted value and the target value. The model outputs pressure P and velocity components U and V at the same time. The prediction errors of the three components are weighted and summed to form the total loss value.

[0110] An early stopping strategy was set during training, and model checkpoints were saved every 5 epochs. After training for 100 epochs, convergence was achieved on the validation set.

[0111] After training, the importance of neurons is determined based on the absolute value of the weights. Low-importance neuron connections are removed according to a preset pruning rate. The pruned network is then fine-tuned and trained. The model accuracy is restored through a learning rate decay strategy, thereby achieving the structured and lightweight processing of the neural network model.

[0112] To verify the model's prediction accuracy and reliability, this embodiment further compares the error distribution between the model's output results and computational fluid dynamics (CAE) simulation results. Flow field data for different symmetrical airfoils of aircraft, including NACA0006, NACA0012, NACA0018, and NACA0024, were selected as research samples, with NACA0024 used as the test set. The dataset covers typical flow field simulation results for Mach numbers ranging from 0.2 to 0.5 and angles of attack ranging from 25° to 55°, including key flow field physical quantities such as pressure and velocity fields.

[0113] This embodiment is based on the integration and deployment of a pre-trained model, and performs flow field predictions for different operating conditions and airfoils. The prediction results are as follows: Figures 4a to 12c As shown, the error analysis compared with the CAE results is as follows: Figures 13 to 15 As shown in the figure. The results show that the average relative error of the neural operator network proposed in this invention remains at a low level under different airfoils and different operating conditions, and the overall error distribution is uniform with no obvious systematic bias. This indicates that the model can significantly improve the prediction efficiency while ensuring high accuracy, and can achieve rapid and stable prediction of complex unsteady flow fields, thus having strong engineering application value.

[0114] In summary, this method, based on the DeepONet network architecture, introduces multi-frequency Fourier position coding. It explicitly encodes complete spectral information from low-frequency global patterns to high-frequency local details using sine and cosine basis functions of different frequencies. Simultaneously, it employs a three-way parallel fusion strategy, concatenating branch features, backbone features, and their element-wise product. The element-wise product term explicitly introduces the nonlinear interaction between conditional parameters and spatial position. The fused feature vector undergoes deep nonlinear transformation through a multilayer perceptron, realizing the transformation from simple linear combination to complex nonlinear mapping. This method maintains accuracy while improving efficiency in flow field prediction, successfully solving the problems of high computational cost and long computation cycle in traditional CFD methods for unsteady flow field prediction. It has broad application prospects, particularly suitable for aircraft aerodynamic design, unsteady flow field modeling, and other engineering fields requiring efficient and accurate flow field prediction.

[0115] Example 2 Based on the same inventive concept, this embodiment provides a symmetric airfoil unsteady flow field prediction system based on deep neural operator networks. The principle of solving the problem is similar to that of the symmetric airfoil unsteady flow field prediction method based on deep neural operator networks, and the repeated parts will not be described again.

[0116] This embodiment provides a symmetric airfoil unsteady flow field prediction system based on deep neural operator networks, including:

[0117] The flow field dataset acquisition module is used to collect flow field datasets for different symmetrical airfoils; wherein, the flow field dataset includes CFD simulation data of different airfoils under different operating conditions;

[0118] The input / output dataset acquisition module is used to preprocess the flow field dataset and then construct the input / output dataset; wherein the input data includes static aerodynamic parameters and corresponding dynamic spatiotemporal coordinate features, and the output data includes pressure field and velocity field data;

[0119] A neural network model construction module is used to construct a neural network model, which includes a static branch network, a dynamic backbone network, and a fusion network. The static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space. The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of the spatiotemporal coordinates in the flow field.

[0120] A high-dimensional feature vector acquisition module is used to fuse the nonlinear features and the potential dynamic features to obtain a high-dimensional feature vector.

[0121] The output prediction value acquisition module is used to input the high-dimensional feature vector into the fusion network for nonlinear mapping and output the output prediction value representing the pressure field and velocity field.

[0122] The training module is used to train the neural network model based on the input and output dataset, using the mean squared error between the output predicted value and the target value as the loss function.

[0123] The flow field prediction module is used to perform lightweight processing on the trained neural network model in order to predict the flow field under different operating conditions and different airfoil conditions.

[0124] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0125] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0126] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0127] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0128] Finally, it should be noted that the above specific embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to examples, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for predicting unsteady flow field of a symmetric airfoil based on deep neural operator network, characterized in that, The method comprises the following steps: Collecting flow field data sets of different symmetrical airfoils; wherein the flow field data sets comprise CFD simulation data of different airfoils under different working conditions; After data preprocessing of the flow field data sets, input-output data sets are constructed; wherein the input data comprises static aerodynamic parameters and corresponding dynamic space-time coordinate features, and the output data comprises pressure field and velocity field data; A neural network model is constructed, which comprises a static branch network, a dynamic main network and a fusion network; the static aerodynamic parameters are taken as the input of the static branch network, and the output reflects the nonlinear features of the static aerodynamic parameters mapped to the latent space; the dynamic space-time coordinate features processed by Fourier position encoding are taken as the input of the dynamic main network, and the output represents the latent dynamic features of the space-time coordinates in the flow field; The nonlinear features and the latent dynamic features are fused to obtain a high-dimensional feature vector; The high-dimensional feature vector is input into the fusion network for nonlinear mapping, and the output prediction value representing the pressure field and the velocity field is output; The mean square error between the output prediction value and the target value is taken as the loss function, and the neural network model is trained based on the input-output data sets; The trained neural network model is subjected to lightweight processing to predict the flow field under different working conditions and different airfoil conditions.

2. The method of claim 1, wherein the method is based on a deep neural operator network for the prediction of unsteady flow fields over a symmetric airfoil. Collecting flow field data sets of different symmetrical airfoils, comprising: Collecting flow field data of different aircraft airfoils as the flow field data sets, and taking the flow field data of at least one of the aircraft airfoils as a validation set; wherein the flow field data sets comprise flow field simulation results with a Mach number range of 0.2-0.5 and an angle of attack range of 25°-55°, and the flow field simulation results comprise pressure field and velocity field variables; Based on the flow field data of different airfoils, grid division is performed, and flow field simulation is performed under different working conditions, wherein the working conditions include changes in Mach number, angle of attack and airfoil parameters; According to the simulation results, the pressure field and velocity field data near the airfoil are obtained.

3. The method of claim 1, wherein, After data preprocessing of the flow field data sets, input-output data sets are constructed, comprising: Data cleaning and abnormality processing are performed on the flow field data sets, and missing values are filled in by linear interpolation to ensure the integrity and continuity of the data; The preprocessed flow field data sets are normalized to map the value range of each physical quantity to the interval [-1, 1], Based on the principles of aerodynamics and data-driven analysis methods, the original static parameters in the flow field data sets are screened; the importance of each original static parameter is evaluated by using a random forest model, and finally the static aerodynamic parameters are selected, which include Mach number, angle of attack, airfoil geometric parameters including maximum thickness and maximum thickness position; The input and output are aligned with the coordinate axis and time index, the static aerodynamic parameters and dynamic space-time coordinate features are taken as the input, and the corresponding pressure field and velocity field components of the flow field are taken as the output to form the input-output data sets for the neural network model.

4. The method of claim 1, wherein, Using the static aerodynamic parameters as input to the static branch network, the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space, including: The static aerodynamic parameters are processed by the static branch network, and the output feature dimension is (B, 256), where B is the batch size. This output contains the nonlinear feature representation of the static aerodynamic parameters mapped to the latent space. The neural network model adopts the Flow-Deeponet neural network framework, and the static branch network includes multiple ResMLP modules; Each of the ResMLP modules includes: a first fully connected layer, a GELU activation function, a Dropout layer, a second fully connected layer, and residual connections; The input dimension of the static branch network is (B, 4), where B is the batch size and 4 is the dimension of the static aerodynamic parameters.

5. The method of claim 1, wherein, The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of spatiotemporal coordinates in the flow field, including: Fourier position encoding is performed on the dynamic spatiotemporal coordinate features; The location-encoded dynamic spatiotemporal coordinate features are used as input to the dynamic backbone network to enhance the neural network model's ability to perceive temporal features. The backbone network adopts an MLP architecture and performs deep feature extraction through multiple fully connected layers; the dynamic spatiotemporal coordinate feature input has a dimension of (B, 19), including two coordinate axes, a time index (x, y, t), and 16 position-encoded frequency features; B is the batch size. After processing by the backbone network, the output feature dimension is (B, 256), which represents the potential dynamic features of the spatiotemporal coordinates in the flow field. The Fourier position encoding of the dynamic spatiotemporal coordinate features includes: The Fourier position encoding uses sine and cosine transforms to encode spatiotemporal coordinates based on multiple frequencies, as shown in the following formula: encoded_coord(x) = [x, sin(2 i πx), cos(2 i πx), …] for I∈{0,1,…,N}; Where x represents the input coordinates; encoded_coord(x) represents Fourier position encoding; i represents different frequency levels; and N represents the maximum frequency index used.

6. The method of claim 1, wherein, The nonlinear features and the latent dynamic features are fused to obtain a high-dimensional feature vector, including: The outputs of the static branch network and the dynamic backbone network are fused by concatenation and element-wise feature product to generate a high-dimensional feature vector with dimension (B, 512). The high-dimensional feature vector is input into the fusion network for nonlinear mapping processing, and the final output dimension is (B,3), which is used to characterize the components of the pressure field and velocity field in the x and y directions of the flow field; The fusion network consists of three fully connected layers, which compress and map the high-dimensional feature vector layer by layer. The final output is constrained to the range of [-1,1] by the tanh activation function to ensure the normalization consistency of each physical quantity.

7. The method of claim 1, wherein, After nonlinearly mapping the high-dimensional feature vectors through the fusion network, the output predicted values ​​representing the pressure field and velocity field are generated, including: The concatenated high-dimensional feature vector is input into the fusion network for nonlinear mapping processing and compressed to the output predicted value; The fusion network comprises three fully connected layers. The first layer compresses the high-dimensional features from 512 dimensions to 256 dimensions. The second layer further compresses the 256 dimensions to 128 dimensions. The third layer outputs three physical quantities, corresponding to the pressure field, the velocity field in the x-direction, and the velocity field in the y-direction of the flow field, respectively. The final output dimension is (B,3), which is used to represent the pressure and velocity of the flow field, where B is the batch size.

8. The method of claim 1, wherein the method further comprises: The loss function is expressed as: MSE = 0.0001 ; wherein is the target value, is the output prediction value, and N is the number of samples.

9. The method of claim 1, wherein, The neural network model is trained based on the input and output datasets, and the trained neural network model is then subjected to lightweight processing, including: The neural network model is trained in a distributed parallel manner based on the PyTorch deep learning framework. During training, mixed precision training is applied to reduce memory usage, and dropout layers with a dropout rate of 0.2 are set in each network layer to prevent overfitting. The network parameters are updated using the AdamW optimizer with an initial learning rate of 1x10 -4 and L1 regularization is introduced to improve the robustness of the model. An early stopping strategy was set during training, and model checkpoints were saved every 5 epochs. After training for 100 epochs, convergence was achieved on the validation set. The trained neural network model is lightweighted using a structured pruning strategy. This includes determining the importance of neurons based on the absolute value of the weights, removing low-importance neuron connections according to a preset pruning rate, fine-tuning the pruned network, and restoring model accuracy through a learning rate decay strategy. This achieves the structured lightweighting of the neural network model.

10. A deep neural operator network based system for the prediction of unsteady flow fields over a symmetric airfoil, characterized in that, include: The flow field dataset acquisition module is used to collect flow field datasets for different symmetrical airfoils; wherein, the flow field dataset includes CFD simulation data of different airfoils under different operating conditions; The input / output dataset acquisition module is used to preprocess the flow field dataset and then construct the input / output dataset; wherein the input data includes static aerodynamic parameters and corresponding dynamic spatiotemporal coordinate features, and the output data includes pressure field and velocity field data; A neural network model construction module is used to construct a neural network model, which includes a static branch network, a dynamic backbone network, and a fusion network. The static aerodynamic parameters are used as input to the static branch network, and the output reflects the nonlinear characteristics of the static aerodynamic parameters mapped to the potential space. The dynamic spatiotemporal coordinate features processed by Fourier position encoding are used as input to the dynamic backbone network, and the output represents the potential dynamic features of the spatiotemporal coordinates in the flow field. A high-dimensional feature vector acquisition module is used to fuse the nonlinear features and the potential dynamic features to obtain a high-dimensional feature vector. The output prediction value acquisition module is used to input the high-dimensional feature vector into the fusion network for nonlinear mapping and output the output prediction value representing the pressure field and velocity field. The training module is used to train the neural network model based on the input and output dataset, using the mean squared error between the output predicted value and the target value as the loss function. The flow field prediction module is used to perform lightweight processing on the trained neural network model in order to predict the flow field under different operating conditions and different airfoil conditions.

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