PIV flow field super-resolution reconstruction method and system based on physical information neural network

Abstract

The invention discloses a PIV flow field super-resolution reconstruction method and system based on a physical information neural network, and belongs to the technical field of fluid mechanics experimental measurement. The method comprises the following steps: generating a training data set: solving a Navier-Stokes equation by adopting a numerical simulation method, and obtaining two-dimensional flow field data under different Reynolds numbers; constructing a multi-modal physical information neural network which comprises a composite network of four parts; training, testing and verifying the multi-modal physical information neural network; and reconstructing experimental data. According to the method, through fusion of physical information and deep learning, the problem of high-resolution flow field reconstruction under PIV hardware limitation is solved, and meanwhile, the method has the advantages of real-time performance, generalization and noise resistance.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of experimental measurement of fluid mechanics, and particularly relates to a PIV flow field super-resolution reconstruction method and system based on a physical information neural network. BACKGROUND

[0002] In order to better study and control fluid, high spatiotemporal resolution data is needed to accurately analyze and understand fluid dynamics. In recent years, with the improvement of hardware capability, the particle image velocimetry (PIV) system is widely used in experimental fluid mechanics research because it can obtain high temporal resolution data and does not interfere with the flow field. PIV is composed of an imaging system, tracer particles and an image processing system, in which a high-speed camera is an important component of the imaging system. However, the high-speed camera has the characteristic that the number of pixels and the frame rate are inversely proportional, that is, the higher the temporal resolution, the lower the spatial resolution. The resolution of the data obtained by PIV is limited by this characteristic, so it is very expensive to obtain high-resolution data in both time and space.

[0003] In the image field, some people use machine learning methods to realize the super-resolution reconstruction of images. Through prior art retrieval, Chinese invention patent No. CN112348745A, entitled "Video super-resolution reconstruction method based on residual convolutional network", reconstructs the video based on the residual convolutional neural network, which can reduce the training parameters while speeding up the model training speed and accuracy. Chinese invention patent No. CN114170085B, entitled "A spatiotemporal super-resolution real-time reconstruction method and system", can fuse two computationally intensive networks together and has good real-time performance, so that high frame rate and high resolution video output can be obtained in real time. In addition, Chinese invention patent No. CN116822326A, entitled "Two-dimensional explosion pressure field super-resolution reconstruction calculation method based on deep learning", can quickly and accurately calculate the two-dimensional super-resolution pressure field and extract the overpressure time history of any measuring point in the pressure field, greatly improving the calculation efficiency of numerical simulation of two-dimensional explosion pressure field. This patent applies machine learning methods to physical research. Chinese invention patent No. CN116843544A, entitled "Method, system and equipment for super-resolution reconstruction of hypersonic flow field by introducing convolutional neural network", uses the high-resolution data of the hypersonic vehicle flow field data obtained by flow field simulation simulation as the standard for verification and evaluation, and has the advantages of high calculation efficiency and high fineness. However, the following problems still exist: 1. Traditional interpolation methods lack physical consistency. For example, image super-resolution reconstruction methods such as bicubic interpolation (BiCubic) ignore the laws of fluid mechanics and reconstruct the flow field based solely on the spatial relationship of pixels. This leads to physically unreasonable phenomena such as velocity field distortion and blurred vortex structure in complex flow scenarios (such as vortex shedding and shear layers). 2. The risk of physical violations in purely data-driven models. Although super-resolution models based on convolutional neural networks improve reconstruction speed, as "black box" models, they rely heavily on simulation data training and are difficult to migrate to experimental scenarios (experimental noise causes a sharp drop in generalization). The output results may violate the law of conservation of mass / momentum (such as non-zero velocity divergence), requiring additional post-processing correction (increasing the computational burden).

[0004] 3. The physical information neural network (PINN) lacks adaptability. The existing PINN framework introduces the NS equation constraints in industrial scenarios, but has limitations in single mapping capabilities when directly applied to PIV super-resolution reconstruction. The high- and low-resolution mapping of the flow field contains linear and nonlinear characteristics. The single-branch structure of the traditional PINN makes it difficult to collaboratively model these two types of relationships, resulting in insufficient accuracy in complex flow reconstruction.

[0005] Compared with convolutional neural networks, physical neural networks use physical information to constrain neural networks so that the output of the model conforms to physical laws. The present invention provides a PIV data super-resolution reconstruction method based on physical information neural networks (PINN). PINN is applied in the field of fluid mechanics to achieve super-resolution reconstruction of experimental data rather than numerical simulation results. Summary of the Invention

[0006] Technical issues to be solved: To overcome the shortcomings of the existing technology, the present invention provides a PIV flow field super-resolution reconstruction method and system based on a physical information neural network. This system comprises a four-part composite network: an artificial neural network (ANN) for low-resolution data fitting, a second ANN for linear mapping, a third ANN for nonlinear mapping, and a fourth PINN module for physical constraints. By separating linear and nonlinear relationships, the system more accurately captures flow field scale characteristics. By integrating physical information with deep learning, the present invention solves the challenge of high-resolution flow field reconstruction within the limitations of PIV hardware, while also offering real-time, generalizable, and noise-resistant advantages.

[0007] The technical solution of the present invention is: a PIV flow field super-resolution reconstruction method based on physical information neural network, the specific steps are as follows: Generate training data sets: Use numerical simulation methods to solve the Navier-Stokes equations and obtain two-dimensional flow field data at different Reynolds numbers; Construct a multimodal physical information neural network, which consists of the following four parts: The first part ANN: input low-resolution time-coordinate information, and output velocity field fitting results; The second part ANN: input low-resolution velocity field and high-resolution time-coordinate information, and output predicted high-resolution velocity components through linear mapping without activation function; The third part ANN: input low-resolution velocity field and high-resolution time-coordinate information, and output predicted high-resolution velocity components through nonlinear mapping with tanh activation function; The fourth part PINN constraint module: based on the vorticity form of Navier-Stokes equation, a physical constraint is constructed, and the control equation is defined as:

[0008] In the formula, is the vorticity; is the velocity, is the time, is the Reynolds number, is the Nabla operator; Training, testing and verifying the multi-modal physical information neural network: the network parameters are initialized by using the Xavier method, the initial training is carried out by using the Adam optimizer to minimize the loss function, after a limited number of iterations, the L-BFGS optimizer is switched to further training, and the training is stopped after a limited number of iterations to obtain a trained model; the low-resolution experimental flow field data are input into the multi-modal physical information neural network, if the error meets the requirement, the model can be used offline, if the error does not meet the requirement, the network parameters are adjusted for continuous training; the verified multi-modal physical information neural network is verified through data of different Reynolds numbers to verify its generalization; Experimental data reconstruction: the low-resolution flow field obtained by PIV experiment is input into the trained multi-modal physical information neural network, and a high-resolution velocity field is output. A further technical scheme of the present application is that the numerical simulation adopts the lattice Boltzmann method (LBM) to solve the flow field, and the control equation expression is as follows:

[0009] In the formula, is the velocity, is the pressure, is the time, is the Reynolds number, is the Nabla operator.

[0010] A further technical solution of the application is: the first part ANN is constructed, and time vector and two-dimensional coordinate vector are received by the input layer and transmitted to the hidden layer; the number of layers of the hidden layer is set to 5 layers, each layer has 50 neurons, and full connection operation is performed between layers; and the output layer is mapped to a two-dimensional velocity field fitting result through a full connection layer.

[0011] A further technical solution of the application is: the second part ANN is constructed, and low-resolution velocity field and high-resolution time-coordinate information are received by the input layer and transmitted to a single hidden layer; the number of neurons of the hidden layer is 5, and the neurons are transformed through a linear mapping relationship without an activation function; and the output layer is mapped to a high-resolution velocity component through a full connection layer.

[0012] A further technical solution of the application is: the third part ANN is constructed, and low-resolution velocity field and high-resolution time-coordinate information are received by the input layer and transmitted to the hidden layer; the number of layers of the hidden layer is set to 7, each layer has 50 neurons, and the neurons are transformed through a nonlinear mapping relationship of a tanh activation function, and the output layer is mapped to a high-resolution velocity component through a full connection layer.

[0013] A further technical solution of the application is: the physical residual of the fourth part PINN constraint module is a numerical derivative R of a control equation, and the derivative format is And The expected value of the derivative format is 0, and the expression is as follows:

[0014] In the formula, is a partial derivative symbol, and are velocity components in and two directions, is a vorticity; is a vorticity component in direction, that is, a component in a plane.

[0015] A further technical solution of the application is: the loss function expression is as follows:

[0016] In the formula, is an error of an ANN used for fitting low-resolution data, and are an output of the network and a corresponding true value respectively, and the true value is low-resolution data; is an error of two ANNs used for fitting a relationship between low-resolution data and high-resolution data, and respectively are the output of the network and the corresponding true value, are high-resolution data; is the error of the physical constraint, and is the sum of the high-resolution-based output and the true value and ; is a regularization weight, is the total number of neurons in the entire network, and the total number of neurons is , then ; represents the number of neurons of the first part ANN, represents the sum of the number of neurons of the second part ANN and the third part ANN, represents the number of neurons of the fourth part PINN constraint module.

[0017] A further technical solution of the present application is that the adjustment strategy of the regularization weight is that when the data is very clean, the regularization term is not needed to reconstruct the data, at this time is set to 0; when the data has noise or the data is experimental data, the is set to 10 -5 , which can achieve effective noise resistance.

[0018] A further technical solution of the present application is that in the process of training, testing and verifying the multi-modal physical information neural network, the initial learning rate is set to 1x10 -3 , the number of iterations using the Adam optimizer is 5x10 4 , and the number of iterations using the L-BFGS optimizer is 5x10 4 .

[0019] A PIV flow field super-resolution reconstruction system based on a physical information neural network comprises: a numerical simulation module for generating a multi-Reynolds number flow field training data set; a network construction module for constructing a four-branch multi-modal physical information neural network; a reconstruction module for loading the trained multi-modal physical information neural network, inputting a low-resolution PIV flow field and outputting a high-resolution velocity field.

[0020] Advantages The present application provides a PIV flow field super-resolution reconstruction method and system based on a physical information neural network, which solves the core contradiction of "incompatible high temporal and spatial resolution" in PIV technology, and provides a high-precision, high-efficiency and strong-robustness reconstruction tool for fluid mechanics experimental research. Compared with the prior art, the present application has the following significant advantages: 1.The present application avoids the dependence on pressure data by using the vorticity form of the Navier-Stokes equation as the PINN constraint, ensures that the reconstructed flow field strictly satisfies the laws of fluid mechanics, solves the problem of vortex structure distortion caused by ignoring physical rules in traditional interpolation methods, and is more consistent with the super-resolution reconstruction requirements in physical problems.

[0021] 2.The multi-modal network structure (linear + nonlinear dual-branch ANN) respectively captures the explicit and implicit features of the flow field scale mapping, improves the reconstruction accuracy of complex flow, and is significantly better than the single-branch PINN model (comparing Figure 5 index). The trained method of the present application can be carried on a real-time processing platform to quickly perform super-resolution reconstruction on the flow field.

[0022] 3.The optimized network structure (fully connected layer + automatic differentiation) supports super-resolution reconstruction response, regularizes noise-resistant weights to effectively suppress experimental data noise, and overcomes the generalization failure problem caused by the sensitivity of pure data-driven models to noise.

[0023] 4.The present application does not require pressure data as any condition, and can perform super-resolution reconstruction on data without increasing additional measurement requirements. BRIEF DESCRIPTION OF DRAWINGS

[0024] Figure 1 is a flowchart of an embodiment of the present application; Figure 2 is a basic structure diagram of a physical information neural network; Figure 3 is a multi-modal physical information neural network structure diagram in an embodiment of the present application; Figure 4 is a specific implementation process in an embodiment of the present application; Figure 5 is a specific effect in simulation data and a comparison diagram with existing methods in an embodiment of the present application; Figure 6 is a specific effect diagram in experimental data of the present application. DETAILED DESCRIPTION

[0025] The embodiments described below with reference to the accompanying drawings are exemplary and are intended to explain the present application, and cannot be understood as a limitation of the present application.

[0026] Based on the problems of physical consistency loss in traditional interpolation methods, physical violation risk of pure data-driven models, and insufficient adaptability of physical information neural networks (PINN), the present application provides a PIV flow field super-resolution reconstruction method based on a physical information neural network, and the specific steps are as follows: Generating a training data set: solving the Navier-Stokes equation using a numerical simulation method to obtain two-dimensional flow field data under different Reynolds numbers; The multi-modal physical information neural network is constructed, and the composite network includes the following four parts: The first part ANN: input low-resolution time-coordinate information, and output velocity field fitting result; The second part ANN: input low-resolution velocity field and high-resolution time-coordinate information, and output predicted high-resolution velocity component through linear mapping without activation function; The third part ANN: input low-resolution velocity field and high-resolution time-coordinate information, and output predicted high-resolution velocity component through nonlinear mapping with tanh activation function; The fourth part PINN constraint module: a physical constraint is constructed based on the vorticity form of the Navier-Stokes equation, and the control equation is defined as:

[0027] In the formula, is the vorticity; is the velocity, is the time, is the Reynolds number, is the Nabla operator; Training, testing and verifying the multi-modal physical information neural network: the network parameters are initialized using the Xavier method, the Adam optimizer is used for initial training to minimize the loss function, and after a limited number of iterations, the L-BFGS optimizer is switched for further training, and the training is stopped after a limited number of iterations to obtain a trained model; the low-resolution experimental flow field data is input into the multi-modal physical information neural network, if the error meets the requirements, the model can be used offline, if the error does not meet the requirements, the network parameters are adjusted for further training; the verified multi-modal physical information neural network is verified for its generalization through data of different Reynolds numbers; Experimental data reconstruction: the low-resolution flow field obtained by PIV experiment is input into the trained multi-modal physical information neural network, and the high-resolution velocity field is output.

[0028] Specifically, the first part ANN is constructed, and the time vector and the two-dimensional coordinate vector are received by the input layer and transmitted to the hidden layer; the number of hidden layer is set to 5 layers, and each layer has 50 neurons, and full connection operation is performed between layers; the output layer is mapped to a 2D velocity field fitting result through a full connection layer.

[0029] Specifically, the second part ANN is constructed, and the low-resolution velocity field and the high-resolution time-coordinate information are received by the input layer and transmitted to a single hidden layer; the number of neurons in the hidden layer is 5, and the linear mapping relationship without activation function is transformed; the output layer is mapped to a high-resolution velocity component through a full connection layer.

[0030] Specifically, the third part ANN is constructed, receives the low-resolution velocity field and the high-resolution time-coordinate information through the input layer and transmits to the hidden layer; the number of hidden layer is set to 7, 50 neurons in each layer, and the output layer is mapped to the high-resolution velocity component through the full connection layer and the nonlinear mapping relationship of the tanh activation function.

[0031] The application also provides a PIV flow field super-resolution reconstruction system based on a physical information neural network, comprising: a numerical simulation module for generating a multi-Reynolds number flow field training data set; a network construction module for constructing a four-branch multi-modal physical information neural network; a reconstruction module for loading the trained multi-modal physical information neural network, inputting a low-resolution PIV flow field and outputting a high-resolution velocity field.

[0032] The above technical solutions are further analyzed in combination with the drawings and examples as follows: In one embodiment, referring to Figure 1 The embodiment of the application is a PIV flow field data super-resolution reconstruction method based on a physical information neural network, comprising the following steps: First, generate a training data set. Compared with experimental data, numerical simulation data is easy to obtain, so the method uses a numerical simulation method to obtain a data set for training the network. In terms of numerical simulation method, first, a solver is built with the lattice Boltzmann (LBM) method as the core, and then the solver is used to directly solve the Navier-Stokes equation, which is as follows:

[0033] wherein, is the velocity, is the pressure, is the time, is the Reynolds number, is the Nabla operator. Two-dimensional flow field data under various Reynolds numbers are obtained by the numerical simulation method, and the data are saved as a total data set for network training.

[0034] Second, for a traditional artificial neural network (ANN), the network uses the deviation of the predicted value as a loss function to achieve the training target. The physical information neural network (PINN) is essentially an ANN coupled with physical information, which can approximate the solution determined by data and partial differential equations by embedding the partial differential equation into the loss function, and uses two errors of data and derivatives together as the loss function. The basic structure is as shown in Figure 2 The numerical derivative can transform the partial differential Obtained by automatic differentiation method. For a given , The expectation value of is 0.

[0035] This embodiment introduces a multi-modal physical information neural network, which is a variant of PINN, and the network structure is as shown in Figure 3 Comparing Figure 2 It can be seen that the network structure of the existing PINN is composed of one ANN and one numerical derivative coding part, while the multi-modal physical information neural network introduced in the embodiment is divided into four parts. The first part is an ANN used to estimate low-resolution data, which inputs the time and coordinate information of the low-resolution data, and the network can output the velocity field. This part of ANN can learn the fitting relationship between the time and coordinates of the low-resolution data and the velocity; the second and third parts are two ANNs used to fit the relationship between the low-resolution and high-resolution data. One of the ANNs learns the linear relationship between the low-resolution and high-resolution data, and there is no activation function in this ANN, which means that the mapping between the data is linear. The other ANN can learn the nonlinear relationship between the low-resolution and high-resolution data by using the nonlinear activation function tanh function as the activation function. The input of the two ANNs is the low-resolution velocity information and the high-resolution time and coordinate data, and the output is the predicted high-resolution flow field data. The two ANNs divide the relationship between the data into linear and nonlinear parts, and then add them together to output the final high-resolution velocity field. The last part is the part shared with PINN, which is used to code the numerical derivative of the control equation . Through the above four parts, the network can realize multi-modal super-resolution reconstruction of the data.

[0036] Since it is difficult to obtain pressure in PIV experiments, the control equation in the network model in the present invention is the velocity-pressure mode Navier-Stokes equation:

[0037] Where is the vorticity. The derivative format is as follows:

[0038] As mentioned above, in PINN, the expectation value of the derivative format and is 0.

[0039] The loss function part of this embodiment, in addition to and In addition to the physical constraint, the ANN of the first part and the ANN of the second and third parts each have their own loss to be calculated. The three loss functions are added together to form the total loss function MSE used to minimize the difference between the predicted value and the true value, as follows:

[0040] In the formula, is the error of the ANN used to fit the low-resolution data, and are the output of the network and the corresponding true value, respectively, and the low-resolution data; is the error of the two ANNs used to fit the relationship between the low-resolution and high-resolution data, and are the output of the network and the corresponding true value, respectively, and the high-resolution data; is the error of the physical constraint, and are the sum of and based on the high-resolution output and the true value; is the regularization weight, then all the neurons that can be trained in the entire network, and the total number of neurons is , then ; represents the number of neurons of the first part ANN, represents the sum of the number of neurons of the second part ANN and the third part ANN, represents the number of neurons of the fourth part PINN constraint module.

[0041] Thirdly, the parameters of the neural network are initialized using the Xavier weight initialization method to avoid the problems of gradient explosion and gradient disappearance. The initial learning rate is set to 1x10 -3 . During the network training process, the Adam optimizer is first used for initial training to minimize the loss function, and the number of iterations is 5x10 4 . Then the L-BFGS optimizer is used for further training, and the number of iterations is 1.5x10 4 . The low-resolution experimental flow field data is input into the network to observe the accuracy of the model. If the error meets the requirements, the model can be used offline, and if the error does not meet the requirements, the parameters of the network are adjusted.

[0042] After the model is trained, the generalization of the network model is verified by data of different Reynolds numbers. The specific effects of the present application and the comparison with the traditional super-resolution method BiCubic interpolation method and the existing PINN model are as follows Figure 5The machine learning model is trained using data of Re=150, and reconstruction of data of higher Reynolds number can be realized.

[0043] In the fourth step, the performance of the model is verified using experimental data. An experimental environment with the same working condition as the simulation is built, and a PIV experiment is performed. Taking the flow around a cylinder as an example, the experiment is carried out on a towed water channel experimental platform. In order to ensure the accuracy and effectiveness of the experimental setup, the present application carries out multiple experiments under the same Reynolds number, and compares the data sampled at the centerline of the cylinder wake with the data in the published literature to ensure the accuracy of the experimental data.

[0044] Subsequently, the network model trained based on the simulation data is used to explore the application effect of the model in experimental data. Figure 6 The specific effect of the present application in experimental data is shown in the table. The results show that the present application has high prediction accuracy and fast prediction speed, and the predicted flow field conforms to the physical law.

[0045] Although the embodiments of the present application have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present application, and those skilled in the art can make changes, modifications, replacements and variations to the above embodiments without departing from the principles and purposes of the present application within the scope of the present application.

Claims

1. A PIV flow field super-resolution reconstruction method based on physical information neural network, characterized by The specific steps are as follows: Generate training data sets: Use numerical simulation methods to solve the Navier-Stokes equations and obtain two-dimensional flow field data at different Reynolds numbers; Construct a multimodal physical information neural network, which consists of the following four parts: The first part of ANN: input low-resolution time-coordinate information and output velocity field fitting results; The second part of the ANN: inputs the low-resolution velocity field and high-resolution time-coordinate information, and outputs the predicted high-resolution velocity component through linear mapping without activation function; The third part of the ANN: inputs the low-resolution velocity field and high-resolution time-coordinate information, and outputs the predicted high-resolution velocity component through the nonlinear mapping of the tanh activation function; The fourth part is the PINN constraint module: physical constraints are constructed based on the Navier-Stokes equations in the form of vorticity. The control equation is defined as: Where, is the vorticity; For speed, For time, is the Reynolds number, is the Nabla operator; The multimodal physical information neural network is trained, tested, and verified: the network parameters are initialized using the Xavier method, and the Adam optimizer is used for initial training to minimize the loss function. After a limited number of iterations, the L-BFGS optimizer is switched for further training. After a limited number of iterations, training is stopped to obtain a trained model. Low-resolution experimental flow field data is input into the multimodal physical information neural network. If the error meets the requirements, the model can be used offline. If the error does not meet the requirements, the network parameters are adjusted to continue training. After the multimodal physical information neural network passes the verification, its generalization is verified using data at different Reynolds numbers. Experimental data reconstruction: The low-resolution flow field obtained from the PIV experiment is input into the trained multimodal physical information neural network to output a high-resolution velocity field.

2. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 1, characterized in that: The numerical simulation uses the lattice Boltzmann method (LBM) to solve the flow field, and the governing equation is expressed as follows: Where, For speed, For pressure, It's time, is the Reynolds number, It is the Nabla operator.

3. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 1, characterized in that: The construction of the first part of the ANN receives the time vector and the two-dimensional coordinate vector through the input layer and passes them to the hidden layer; the number of hidden layers is set to 5, with 50 neurons in each layer, and fully connected operations are performed between layers; the output layer maps the 2D velocity field fitting results through the fully connected layer.

4. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 3, characterized in that: The construction of the second part of the ANN receives the low-resolution velocity field and high-resolution time-coordinate information through the input layer and transmits it to a single hidden layer; the number of neurons in the hidden layer is 5, which is transformed through a linear mapping relationship without an activation function; the output layer maps the high-resolution velocity component through a fully connected layer.

5. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 4, characterized in that: The construction of the third part of the ANN receives the low-resolution velocity field and high-resolution time-coordinate information through the input layer and transmits it to the hidden layer; the number of hidden layers is set to 7, with 50 neurons in each layer. Through the nonlinear mapping relationship transformation of the tanh activation function, the output layer maps the high-resolution velocity component through the fully connected layer.

6. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 5, characterized in that: The physical residual of the PINN constraint module in the fourth part is the numerical derivative R of the control equation, and the derivative format is and The expected value of is 0, and the expression is as follows: Where, is the symbol of partial derivative, and For speed exist and The components in two directions, is the vorticity; Vorticity exist Direction, that is The weight of the plane.

7. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 6, characterized in that: The loss function expression is as follows: Where, is the error of the ANN used to fit the low-resolution data, and are the output of the network and the corresponding true value, which are low-resolution data; is the error between the two ANNs used to fit the relationship between low-resolution and high-resolution data, and are the output of the network and the corresponding true value, which are high-resolution data; is the error of the physical constraint, and It is obtained based on high-resolution output and true value and of and; is the regularization weight, is all the trainable neurons in the entire network, and the total number of neurons is ,but ; represents the number of neurons in the first part of the ANN, represents the sum of the number of neurons in the second part ANN and the third part ANN, Indicates the number of neurons in the fourth part PINN constraint module.

8. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 7, characterized in that: The regularization weight The adjustment strategy is that when the data is very clean, no regularization term is needed to reconstruct the data. Set to 0; when the data is noisy or experimental data, Set to 10 -5 Effective noise immunity can be achieved.

9. The method for PIV flow field super-resolution reconstruction based on physical information neural network according to claim 1, characterized in that: During the training, testing, and verification of the multimodal physical information neural network, the initial learning rate is set to 1×10 -3 , the number of iterations using the Adam optimizer is 5×10 4 , the number of iterations using the L-BFGS optimizer is 5×10 4 .

10. A PIV flow field super-resolution reconstruction system based on a physical information neural network, used to implement the PIV flow field super-resolution reconstruction method based on a physical information neural network according to any one of claims 1 to 9; characterized in that: include: Numerical simulation module: used to generate multi-Reynolds number flow field training data set; Network construction module: used to construct a four-branch multimodal physical information neural network; Reconstruction module: load the trained multimodal physical information neural network, input the low-resolution PIV flow field and Output high-resolution velocity field.

Images