Data-driven cfd acceleration method based on multilayer perceptron and transfer learning

Abstract

This application discloses a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning, belonging to the field of interdisciplinary application technology of computational fluid dynamics and machine learning. The method includes: constructing a dataset containing grid coordinates, operating parameters, and flow field physical quantities using a CFD solver written in OpenFOAM based on the actual working conditions of the target flow field, and performing global normalization; training a multilayer perceptron neural network consisting of a 9-layer fully connected structure to establish a nonlinear mapping relationship between the operating parameters and the converged flow field; using the trained multilayer perceptron model to predict the flow field results of the target working condition, and using the prediction results as the initial field input to a traditional CFD solver for iterative solution to accelerate the computational convergence process; when facing new working conditions, loading the pre-trained model weights through a transfer learning strategy, freezing some hidden layers and adding an adaptation layer structure, and retraining the model with a small amount of new working condition data to further improve the model's generalization ability.

Description

Technical Field

[0001] This application relates to the field of interdisciplinary application technology of computational fluid dynamics and machine learning, and in particular to a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning. Background Technology

[0002] Computational fluid dynamics (CFD), as a core tool for studying fluid motion, has been widely applied in engineering fields such as aircraft aerodynamic design, engine internal flow analysis, and vehicle aerodynamic optimization. By numerically solving the Navier-Stokes (NS) or Euler equations, CFD can provide high-precision flow field information, offering crucial support for engineering design. However, traditional CFD methods face significant technical bottlenecks in practical engineering applications. First, typical CFD solutions require thousands or even tens of thousands of iterations to converge, especially in complex flows with high Reynolds numbers or shock waves, where the convergence process is even slower, resulting in long computation cycles and high resource consumption, making it difficult to support rapid iterative design and multi-condition optimization. Second, although hardware acceleration technologies such as GPU parallel computing and algorithm optimization methods such as improved time-progression schemes and multigrid methods can improve computational efficiency to some extent, the former typically requires expensive equipment, and the latter's acceleration effect under complex multi-parameter conditions remains limited, still failing to meet the rapid computational needs of engineering scenarios with wide parameter variations and frequent condition switching.

[0003] In recent years, machine learning methods have been gradually introduced into the field of CFD acceleration computing, aiming to establish a nonlinear mapping relationship between operating parameters and convergent solutions by learning flow field data. However, existing methods still have many key shortcomings. First, the generalization ability of most models is relatively limited, achieving high prediction accuracy only within the training operating conditions. When faced with new operating conditions where parameters such as Mach number and angle of attack exceed the coverage of the training data, the model output error may increase significantly, making it difficult to guarantee prediction reliability. Second, model training is costly. Adapting to new operating conditions often requires retraining the entire model, which is time-consuming and prone to overfitting, making it difficult to quickly respond to engineering design requirements. Third, existing methods often do not fully consider physical constraints such as mass, momentum, and energy conservation during prediction, resulting in prediction results that, while fitting well with the training data, may deviate from the basic flow laws, making it difficult to meet the physical consistency requirements of engineering applications. Fourth, current methods struggle to simultaneously achieve computational acceleration efficiency and solution accuracy in practical applications. When pursuing high speedup ratios, key flow field features such as shock wave location and separation zone structure are easily distorted, limiting their practical application in refined design.

[0004] Therefore, there is an urgent need to develop a flow field acceleration solution method that combines solution efficiency, prediction accuracy, and working condition generalization ability. Summary of the Invention

[0005] The purpose of this application is to address the problem that existing solution methods cannot simultaneously achieve solution efficiency, prediction accuracy, and working condition generalization ability. To solve the above technical problems, a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning is provided, which combines solution efficiency, prediction accuracy, and working condition generalization ability.

[0006] To achieve the above objectives, the embodiments of this application adopt the following technical solution: a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning, which is suitable for rapid numerical simulation of two-dimensional compressible flow fields across working conditions in engineering. The method is characterized by: using MLP to establish a nonlinear mapping relationship between working condition parameters and convergent flow field, introducing transfer learning to improve the generalization ability of unknown working conditions, and shortening the iteration number of the CFD solver through the optimized initial field. The steps are as follows: (1) Solve the two-dimensional Euler equations using a CFD solver written in OpenFOAM to generate a dataset containing grid coordinates, working condition parameters and convergent flow field physical quantities, and perform data processing. (1) Global linear normalization processing; (2) Construct an MLP model consisting of a 9-layer fully connected structure, with spatial coordinates, Mach number, and angle of attack as inputs and flow field physical quantities as outputs. Train the model using the Adam three-stage optimization strategy and establish a nonlinear mapping relationship between working condition parameters and converged flow field; (3) Use the trained MLP model to predict the target working condition flow field and use it as a new initial field. Input it into the traditional CFD solver to iterate until convergence, thereby shortening the number of iterations; (4) Load the weights of the pre-trained MLP model, freeze some hidden layers, add an adaptation layer and train it with a small amount of new working condition data to improve the model's adaptability to unknown working conditions.

[0007] Further, according to an embodiment of this application, the network structure of the LP neural network is as follows: it includes one input layer, eight hidden layers, and one output layer, wherein the input layer has four neurons, each hidden layer has 200 neurons, and the output layer has eight neurons; the activation function of this neural network is the Tanh function, whose mathematical expression is: .

[0008] Furthermore, according to the embodiments of this application, the operating condition range of the dataset used in step (1) is Ma=0.2~0.5, AoA=0°~6°; the flow field physical quantities included in the dataset are turbulent kinetic energy. Flow field velocity ,pressure Turbulent kinematic viscosity Specific dissipation energy ,density Turbulent thermal diffusivity The flow field velocity and density are normalized using reference values, and the formula is as follows: ,in For the reference data of the corresponding physical quantities, the other flow field physical quantities are normalized using maximum and minimum values, as shown in the formula: ,in and These are the global minimum and global maximum values ​​of the physical quantity in the sample data, respectively.

[0009] Further, according to the embodiments of this application, the three-stage optimization strategy adopted in step (2) is as follows: in the first stage, the Adam optimizer is used to train 50,000 times with a learning rate of 0.001; in the second stage, the Adam optimizer is used again to train 25,000 times with a learning rate of 0.0001; in the third stage, the learning rate is adjusted to 0.00001 and the training is continued for 25,000 times for high-precision fine-tuning; the loss function used in the training process is mean squared error (MSE).

[0010] Further, according to the embodiments of this application, in step (4), the transfer learning adopts the method of freezing the first 8 hidden layers of the original MLP neural network and adding 2 hidden layers (with 128 and 64 neurons respectively); the learning rate is set differentially during training, the frozen structural parameters are kept frozen or fine-tuned with a learning rate of 1e-5, and the learning rate of the newly added layer is 1e-3.

[0011] Furthermore, according to the embodiments of this application, the two-dimensional conserved Euler equations are spatially discretized using the finite volume method; the gradient terms are presented using the Gaussian linear scheme with a cellLimited limiter; the divergence terms of the momentum and energy equations are presented using the bounded Gaussian linear upwind limited scheme; the divergence terms of the turbulence equations are presented using the bounded Gaussian upwind scheme; and the Laplace terms are presented using the Gaussian linear corrected scheme. Since the steady-state flow field is being solved, the time derivative terms are discarded using the steady-state solution scheme, eliminating the need for time propagation. Bounded constraints and gradient limiting terms are also introduced to enhance the stability of the numerical calculation.

[0012] This invention's method boasts significant acceleration efficiency, high accuracy, strong generalization ability, excellent convergence stability, and outstanding engineering practicality. In CFD solving, the number of iterations can be reduced to as low as 35% of traditional methods, significantly shortening computation time without additional hardware upgrades. The MLP model's average prediction error for flow field physical quantities within the training set is no more than 0.88%, and the average error for the test set is no more than 4.23%, with the flow field distribution highly consistent with the CFD calculation results, meeting engineering accuracy requirements. Simultaneously, leveraging transfer learning strategies, the model can control the prediction error for new conditions outside the training set to within 5%, adapting to a wider range of conditions without retraining the entire model. This method maintains good convergence speed under different relaxation factors, effectively avoiding the slow convergence problem of traditional methods under low relaxation factors. Furthermore, it is compatible with solving two-dimensional compressible flow fields using structured meshes, requiring no modification to the core logic of traditional CFD solvers, and can be directly integrated into engineering design processes, lowering the barrier to entry and demonstrating strong engineering applicability. Attached Figure Description

[0013] The present application will be further described below with reference to the accompanying drawings and embodiments.

[0014] Figure 1 This is a flowchart of the CFD accelerated solution process of this application (comparison of the iterative process of the traditional method and the method of this invention).

[0015] Figure 2 This is a schematic diagram of the MLP neural network structure of this application.

[0016] Figure 3 This is a flowchart illustrating the transfer learning model structure and training process of this application.

[0017] Figure 4 This is a distribution diagram of flow field prediction errors under different operating conditions in this application.

[0018] Figure 5 This is a residual convergence curve of this application (comparison of residual descent trends between traditional CFD methods and the method of this invention). Detailed Implementation

[0019] To make the objectives, technical solutions, and advantages of the present invention clear and complete, the embodiments of the present invention will be further described in detail below with reference to the accompanying drawings. It should be understood that the specific embodiments described herein are only some, not all, embodiments of the present invention, and are merely illustrative of the embodiments of the present invention. They are not intended to limit the embodiments of the present invention. All other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] In the description of this invention, it should be noted that the terms "center," "middle," "upper," "lower," "left," "right," "inner," "outer," "top," "bottom," "side," "vertical," and "horizontal," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "a," "first," "second," "third," "fourth," "fifth," and "sixth" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0021] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0022] For purposes of simplicity and illustration, the principles of the embodiments are described primarily by way of example. In the following description, numerous specific details are set forth to provide a thorough understanding of the embodiments. However, it will be apparent to those skilled in the art that these embodiments may not be limited to these specific details in practice. In some instances, well-known methods and structures have not been described in detail to avoid unnecessarily obscuring these embodiments. Furthermore, all embodiments can be used in combination with each other. Example

[0023] To address the aforementioned shortcomings, such as Figure 1-5As shown, this embodiment provides a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning, which is suitable for fast numerical simulation of two-dimensional compressible flow fields across working conditions in engineering. Among them: MLP is used to establish the nonlinear mapping relationship between working condition parameters and convergent flow field, transfer learning is introduced to improve the generalization ability of unknown working conditions, and the number of iterations of CFD solver is shortened by the optimized initial field. The steps are as follows: (1) Solve the two-dimensional Euler equations using the CFD solver written in OpenFOAM to generate a dataset containing grid coordinates, working condition parameters and convergent flow field physical quantities, and perform global linear normalization on the data. (2) Construct a 9-layer fully connected MLP model, with spatial coordinates, Mach number, and angle of attack as inputs and flow field physical quantities as outputs. Use the Adam three-stage optimization strategy to train the model and establish a nonlinear mapping relationship between working condition parameters and converged flow field. (3) Use the trained MLP model to predict the target working condition flow field and use it as a new initial field. Input it into the traditional CFD solver to iterate until convergence, thereby shortening the number of iterations. (4) Load the weights of the pre-trained MLP model, freeze some hidden layers, add an adaptation layer and train it with a small amount of new working condition data to improve the model's adaptability to unknown working conditions.

[0024] Further, according to an embodiment of this application, the network structure of the LP neural network is as follows: it includes one input layer, eight hidden layers, and one output layer, wherein the input layer has four neurons, each hidden layer has 200 neurons, and the output layer has eight neurons; the activation function of this neural network is the Tanh function, whose mathematical expression is: .

[0025] Furthermore, according to the embodiments of this application, the operating condition range of the dataset used in step (1) is Ma=0.2~0.5, AoA=0°~6°; the flow field physical quantities included in the dataset are turbulent kinetic energy. Flow field velocity ,pressure Turbulent kinematic viscosity Specific dissipation energy ,density Turbulent thermal diffusivity The flow field velocity and density are normalized using reference values, and the formula is as follows: ,in For the reference data of the corresponding physical quantities, the other flow field physical quantities are normalized using maximum and minimum values, as shown in the formula: ,in and These are the global minimum and global maximum values ​​of the physical quantity in the sample data, respectively.

[0026] Further, according to the embodiments of this application, the three-stage optimization strategy adopted in step (2) is as follows: in the first stage, the Adam optimizer is used to train 50,000 times with a learning rate of 0.001; in the second stage, the Adam optimizer is used again to train 25,000 times with a learning rate of 0.0001; in the third stage, the learning rate is adjusted to 0.00001 and the training is continued for 25,000 times for high-precision fine-tuning; the loss function used in the training process is mean squared error (MSE).

[0027] Further, according to the embodiments of this application, in step (4), the transfer learning adopts the method of freezing the first 8 hidden layers of the original MLP neural network and adding 2 hidden layers (with 128 and 64 neurons respectively); the learning rate is set differentially during training, the frozen structural parameters are kept frozen or fine-tuned with a learning rate of 1e-5, and the learning rate of the newly added layer is 1e-3.

[0028] Furthermore, according to the embodiments of this application, the two-dimensional conserved Euler equations are spatially discretized using the finite volume method; the gradient terms are presented using the Gaussian linear scheme with a cellLimited limiter; the divergence terms of the momentum and energy equations are presented using the bounded Gaussian linear upwind limited scheme; the divergence terms of the turbulence equations are presented using the bounded Gaussian upwind scheme; and the Laplace terms are presented using the Gaussian linear corrected scheme. Since the steady-state flow field is being solved, the time derivative terms are discarded using the steady-state solution scheme, eliminating the need for time propagation. Bounded constraints and gradient limiting terms are also introduced to enhance the stability of the numerical calculation.

[0029] Specifically, this invention provides a data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning. By constructing a nonlinear mapping relationship between operating parameters and the convergent flow field, it provides an optimized initial flow field for traditional CFD solvers. Furthermore, it combines transfer learning to enhance cross-operating condition adaptability, thereby accelerating the CFD calculation process. The specific steps are as follows: 1. Dataset Construction: (1) Using the CFD solver written in OpenFOAM, the steady-state two-dimensional viscous compressible fluid Navier-Stokes equations are solved for the target flow field (such as the flow around the NACA0012 airfoil) within the preset working conditions (Mach number 0.2-0.5, angle of attack 0°-6°); (2) Generate computational grid coordinates, operating parameters (Mach number Ma, angle of attack AoA) and convergent flow field physical quantities (turbulent kinetic energy). Flow field velocity ,pressure Turbulent kinematic viscosity Specific dissipation energy ,density Turbulent thermal diffusivity The dataset, which covers 49 working conditions, is divided into training and testing sets in a 40:9 ratio. (3) Perform global linear normalization on the dataset to eliminate dimensional differences. The normalization formulas for flow field velocity and density are: ,in For reference, the normalization formulas for the other flow field physical quantities are: ,in , These are the global minimum and maximum values ​​of the sample data, respectively.

[0030] The steps for CFD solver configuration and dataset generation are as follows: Solving the two-dimensional conserved Euler equations, the gradient terms are solved using a Gaussian linear scheme with cellLimited constraints, the divergence terms of the momentum and energy equations using a bounded Gaussian linear upwind limited scheme, the divergence terms of the turbulence equations using a bounded Gaussian upwind scheme, and the Laplace terms using a Gaussian linear corrected scheme. The solution uses a steady-state scheme, discarding the time derivative terms and eliminating the need for time propagation. Bounded constraints and gradient constraints are introduced to enhance numerical stability.

[0031] The mesh type is a quadrilateral structured mesh. Near-wall mesh refinement is performed for the NACA0012 airfoil. The boundary conditions are set as free-flow boundaries and the far-field velocity is set using the freestreamVelocity format. The wall boundary is subject to no-slip constraints using the noSlip format.

[0032] Seven Mach numbers (0.20, 0.25, ..., 0.50) and seven angles of attack (0°, 1°, ..., 6°) were set, resulting in a total of 49 working conditions. The residuals of each flow field physical quantity were solved to be lower than a preset threshold. The grid coordinates, working condition parameters, and flow field physical quantities (u, v, ρ, p, etc.) were recorded to generate a dataset.

[0033] Residual threshold 1e-6 1e-5 1e-5 1e-5 1e-5 Global normalization is performed on the input features (x, y, Ma, AoA) and the output physical quantities (u, v, ρ, p, etc.), and the normalization parameters are saved. , , , ).

[0034] 2. MLP Neural Network Training: (1) Construct an MLP neural network consisting of 9 fully connected layers. The input layer contains 4 neurons (spatial coordinates x, y, Mach number Ma, angle of attack AoA), the output layer contains 8 neurons (corresponding to 8 flow field physical quantities), and the middle 8 hidden layers each contain 200 neurons. The Tanh activation function is used to enhance the nonlinear modeling capability. The specific network structure is as follows: input layer (4 neurons) → 8 hidden layers (200 neurons each, Tanh activation function) → output layer (8 neurons).

[0035] (2) During the training process, the mean squared error is used as the loss function and the Adam optimizer is used. First, the initial training is performed by iterating 50,000 times with a learning rate of 0.001. Then, the learning rate is reduced to 0.0001 and iterated for 25,000 times to refine the training. Finally, the learning rate is further reduced to 0.00001 and iterated for 25,000 times to perform high-precision fine-tuning. The mean squared error is used as the loss function for high-precision fine-tuning. (3) After training, save the model weights and normalization parameters for subsequent flow field prediction.

[0036] Furthermore, the model was validated by using a test set (9 working conditions) to evaluate the prediction accuracy, ensuring that the average error of each physical quantity does not exceed 4.23% and the maximum error does not exceed 9.35%.

[0037] 3. CFD accelerated solution process: (1) For the target working conditions Ma=0.6, AoA=0° and Ma=0.375, AoA=2.5°, the working condition parameters and grid coordinates are input into the trained MLP model to predict the approximate convergent flow field, and the prediction result is used as the initial flow field of the CFD solver. (2) The initial field data predicted by the MLP model is input into the OpenFOAM solver. The finite volume method is used for spatial discretization, and a steady-state solution scheme is used to discard the time derivative term, thus eliminating the need for time advancement. At the same time, bounded constraints and gradient limiting terms are introduced to enhance numerical stability. The solution is iterated until the residuals of each flow field physical quantity are lower than the preset threshold. (3) The new initial field predicted by the MLP model can skip the initial redundant iteration steps of the original initial field and shorten the solution time. The iteration relaxation factor of velocity U can be set to 0.5, 0.6 or 0.7. Under the influence of different relaxation factors, the solution can still ensure fast convergence.

[0038] Three types of working conditions were selected for verification: working conditions within the training set (Ma=0.5, α=4°, Ma=0.45, α=0°), unknown working conditions within the training set (Ma=0.375, α=2.5°), and working conditions outside the training set (Ma=0.6, α=0°). For these target operating conditions, the flow field is predicted by the MLP model and used as the new initial field. The result is then input into the CFD solver for iterative solution. The number of iterations is counted and compared with the solution results of the traditional uniform initial field. The accelerated model has the fewest iterations, which is 35% of the traditional method.

[0039] 4. Transfer learning generalization optimization: (1) For high-precision flow field data of unknown working conditions outside the training set obtained by traditional CFD solvers, the weights of the pre-trained MLP model are loaded and the first 8 layers of the general feature extraction layer responsible for learning the global physical laws of the flow field are frozen. (2) Two hidden layers are added on the basis of the original model structure, with 128 and 64 neurons respectively, and the output layer structure is retained. A layered training strategy is adopted: the parameters of the original frozen layer structure are kept frozen or fine-tuned with a small learning rate of 1e-5, and the new layer is trained with a larger learning rate of 1e-3, thereby reducing the model training cost and improving the training efficiency. (3) Generate a small amount of CFD converged flow field data for new working conditions, combine it with the original training set to construct an extended dataset, and keep the data normalization scale consistent. Train the transfer learning model on this extended dataset to improve the model's prediction accuracy for unknown working conditions.

[0040] Load the weights of the pre-trained MLP model, freeze the first 8 hidden layers, add 2 new hidden layers (128 and 64 neurons, Tanh activation) and retain the original output layer structure; Generate CFD converged flow field data for new operating conditions (Ma=0.6, α=0°, Ma=0.375, α=2.5°), add them to the original training set to construct an expanded training set, and use hierarchical learning rate training, that is, keep the structural parameters of the original frozen layer frozen or fine-tune them with a learning rate of 1e-5, and train the new layer with a learning rate of 1e-3. After training, save the transfer learning model and verify its prediction accuracy and acceleration effect for new working conditions, ensuring that the prediction error is less than 5%.

[0041] The experimental results show that when the initial field data is solved before and after prediction, the number of iterations is reduced by up to 65% when U_RF = 0.6 and by up to 49% when U_RF = 0.7. The average L2 error of the flow field physical quantities is less than 0.72%. At the same time, when comparing the number of iterations of the transfer learning model with the original model, the number of iterations of the transfer learning model is reduced by 20%, and the prediction error is no more than 1.73%. The transfer learning training on the CPU takes only 53 minutes, which is much less than the more than 200 hours required to retrain the complete model, and the model training efficiency is significantly improved.

[0042] The method of this invention possesses significant acceleration efficiency, high accuracy, strong generalization ability, excellent convergence stability, and outstanding engineering practicality. In CFD solving, the number of iterations can be reduced to as low as 35% of traditional methods, significantly shortening computation time without additional hardware upgrades; the MLP model averages the predicted physical quantities of the flow field under working conditions within the training set. The error does not exceed 0.88%, and the average value of the test set under operating conditions is [not specified]. The error is no more than 4.23%, and the flow field distribution is highly consistent with the CFD calculation results, meeting engineering accuracy requirements. Furthermore, by leveraging transfer learning strategies, the model can control the prediction error for new operating conditions outside the training set to within 5%, adapting to a wider range of operating conditions without requiring retraining the entire model. This method maintains good convergence speed under different relaxation factors, effectively avoiding the slow convergence problem of traditional methods under low relaxation factors. It is also compatible with solving two-dimensional compressible flow fields using structured grids, requiring no modification to the core logic of traditional CFD solvers, and can be directly integrated into engineering design processes, lowering the barrier to entry and demonstrating strong engineering applicability.

[0043] Although the illustrative specific embodiments of this application have been described above to enable those skilled in the art to understand this application, this application is not limited to the scope of the specific embodiments. For those skilled in the art, all applications utilizing the concept of this application are protected as long as various variations are within the spirit and scope of this application as defined and determined by the appended claims.

Claims

1. A data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning, applicable to rapid numerical simulation of two-dimensional compressible flow fields across working conditions in engineering, characterized by: A nonlinear mapping relationship between operating parameters and the convergent flow field is established using MLP. Transfer learning is introduced to improve the generalization ability for unknown operating conditions. The number of iterations of the CFD solver is shortened by using the optimized initial field. The steps are as follows: (1) Solve the two-dimensional Euler equations using the CFD solver written in OpenFOAM to generate a dataset containing grid coordinates, working parameters and convergent flow field physical quantities, and perform global linear normalization on the data. (2) Construct an MLP model consisting of a 9-layer fully connected structure, with spatial coordinates, Mach number, and angle of attack as inputs and flow field physical quantities as outputs. Use the Adam three-stage optimization strategy to train the model and establish the nonlinear mapping relationship between the working parameters and the convergent flow field. (3) The trained MLP model is used to predict the target working condition flow field and is used as the new initial field. The traditional CFD solver is then input to iterate until convergence, thereby shortening the number of iterations. (4) Load the pre-trained MLP model weights, freeze some hidden layers, add an adaptation layer and train it with a small amount of new working condition data to improve the model's adaptability to unknown working conditions.

2. The data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning according to claim 1, characterized in that, The MLP neural network has the following structure: it consists of one input layer, eight hidden layers, and one output layer. The input layer has four neurons, each hidden layer has 200 neurons, and the output layer has eight neurons. The activation function of this neural network is the Tanh function, whose mathematical expression is: .

3. The data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning according to claim 1, characterized in that, The operating conditions of the dataset used in step (1) are Ma = 0.2~0.5 and AoA = 0°~6°; The dataset contains flow field physical quantities of turbulent kinetic energy. Flow field velocity ,pressure Turbulent kinematic viscosity Specific dissipation energy ,density Turbulent thermal diffusivity ; The flow field velocity and density are normalized using reference values, and the formula is as follows: ,in For the reference data of the corresponding physical quantities, the other flow field physical quantities are normalized using maximum and minimum values, as shown in the formula: ,in and These are the global minimum and global maximum values ​​of the physical quantity in the sample data, respectively.

4. The data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning according to claim 1, characterized in that, The three-stage optimization strategy used in step (2) is as follows: In the first stage, the Adam optimizer is used to train 50,000 times with a learning rate of 0.001; in the second stage, the Adam optimizer is used again to train 25,000 times with a learning rate of 0.0001; in the third stage, the learning rate is adjusted to 0.00001 and the training is continued for 25,000 times for high-precision fine-tuning. The loss function used during training is mean squared error (MSE).

5. The data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning according to claim 1, characterized in that, In step (4), the transfer learning adopts the method of freezing the first 8 hidden layers of the original MLP neural network and adding 2 new hidden layers (with 128 and 64 neurons respectively); the learning rate is set differentially during training, the structural parameters are kept frozen or fine-tuned with a learning rate of 1e-5, and the learning rate of the newly added layers is 1e-3.

6. The data-driven CFD acceleration solution method based on multilayer perceptron and transfer learning according to claim 1, characterized in that, The two-dimensional conserved Euler equations are spatially discretized using the finite volume method. The gradient terms are represented by the Gauss-linear scheme with a cellLimited limiter. The divergence terms of the momentum and energy equations are represented by the boundedGauss-linearUpwind limited scheme, the divergence terms of the turbulence equations are represented by the boundedGauss-upwind scheme, and the Laplace terms are represented by the Gauss-linear corrected scheme. To solve the steady-state flow field, the steady-state solution scheme is used, discarding the time derivative terms and eliminating the need for time propagation. Bounded constraints and gradient limiting terms are introduced to enhance the stability of the numerical calculation.

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