Small-scale turbulent flow field reconstruction method based on physical information neural network

Abstract

The invention discloses a small-scale turbulent flow field reconstruction method based on a physical information neural network. According to the method, small-scale and high-resolution velocity field and pressure distribution can be reconstructed from sparse turbulent flow field data. According to the method, firstly, residual points and data frames are extracted from a computational domain, and a data set is constructed and sparsification is carried out on the data set; secondly, performing maximum and minimum normalization preprocessing on input and output data of the model, and performing linear transformation in combination with a calculus method and a Navier-Stokes equation so as to apply physical constraints; secondly, turbulence data are input into an improved PINN framework for flow field reconstruction, and a predicted speed and a predicted pressure field are obtained; and finally, fitting real flow field data by using an automatic differential technology and a minimization loss function. According to the scheme, the expansibility of the model is improved, the hydrodynamic equation can be rapidly solved from sparse data, the small-scale structure in the flow field is effectively captured, and reconstruction of high-precision flow field information is achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of high-resolution reconstruction of turbulent flow fields and deep learning, and specifically relates to a method for reconstructing small-scale turbulent flow fields based on physics-informed neural networks. Background Art

[0002] Due to the limitations of measurement techniques, it is difficult to obtain complete fluid data in traditional fluid mechanics experiments, and usually only some sparse velocity information can be obtained. However, in the engineering field, it is very necessary to understand the physical characteristics and physical mechanisms of the flow field. Therefore, it is very important to obtain the flow field information of the complete fluid from sparse data.

[0003] The Navier-Stokes equation in the control equations of fluid mechanics is a typical nonlinear partial differential equation that can describe the motion law of fluids. By solving the fluid mechanics equations, the physical characteristics of fluids can be obtained. Currently, common numerical methods for solving partial differential equations include the finite element method and the finite volume method, etc. These methods are widely used in various fields due to their high accuracy. However, these methods usually require a large amount of computing time and high costs, and will face challenges of low efficiency in practical applications.

[0004] With the continuous progress of machine learning technology, deep learning has been widely applied in many fields such as natural language processing, weather prediction, bioinformatics, and fluid mechanics. In recent research, physics-informed neural networks (PINN) as an emerging deep learning method has shown remarkable effects in solving the forward and inverse problems of partial differential equations. PINN can not only learn data but also satisfy potential physical laws. Due to the control equations embedded in PINN, it can also show advantages in the absence of observable data. However, the model scalability of PINN based on fully connected networks is limited and the accuracy is low. Compared with fully connected PINN, convolutional neural network (CNN) has been increasingly applied in PINN due to its advantages such as parameter sharing, spatial feature extraction, low inference cost, and high scalability. Currently, there have been studies applying physics-informed networks based on CNN to the solution of fluid mechanics control equations and successfully modeling fluids. Since traditional CNN needs to transform the input training data into a regular matrix, this often leads to the loss of detailed information, which may cause the direct coupling of differential equation residuals, resulting in a reduction in the accuracy of the flow field reconstruction of the model of PINN based on CNN.

[0005] In summary, there are certain limitations in the existing methods for reconstructing flow field information from sparse data, mainly reflected in aspects such as insufficient model scalability, limited computational accuracy, and coupling problems of residual equations. The existence of these problems makes it difficult to reconstruct high-quality flow fields in practical applications. To address the above challenges, the present invention proposes a physical convolutional information neural network model based on feature fusion (FFPICN). This model can significantly improve the scalability of the model, enabling it to quickly solve the hydrodynamic equations from sparse data and effectively capture small-scale structures in the flow field, thereby realizing the reconstruction of high-precision flow field information. Summary of the Invention

[0006] To solve the problems of still insufficient accuracy and poor scalability in the current methods for solving hydrodynamic equations, the present invention provides a method for reconstructing small-scale turbulent flow fields based on a physics-informed neural network to improve the accuracy and scalability of the simulated flow field to a certain extent. To achieve the above object, the present invention provides the following technical solutions:

[0007] Step 1: Select data frames and residual points for training within the computational domain to construct a data set;

[0008] Step 2: Perform data point sparsification processing;

[0009] Step 3: Perform data preprocessing, and at the same time combine the hydrodynamic control equations and calculus techniques for equation transformation;

[0010] Step 4: Construct the FFPICN model according to the improved physics-informed neural network, input the spatial coordinates and time coordinates of the training data, and output the corresponding flow field data through the neural network to establish a hydrodynamic model;

[0011] Step 5: Use the automatic differentiation technique to calculate the residuals of the hydrodynamic control equations;

[0012] Step 6: Calculate the control equation loss and data loss, and obtain the loss function in the form of mean square error.

[0013] Further, in Step 1, the Latin hypercube sampling method is used to extract residual points from the computational domain within the computational region; in the forced isotropic turbulence data set, 200 velocity data frames with a resolution of 128×128×3 and 200 pressure-velocity frames are extracted from the 1024 3 periodic grid in the domain of [0, 2π] 3 Only the data frames with only velocity information and 1 million residual points generated by Latin hypercube sampling are used as the training set, and the data frames including velocity and pressure are used as the validation set.

[0014] Further, the method adopted in Step 2 is a sparsification method based on coordinate selection, which reduces the data volume by selecting some data point coordinates in the original data; coordinates are selected according to the ratios of 100%, 50%, 25% and 1%, and corresponding data subsets are extracted according to the selected coordinates, and a sparse data set is jointly constructed with the sparse data points selected at different ratios and the residual points generated by Latin hypercube sampling.

[0015] Further, the preprocessing method in Step 3 is to normalize the input and output by using the maximum-minimum normalization; the dependent and independent variables ψ = [x, y, z, t, u, v, w, p] in the dimensionless Navier-Stokes equations T are respectively applied with the maximum-minimum normalization; the formula of the incompressible dimensionless Navier-Stokes equations is defined as:

[0016] E m = u x + v y + w z

[0017]

[0018] where E m is the continuity equation, and E x , E y , E z are the momentum equations, x, y and z are the spatial coordinates, t is the time coordinate, u, v, w are the three components of the velocity, namely the flow velocity, the spanwise velocity and the vertical velocity respectively, and p is the pressure; u x , v x , w x , p x , u y , v y , w y , p y , u z , v z , w z and p z are respectively the first-order partial derivatives of the three velocity components and the pressure with respect to the spatial coordinates x, y and z, and u t , v t and w t are the first-order partial derivatives of the velocity with respect to the time coordinate t, and u xx , u yy , u zz , v xx , v yy , v zz , w xx , w yy and w zzare the second-order partial derivatives of the three velocity components with respect to the spatial coordinates x, y, and z, and Re is the Reynolds number.

[0019] Calculate the maximum value ψ calculated separately according to the training set max = [x max , y max , z max , t max , u max , v max , w max , p max T and the minimum value ψ min = [x min , y min , z min , t min , u min , v min , w min , p min T , Map the data to the range [0, 1] through max-min normalization to obtain the normalized variable ψ * = [x * , y * , z * , t * , u * , v * , w * , p * T , The max-min normalization formula is defined as follows:

[0020]

[0021] Where X * represents the normalized spatial coordinates (x * , y * , z * ) and the time coordinate t * , X min and X max represent the minimum and maximum values of X respectively; in addition, while normalizing the data, the Navier-Stokes equation is transformed by combining calculus techniques, and the formula of its calculus technique is defined as follows:

[0022]

[0023] Substitute the transformed variables into the Navier-Stokes equation through calculus techniques, and the transformed Navier-Stokes formula is:

[0024] ​​​

[0025] Among them and Δu, Δv, and Δw are the differences between the maximum and minimum values of the velocity in three directions, Δp is the difference between the maximum and minimum values of the pressure, and Δx, Δy, Δz, and Δt respectively represent the differences between the maximum and minimum values of the time coordinate and the spatial coordinates, and are respectively the first-order and second-order partial derivatives of the output velocities u * , v * , w * and the pressure p * with respect to the spatial coordinates (x * , y * , z * ) of the residual points and the time coordinate t * * are respectively the transformed continuity equation and momentum equation.

[0026] Furthermore, the basic structure of the neural network in Step 4 is a model constructed on the Pycharm platform using the Python programming language based on the PyTorch deep learning framework; the model consists of two main modules: a feature extraction module and a physical information module. Among them, the feature extraction module focuses on the deep extraction of fluid data and the fusion of multi-level non-linear information of complex fluids, and realizes the extraction of fluid small-scale structures through the combination of convolution and feature fusion; the physical information module satisfies physical constraints through automatic differentiation and minimization of losses; the feature extraction module first extracts the shallow fluid representation from the preprocessed data, and uses three 1×1 convolutions in parallel to perform shallow feature extraction. The definition formula of its 1×1 convolution structure is expressed as:

[0027] z = F(X * , W, b)

[0028] where the input X * of the convolution represents the normalized spatial coordinates (x * , y * , z * ) and the time coordinate t * , W and b are respectively the weight and bias term of the convolution kernel, F(·) is the mapping function, and z is the output of the convolution layer; z is non-linearly transformed through the adaptive swish activation function, and its formula is defined as:

[0029] f Swish (z) = z·σ(βz)

[0030] where f Swish (z) represents the output of the adaptive Swish activation function, is the Sigmoid activation function, β ∈ R is a trainable parameter that can be adjusted through training; the shallow feature representations generated by passing through 3 parallel 1×1 CNN layers and the adaptive swish activation function are fused to construct a multi-level fluid feature set, and the feature fusion formula is defined as:

[0031] a3 = w·a1+(1 - w)·a2

[0032] where a1, a2, and w are the feature representations obtained through convolution and the adaptive activation function respectively, and w and (1 - w) are the weights for fusing the feature representations a1 and a2; the fused feature representation a3 is used to deeply extract fluid features using a convolutional fusion (FCNN) module; the FCNN module is mainly composed of a combination of 1×1 convolution and a feature fusion layer, and by stacking K FCNN modules, the deep feature representation of the input data is gradually learned; the output of the Kth FCNN module is defined as:

[0033] w k+1 = f Swish (F(a k+2 , W k , b k ))

[0034] a k+3 = w k+1 a1+(1 - w k+1 )a2

[0035] where w k+1 represents the new feature fusion weight obtained by the Kth FCNN module, W k and b k are the weights and biases of the convolution in the kth FCNN module, a k+2 and a k+3 are the input and output of the Kth FCNN module respectively, and then they are compressed and processed through 1×1 convolution to predict the velocity and pressure of the fluid;

[0036] Furthermore, in the fourth step, the input is composed of spatial coordinates (x * , y * , z * ) and time coordinate t * , and the output data stream includes the flow velocity u * , spanwise velocity v * , vertical velocity w * and pressure p * .

[0037] Furthermore, in the fifth step, the partial derivative calculation can be performed using the automatic differentiation technique in the physical information module, and the output velocities u * , v * , w* and the pressure p * Regarding the first-order partial derivatives with respect to the spatial coordinates (x * , y * , z * ) and the time coordinate t * The first-order partial derivatives and the second-order partial derivatives The partial derivatives can be calculated using the torch.autograd.grad() function in PyTorch. Furthermore, the loss in step six includes the data loss and the control equation loss in the physical information module. The data loss is the root mean square error (MSE) between the velocity obtained in step four and the given data; the control equation loss is the root mean square error of the residuals of the continuity equation and the momentum equation; the loss is defined as follows:

[0038] L

[0039] L data = MSE(u pre * , u * ) + MSE(v pre * , v * ) + MSE(w pre * , w * )

[0040]

[0041] L = L data + αL eqn

[0042] where L data is to measure the error between the predicted velocity components u pre * , v pre * and w pre * and the true values u * , v * and w * , L eqn reflects the control equation residuals and is used to describe the deviation degree of the network output from satisfying the control equation and ; L is the total loss value, and the weight coefficient α in the loss function is 1. Finally, the original variables are restored through the non-normalization process to obtain the final predicted values of velocity and pressure, and its formula is defined as:

[0043] ψ = ψ * ⊙ (ψ max - ψ min ) + ψmin

[0044] Among them, ⊙ represents pointwise element multiplication.

[0045] A computer-readable storage medium stores a computer program, wherein the computer program, when executed by a processor, implements the steps of the method according to any one of claims 1-8.

[0046] The beneficial effects of the present invention are as follows:

[0047] 1. A method for reconstructing a small-scale turbulent flow field based on a physics-informed neural network proposed by the present invention can use deep learning technology to train and predict the flow field. Compared with the traditional finite element method and finite volume method, the present invention avoids the cumbersome mesh generation and calculation process, thus significantly reducing the calculation time and cost.

[0048] 2. The feature fusion layer proposed by the present invention can effectively fuse the extracted shallow fluid features to obtain a feature representation with multi-level turbulent information. Combining it with convolution can extract deep turbulent features, significantly enhancing the scalability of the PINN model. Compared with the traditional PINN model based on a fully connected network, the improved model can capture the small-scale structure of turbulence with higher accuracy, and the average prediction error of the flow field is reduced by 33%.

[0049] 3. The preprocessing technology proposed by the present invention performs maximum-minimum normalization on the input and output data, and at the same time linearly transforms the Navier-Stokes equation using calculus technology. By combining convolution and feature fusion technologies, it can efficiently extract deep non-linear features in the turbulent flow field, avoiding the problem of detail loss in the traditional CNN method. Compared with the PINN model based on CNN, it can ensure that the data and the equation are linearly transformed simultaneously during the training process to satisfy the physical laws, promote the direct coupling of the differential equation residuals, improve the ability to capture the small-scale structure of turbulence, and reduce the average prediction error of the flow field by about 31%. Brief Description of the Drawings

[0050] Figure 1 It is a schematic diagram of the system of the present invention;

[0051] Figure 2 It is a schematic diagram of the overall structure of the high-resolution reconstruction of turbulence based on the PINN model;

[0052] Figure 3 The loss graph of the reconstruction result of the present invention using 100% isotropic turbulence data;

[0053] Figure 4 The loss graph of the reconstruction result of the present invention using 50% isotropic turbulence data;

[0054] Figure 5 The present invention uses the loss map of the reconstruction result of 25% isotropic turbulence data;

[0055] Figure 6 The present invention uses the loss map of the reconstruction result of 1% isotropic turbulence data; Specific embodiments

[0056] The technical solutions in the present invention will be clearly and completely described below with reference to the accompanying drawings in the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments in the present invention without creative efforts shall fall within the protection scope of the present invention.

[0057] In this Embodiment 1, the forced isotropic turbulence data with Re≈433 is used as a reference, the Z-axis is set to 0, and it is stored at intervals of a time step Δt = 0.002. The total time of the intercepted velocity and pressure is 0.4 seconds respectively, as Figure 1 shown, and a small-scale turbulent flow field reconstruction method based on a physics-informed neural network is provided. The method includes the following steps:

[0058] Step 1: Select data frames and residual points for training in the computational domain to construct a data set;

[0059] The Latin hypercube sampling method is used to extract residual points. First, the minimum and maximum values of the time coordinate and the space coordinate are used as the lower and upper bounds to determine the sampling range of the residual points. The Latin hypercube sampling method is used to generate uniformly distributed sample points within the range, and the generated sample points are randomly arranged to obtain 1 million residual points; in the forced isotropic turbulence data set, from [0, 2π] 3 domain, 1024 3 200 velocity data frames with a resolution of 128x128x3 and 200 pressure-velocity frames are extracted from the periodic grid. Only the data frames with velocity information and the 1 million residual points generated by the Latin hypercube sampling are used as the training set, and the data frames including velocity and pressure are used as the validation set.

[0060] Step 2: Sparsification processing of data points;

[0061] The sparsification is performed by a method based on coordinate selection. The amount of data is reduced by selecting some data point coordinates in the original data. Coordinates are selected according to the ratios of 100%, 50%, 25% and 1%, and the corresponding data subsets are extracted according to the selected coordinates. The sparsified data points with different selected ratios and the residual points generated by the Latin hypercube sampling are used to jointly construct a sparse data set.

[0062] Step 3: Data preprocessing, perform equation transformation by combining the hydrodynamic control equations and calculus techniques;

[0063] Perform normalization on the input and output respectively using min-max normalization; for the dependent and independent variables ψ = [x, y, z, t, u, v, w, p] in the dimensionless Navier-Stokes equations T Apply min-max normalization respectively. The formula for the incompressible dimensionless Navier-Stokes equations is defined as:

[0064] E m = u x + v y + w z

[0065]

[0066] where E m is the continuity equation, and E x , E y , E z are the momentum equations, x, y, and z are spatial coordinates, t is the time coordinate, u, v, w are the three components of velocity, namely the flow velocity, spanwise velocity, and vertical velocity respectively, and p is the pressure; u x , v x , w x , p x , u y , v y , w y , p y , u z , v x , w z and p z are the first-order partial derivatives of the three velocity components and pressure with respect to the spatial coordinates x, y, and z respectively, u t , v t and w t are the first-order partial derivatives of velocity with respect to the time coordinate t, u xx , u yy , u zz , v xx , v yy , v zz , w xx , w yy and w zz are the second-order partial derivatives of the three velocity components with respect to the spatial coordinates x, y, and z, and Re is the Reynolds number.

[0067] Calculate the maximum values ψ max = [x max , y max , z max , tmax ,u max ,v max ,w max ,p max T and the minimum value ψ min = [x min ,y min ,z min ,t min ,u min ,v min ,w min ,p min T ,map the data to the range [0, 1] through min - max normalization to obtain the normalized variable ψ * = [x * ,y * ,z * ,t * ,u * ,v * ,w * ,p * T ,the normalization formula is defined as follows:

[0068]

[0069] where X * represents the spatial coordinates (x * ,y * ,z * ) and the time coordinate t * ,X min and X max represent the minimum and maximum values of X respectively. In addition, while normalizing the data, the Navier - Stokes equation is transformed by combining calculus techniques, and the formula of the calculus technique is defined as follows:

[0070]

[0071] Substitute the transformed variables into the Navier - Stokes equation through calculus, and the transformed Navier - Stokes formula is:

[0072]

[0073]

[0074] where and ​​​Δu, Δv, and Δw are the differences between the maximum and minimum values of the velocity in three directions, Δp is the difference between the maximum and minimum values of the pressure, and Δx, Δy, Δz, and Δt represent the differences between the maximum and minimum values of the time coordinate and the spatial coordinates, respectively. and are the first-order and second-order partial derivatives of the output velocity u * 、v * 、w * and pressure p * with respect to the spatial coordinates (x * ,y * ,z * ) and the time coordinate t * , respectively, and are the transformed continuity equation and momentum equation, respectively.

[0075] Step 4: Construct the FFPICN model based on the improved physics-informed neural network. Input the spatial coordinates and time coordinates of the training data, and output the corresponding flow field data through the neural network to establish a fluid mechanics model.

[0076] The model is constructed on the Pycharm platform using the Python programming language based on the PyTorch deep learning framework as Figure 2 shown. The model input consists of the spatial coordinates (x * ,y * ,z * ) and the time coordinate t * . The output data stream includes the flow velocity u * 、spanwise velocity v * 、vertical velocity w * and pressure p * . The model mainly consists of two modules: a feature extraction module and a physics information module. Among them, the feature extraction module focuses on the deep extraction of fluid data and the fusion of multi-level non-linear information of complex fluids, and realizes the extraction of fluid small-scale structures through the combination of convolution and feature fusion. The physics information module satisfies the physical constraints through automatic differentiation and minimizing the loss. First, the feature extraction module extracts the shallow fluid representation from the preprocessed data. In the present invention, three 1x1 convolutions are performed in parallel for shallow feature extraction, and the definition formula of its 1x1 convolution structure is expressed as:

[0077] z = F(X * , W, b)

[0078] where the input of the convolution is X * representing the normalized spatial coordinates (x * ,y * ,z* ) and the time coordinate t * , where W and b are the weights and bias terms of the convolutional kernel respectively, F(·) is the mapping function, and z is the output of the convolutional layer. The z is non-linearly transformed through the adaptive swish activation function, and its formula can be defined as:

[0079] f Swish (z) = z·σ(βz)

[0080] where f Swish (z) represents the output of the adaptive Swish activation function, is the Sigmoid activation function, and β ∈ R is a trainable learning parameter that is adjusted through training. The shallow feature representations generated by passing through 3 parallel 1x1 CNN layers and the adaptive swish activation function are fused to construct a multi-level fluid feature set, and the feature fusion formula is defined as:

[0081] a3 = w·a1 + (1 - w)·a2

[0082] where a1, a2, and w are the feature representations obtained through convolution and the adaptive activation function respectively, w and (1 - w) are the weights for the fusion of the feature representations a1 and a2. Since the input data ranges from [0, 1] after max-min normalization, w is obtained by non-linearly transforming the feature representation through the adaptive activation function, and the value of w obtained by the change of the parameter β of the adaptive activation function during each training process will also be optimized; the fused feature representation a3 is used to deeply extract the fluid features using the convolutional fusion (FCNN) module; the FCNN module is mainly composed of a combination of 1×1 convolution and the feature fusion layer, and by stacking K FCNN modules, the deep feature representation of the input data is gradually learned; the output of the Kth FCNN module is defined as:

[0083] w k+1 = f Swish (F(a k+2 , W k , b k ))

[0084] a k+3 = w k+1 a1 + (1 - W k+1 )a2

[0085] where w k+1 represents the new feature fusion weight obtained by the Kth FCNN module, W k and b k are the weights and bias of the convolution in the kth FCNN module, a k+2 and a k+3are the input and output of the Kth FCNN module, respectively, and then compressed and processed by 1×1 convolution to predict the velocity and pressure of the fluid;

[0086] Step 5: Calculate the residuals of the fluid dynamics governing equations using automatic differentiation techniques;

[0087] The partial derivatives can be calculated using the automatic differentiation technology in the physical information module to calculate the output speed u * 、v * 、w * and pressure p * Regarding the spatial coordinates of the residual point (x * ,y * , z * ) and time coordinate t * The first partial derivative of and the second-order partial derivatives Partial derivatives can be calculated using PyTorch’s torch.autograd.grad() function.

[0088] Step 6: Calculate the control equation loss and data loss, and use the mean square error form to obtain the loss function.

[0089] The loss consists of the data loss and the control equation loss in the physical information module. The data loss is the root mean square error (MSE) between the velocity obtained in step 4 and the given data. The control equation loss is the root mean square error of the residuals of the continuity equation and the momentum equation. The loss is defined as follows:

[0090] L data =MSE(u pre * ,u * )+MSE(v pre * , v * )+MSE(w pre * , w * )

[0091]

[0092] L=L data +αL eqn

[0093] Among them, L data To measure the predicted velocity component u pre * 、v pre * and w pre * With the true value u* , v * and w * The error between them, L eqn reflects the residual of the governing equation and is used to describe the deviation degree of the network output from satisfying the governing equation and . L is the total loss value, the weight coefficient α in the loss function is 1, and finally the original variables are restored through the denormalization process to obtain the final velocity and pressure prediction values, and its formula is defined as:

[0094] ψ = ψ * ⊙(ψ max - ψ mih ) + ψ min

[0095] where ⊙ represents element-wise multiplication.

[0096] In this Example 2, the results of using sparse turbulent data to reconstruct the high-resolution flow field for the model with isotropic turbulence of Re≈433 are evaluated and compared with the traditional PINN model and the Res-PINN model. The traditional PINN is a model for solving partial differential equations based on a fully connected network, and the Res-PINN model is a deep convolutional network structure composed of multiple residual blocks and physical constraints, aiming to predict the velocity and pressure distributions of fluids from data. The relative L2 error is used to quantitatively evaluate the results of the FFPICN model for reconstructing the velocity and pressure of data with different sparsities. The relative L2 error is defined as:

[0097]

[0098] where and represent the predicted value and the true value of the flow field respectively, and N represents the total number of reference points. The lower the value of the L2 error, that is, the smaller the average deviation between the predicted value and the actual value, the more accurate the result of the deep learning model.

[0099] Table 1 lists the relative L2 errors of the velocity components in three directions and the pressure of u, v, w, and p after high-resolution reconstruction of the isotropic turbulent data sets with training data of 100%, 50%, 25%, and 1% respectively. It can be found from the table that all methods can reconstruct high-resolution flow fields from sparse data. Specifically, among the reconstruction results of the data sets with four different sparsities, PINN performs the worst, and the FFPICN model has the best reconstruction effect, especially under the condition of only using 1% sparse data. Compared with PINN and Res-PINN, the FFPICN model has significantly reduced the reconstruction errors of each flow field variable: the error in the u direction is reduced by 41% and 39% compared with PINN and Res-PINN respectively; the error in the v direction is reduced by 38% and 36% respectively; the error in the w direction is reduced by 31% and 27% respectively; the error in the pressure field p is reduced by 24% and 22% respectively. By significantly reducing the reconstruction errors of each variable, the FFPICN model demonstrates better equation fitting ability and better reconstruction accuracy, which benefits from the superposition of the feature fusion layer and multiple convolutional fusion layers in the proposed FFPICN model, which can deeply extract the non-linear small-scale structures of turbulence, thereby capturing more turbulence information and significantly improving the accuracy and reconstruction ability of the model. In addition, the present invention further verifies the stability and convergence performance of the FFPICN model through the analysis of the training losses of turbulent data with different sparsities. The training losses of multiple methods for turbulent data with different sparsities are shown in Figure 3 , 4 , 5, and 6. It can be seen from the figure that as the sparsity decreases, the training loss curves of PINN and Res-PINN increase significantly, which is due to the insufficient model learning caused by the reduction of the data volume, resulting in performance degradation and increased loss. However, the training loss of the FFPICN model is hardly affected by the data sparsity and always remains at a low level (about 3×10 -2 ), which indicates that the training performance of the proposed method is better than other methods. In summary, the proposed FFPICN model shows significant advantages under sparse data conditions, has stronger stability and better reconstruction accuracy, and can better solve the Navier-Stokes equation.

[0100] Table 1 Relative L2 errors of PINN, Res-PINN, and FFPICN models reconstructed using 100%, 50%, 25%, and 1% forced isotropic turbulent data

[0101]

[0102] Although some embodiments of the present invention have been shown and described, for those of ordinary skill in the art, various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention, and the resulting solutions all fall within the protection scope of the present invention.

Claims

1. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network, characterized in that, The steps of flow field reconstruction are as follows: Step 1: Select data frames and residual points for training within the computational domain to construct a data set; Step 2: Perform data point sparsification; Step 3: Conduct data preprocessing and perform equation transformation by combining the fluid mechanics control equations and calculus techniques; Step 4: Construct the FFPICN model according to the improved physics-informed neural network. Input the spatial coordinates and time coordinates of the training data, and output the corresponding flow field data through the neural network to establish a fluid mechanics model; Step 5: Use the automatic differentiation technique to calculate the residuals of the fluid mechanics control equations; Step 6: Calculate the control equation loss and data loss, and obtain the loss function in the form of mean square error.

2. The small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that In Step 1, the Latin hypercube sampling method is adopted in the computational domain to extract residual points from the computational domain; in the forced isotropic turbulence data set, 200 velocity data frames with a resolution of 128×128×3 and 200 pressure-velocity frames are extracted from the 10243 periodic grid in the [0,2π]3 domain. Among them, only the data frames with velocity information and 1 million residual points generated by the Latin hypercube sampling are jointly used as the training set, and the data frames including velocity and pressure are used as the validation set.

3. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that, The sparsification method adopted in Step 2 is the coordinate selection-based sparsification method, which reduces the data volume by selecting some data point coordinates in the original data; select coordinates according to the ratios of 100%, 50%, 25%, and 1%, and extract the corresponding data subsets according to the selected coordinates. The sparsely data points with different selected ratios and the residual points generated by the Latin hypercube sampling method are jointly used to construct a sparse data set.

4. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that, The preprocessing method in the third step is to normalize the input and output using the maximum-minimum normalization; the dependent and independent variables ψ = [x, y, z, t, u, v, w, p] in the dimensionless Navier-Stokes equations T are respectively applied with the maximum-minimum normalization; the formula definition of the incompressible dimensionless Navier-Stokes equations is as follows: E m = u x + v y + w z Among them, E m is the continuity equation, and E x 、E y 、E z are the momentum equations, where x, y, and z are spatial coordinates, t is the time coordinate, u, v, and w are the three components of velocity, namely the streamwise velocity, the spanwise velocity, and the vertical velocity respectively, and p is the pressure; u x 、v x 、w x 、p x 、u y 、v y 、w y 、p y 、u z 、v z 、w z and p z are the first-order partial derivatives of the three velocity components and the pressure with respect to the spatial coordinates x, y, and z respectively, u t 、v t and w t are the first-order partial derivatives of the velocity with respect to the time coordinate t, u xx 、u yy 、u zz 、v xx 、v yy 、v zz 、w xx 、w yy and w zz are the second-order partial derivatives of the three velocity components with respect to the spatial coordinates x, y, and z; Re is the Reynolds number; Calculate the maximum value ψ respectively according to the training set max =[x max ,y max ,z max ,t max ,u max ,v max ,w max ,p max T and the minimum value ψ min =[x min ,y min ,z min ,t min ,u min ,v min ,w min ,p min T , map the data to the range [0, 1] through min-max normalization to obtain the normalized variable ψ * =[x * ,y * ,z * ,t * ,u * ,v * ,w * ,p * T , the min-max normalization formula is defined as follows:​​​ where X * represents the normalized spatial coordinates (x * , y * , z * ) and the time coordinate t * , X min and X max represent the minimum and maximum values of X respectively; in addition, while normalizing the data, the Navier-Stokes equation is transformed by combining calculus techniques, and the formula definition of the calculus techniques is as follows: The variables are transformed and substituted into the Navier-Stokes equation through calculus techniques, and the transformed Navier-Stokes formula is: Among them and Δu, Δv, and Δw are the differences between the maximum and minimum values of the velocity in three directions, Δp is the difference between the maximum and minimum values of the pressure, and Δx, Δy, Δz, and Δt respectively represent the differences between the maximum and minimum values of the time coordinate and the spatial coordinates and are respectively the first-order partial derivatives and second-order partial derivatives of the output velocity u * , v * , w * and the pressure p * with respect to the spatial coordinates (x * , y * , z * ) and the time coordinate t * , and and are respectively the transformed continuity equation and momentum equation are respectively the transformed continuity equation and momentum equation 5. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that, The basic structure of the neural network in Step 4 is a model constructed on the Pycharm platform using the Python programming language based on the PyTorch deep learning framework; The model consists of two main modules: the feature extraction module and the physics information module. Among them, the feature extraction module focuses on deeply extracting fluid data and fusing multi-level non-linear information of complex fluids, and realizes the extraction of fluid small-scale structures through the combination of convolution and feature fusion; the physics information module satisfies physical constraints through automatic differentiation and minimizing the loss; the feature extraction module first extracts the shallow fluid representation of the preprocessed data, and three 1×1 convolutions are used in parallel to perform shallow feature extraction. The definition formula of its 1×1 convolution structure is expressed as: z = F(X * , W, b) Among them, the input X of the convolution * represents the spatial coordinates (x * , y * , z * ) and the time coordinate t * , where W and b are the weights and bias terms of the convolution kernel respectively, F(·) is the mapping function, and z is the output of the convolution layer; z is non-linearly transformed through the adaptive swish activation function, and its formula is defined as: f Swish (z) = z·σ(βz) where f Swish (z) represents the output of the adaptive Swish activation function, is the Sigmoid activation function, β ∈ R is a trainable learnable parameter that is adjusted through training; the shallow feature representations generated by passing through 3 parallel 1×1 CNN layers and the adaptive swish activation function are fused to construct a multi-level fluid feature set, and the feature fusion formula is defined as: a3 = w·a1+(1 - w)·a2 where a1, a2, and w are the feature representations obtained through convolution and the adaptive activation function respectively, and w and (1 - w) are the weights for fusing the feature representations a1 and a2; the fused feature representation a3 is used to deeply extract fluid features using the convolutional fusion (FCNN) module; the FCNN module is mainly composed of the combination of 1×1 convolution and the feature fusion layer. By stacking K FCNN modules, the deep feature representation of the input data is gradually learned; the output of the Kth FCNN module is defined as: w k+1 = f Swish (F(a k+2 , W k , b k )) a k+3 = w k+1 a1 + (1 - w k+1 )a2 Among them, w k+1 represents the new feature fusion weight obtained by the K-th FCNN module, W k and b k are the weights and biases of the convolution in the k-th FCNN module, a k+2 and a k+3 are the input and output of the K-th FCNN module respectively, and then are compressed and processed by 1×1 convolution to predict the velocity and pressure of the fluid.

6. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that, In step 4, the input consists of the spatial coordinates (x * , y * , z * ) and the temporal coordinate t * . The output data stream includes the flow velocity u of the fluid * , the spanwise velocity v * , the vertical velocity w * , and the pressure p * .

7. A small-scale turbulent flow field reconstruction method based on a physics-informed neural network according to claim 1, characterized in that The partial derivative calculation in the fifth step can be performed using the automatic differentiation technology in the physical information module, and the output velocities u * , v * , w * and pressure p * with respect to the spatial coordinates (x * , y * , z * ) of the residual points and the time coordinate t * are calculated for the first-order partial derivatives and the second-order partial derivatives The partial derivatives can be calculated using the torch.autograd.grad() function of PyTorch.

8. A method for reconstructing a small-scale turbulent flow field based on a physics-informed neural network according to claim 1, characterized in that, The losses in step six include the data loss in the physical information module and the control equation loss. The data loss is the root mean square error (MSE) between the velocity obtained in step four and the given data; the control equation loss is the root mean square error of the residuals of the continuity equation and the momentum equation; the loss is defined as follows: L data = MSE(u pre * , u * ) + MSE(v pre * , v * ) + MSE(w pre * , w * ) L eqn = MSE(E m* ) + MSE(E x* ) + MSE(E y* ) + MSE(E z* ) L = L data + αL eqn Among them, L data is to measure the predicted velocity components u pre * , v pre * and w pre * and the error between the true values u * , v * and w * . L eqn reflects the residual of the control equation and is used to describe the deviation degree of the network output from satisfying the control equation and . L is the total loss value, the weight coefficient α in the loss function is 1, and finally the original variables are recovered through the non-normalization process to obtain the final predicted values of velocity and pressure. Its formula is defined as: ψ = ψ * ⊙(ψ max - ψ min ) + ψ min where ⊙ represents element-wise multiplication.

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