A turbulent flow field missing completion method based on a conditional latent diffusion model

Abstract

This invention belongs to the field of marine engineering technology and discloses a method for completing missing turbulent flow fields based on a conditional latent diffusion model. The method constructs a conditional latent diffusion model based on multi-source constraints, introducing a multi-source conditional constraint guidance module during the inverse diffusion process. By fusing global information and prior physical knowledge, it enhances the model's completion capability in large-scale continuous missing regions. A context-adaptive difference mechanism is designed to dynamically correct the gradient during the generation process, enhancing the structural alignment between the completed region and the known region, suppressing over-smoothing, and improving the recovery of high-frequency details. A mask consistency mechanism is introduced to explicitly constrain the known region, ensuring data consistency and boundary continuity of the generated results. This method can be applied to scenarios such as marine observation systems, environmental pollution diffusion analysis, and meteorological data processing, achieving high-precision flow field reconstruction under complex missing conditions.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering technology and involves the intersection of computational fluid dynamics and deep learning methods. Specifically, it is a method for completing missing turbulent flow fields based on a conditional latent diffusion model. Background Technology

[0002] The study of turbulence evolution is a crucial foundation for marine engineering equipment, ecological environment monitoring, and meteorological observation, playing a vital role in engineering applications such as marine meteorological observation systems, marine ecological environment monitoring systems, and offshore platform observation equipment. Complete and high-resolution flow field data can provide a theoretical basis and data support for marine meteorological observation sensors and marine ecology and environmental governance. However, limitations in measurement equipment accuracy, the complexity of the observation environment, and data transmission and storage costs make it difficult to obtain complete turbulence flow field data in actual observations, resulting in varying degrees of data gaps.

[0003] Existing flow field completion methods mainly include spatial interpolation, orthogonal decomposition, and deep learning. Among them, deep learning-based methods have shown good performance in flow field missing data completion tasks due to their excellent nonlinear representation capabilities. However, when facing complex application scenarios such as marine meteorological observation and environmental monitoring, existing methods still have the following shortcomings: First, existing methods are not adaptable enough to handling complex boundary conditions and large-scale continuous missing regions. In actual turbulent scenarios, when dealing with large-scale, irregular missing regions caused by observation blind spots or equipment failures, existing methods struggle to maintain structural consistency between the reconstructed region and the real flow field, easily leading to structural distortion. Second, existing methods mainly rely on data-driven mechanisms and lack effective physical constraints, resulting in over-smoothing of the reconstruction results, which is difficult to meet the accuracy and reliability requirements of marine meteorological observation and environmental monitoring. In addition, existing generative models in flow field completion tasks typically lack explicit constraint mechanisms for the observation area and effective modeling capabilities for global structural information. When dealing with large-scale missing data, they are prone to structural shifts and boundary discontinuities, thus affecting the reliability of the model in engineering inspection, environmental assessment, and observation data services. Therefore, there is an urgent need for a method to complete missing turbulent flow fields for engineering applications such as marine meteorological observation and environmental monitoring, so as to achieve high-precision reconstruction of flow field data under complex observation conditions and improve the integrity, physical consistency and reliability of flow field data in engineering applications. Summary of the Invention

[0004] The purpose of this invention is to provide a method for completing missing turbulent flow fields based on a conditional potential diffusion model, in order to solve the problem of missing flow field data caused by factors such as complex sea conditions, limited sensor deployment, and incomplete data acquisition in marine observation equipment, meteorological observation systems, and environmental monitoring. This overcomes the problems of structural distortion, insufficient physical constraints, and excessive smoothing of details in the reconstruction results caused by complex boundary conditions and large-scale irregularities in the prior art.

[0005] To achieve the above objectives, this invention proposes a Multi-Source Constrained Conditional Latent Diffusion (MSCLDM) model for high-precision completion of turbulent flow field data in ocean observation and environmental monitoring. Specifically, it includes the following steps:

[0006] Step 1: Acquire raw turbulence data, preprocess the data, and build a dataset. Follow these steps:

[0007] Step 1.1: Obtain two-dimensional turbulent velocity and pressure field data for T time steps from the database, and map the data into two-dimensional flow field data in the form of a two-dimensional grid of size H×W, with each grid point containing two velocity components. (representing velocity in the x and y directions respectively) and pressure .

[0008] Step 1.2: Perform missing processing on the two-dimensional flow field data described in Step 1.1. Generate masks by setting different missing ratios to construct a missing flow field dataset, and divide the missing flow field dataset into a training set and a test set in a 4:1 ratio.

[0009] Step 2: Design and construct the flow field completion model. The conditional latent diffusion model framework based on multi-source constraints constructed in this invention is as follows: Figure 1 As shown, the model includes: a multi-scale Fourier autoencoder (MFAE) module, a multi-source conditional constraint guidance module, and a diffusion model module.

[0010] Step 2.1: Construct the input layer, which will contain the complete flow field data. and low-resolution flow field data obtained after downsampling. Organized as tensors, both of these tensors have dimensions (N, H, W, C), where: N represents the number of flow fields, H and W are the number of grids in the x and y spatial directions of the two-dimensional flow field at each time step, respectively, and C represents the number of channels, containing two velocity components. and pressure .

[0011] Step 2.2: Construct a multi-source condition constraint guidance module, which includes a global information constraint unit, a physical information encoding unit, and an energy function guidance unit.

[0012] The global information constraint unit in step 2.2 includes the MFAE encoder and the Fast Fourier Convolution (FFC) module. Low-resolution data. First, the encoder module of MFAE extracts global features, and then the FFC module performs frequency domain feature enhancement to obtain the global variable z representing the global structure information of the flow field. g The FFC module adopts a dual-branch architecture, such as... Figure 2 As shown, the left side includes two parallel convolutional layers and one ReLU activation function, while the right side includes one convolutional layer and one Spectral Transform Block (STB). The STB consists of one convolutional layer, one Fast Fourier Transform layer, one Inverse Fast Fourier Transform layer, and one residual connection. Subsequently, the global variable z... g The input is fed into the cross-attention layer of the denoising network U-Net, which is used to guide the global structure of the generated results at each stage of the backdiffusion.

[0013] The physical information encoding unit in step 2.2 consists of an input layer and a FiLM encoding module. First, the input Reynolds number scalar parameter Re is normalized, and then input into the FiLM encoding module, which consists of three multilayer perceptrons, for nonlinear mapping to obtain the corresponding physical feature vector g. The feature vector g is then input into the energy function guiding unit as a condition variable to participate in the reverse diffusion process of the diffusion model, thereby establishing the correlation between physical parameters and the multi-scale structure of the flow field.

[0014] The energy function guiding unit in step 2.2 is based on the energy function. Build, It consists of 3 convolutional layers and 1 multilayer perceptron, and the flow field latent variable z corresponds to each time step t of the diffusion sampling. t The concatenated vector is obtained by concatenating the physical feature vector g with the concatenated vector, and then inputting the concatenated vector into the energy function. Energy value obtained from The energy value is obtained by utilizing the automatic differentiation mechanism of the deep learning computing framework. Regarding its input item z t By taking the derivative, we obtain the expression in the spatial dimension that is related to z. t Same physical constraint gradient term This is introduced into the inverse sampling process of the diffusion model to guide the denoising process with physical conditions. The formula for the above process is:

[0015]

[0016] in, This is the guided mean after correction for the physical gradient of the energy function. The original mean value of the noise predicted by the denoising network U-Net at time step t is derived; z t Represents the noisy flow field latent variables at time step t; g represents the physical eigenvectors; z g Represents a global variable; These are weighting coefficients that change over time.

[0017] The MFAE module in step 2.2 consists of an encoder and a decoder. The encoder includes one position encoding layer, three cascaded downsampling modules, and one multilayer perceptron. The downsampling module comprises three parallel Fourier convolutional modules of different scales, one downsampling layer, and one residual connection. The Fourier convolutional module has two branches: the left branch consists of one 2D convolutional layer, one normalization layer, one ReLU activation function, and one residual connection; the right branch consists of one fast Fourier transform, one inverse Fourier transform, one residual connection, and one 2D convolution. The decoder consists of one multilayer perceptron, three sets of upsampling modules, and one shape transformation layer. The upsampling module consists of three parallel Fourier convolutional modules of different scales, one upsampling layer, and one residual connection.

[0018] Step 2.3: Construct the diffusion model module. This module is used to generate and complete the missing flow field in the latent feature space, including latent space mapping, forward diffusion process, and reverse diffusion process guided by the multi-source condition constraint guidance module. The reverse diffusion process includes a mask consistency mechanism, a context-adaptive discrepancy (CAD) constraint mechanism, and a denoising network U-Net. Its specific implementation is as follows:

[0019] The input complete flow field data is processed by a pre-trained encoder of MFAE. Nonlinear feature extraction is performed to map the physical space to a latent feature space, obtaining an initial latent variable z0. A forward diffusion process is then performed in the latent space, gradually injecting Gaussian noise to add noise to the latent variable z0 until a noise latent variable z conforming to a standard Gaussian distribution is obtained at the T-th time step. T .

[0020] During the inference phase, this invention uses randomly initialized Gaussian noise z TThe reverse diffusion process begins, gradually eliminating noise through the U-Net denoising network to recover the complete flow field latent variables. During each time step t, the generated trajectory is collaboratively guided by a multi-source condition constraint guidance module. First, the global variable z... g Injecting cross-attention layers into the denoising network U-Net to achieve global structure guidance; synchronously constructing the energy function. Using an automatic differentiation mechanism to determine the energy value Regarding its input item z t The gradient is obtained by taking the derivative. The gradient is then used as a correction term in the sampling equation to correct the sampling direction, ensuring that the generated results converge to the physically feasible region. A CAD mechanism based on optimal transport theory is constructed, which uses the features of the known observation area mapped to the latent space as a distribution reference to calculate the Brenier potential function and extract its gradient with respect to the latent variables. The process involves dynamically correcting the spatial distribution consistency between the generated flow field and the known flow field; finally, combining mask consistency operations to apply hard data constraints to the known region. The specific process is uniformly represented by the following formula:

[0021]

[0022]

[0023] in, This represents the intermediate latent variables generated at time step t-1 after the above multi-source constraint guidance, before the mask consistency operation is performed; This represents the updated latent variables that proceed to the next time step sampling operation after the mask consistency operation; μ θ (z t , t | z g The original mean of the noise predicted by the denoising network U-Net at time step t is derived. and These are the weight coefficients of the context-adaptive difference constraint mechanism and the physical constraint term of the energy function, respectively. To define the variance control parameter for the magnitude of random noise injected at time step t, MaskConsist represents the true standard Gaussian noise of the sample. The function representing the mask consistency operation yields the fully denoised flow field latent variables after T time steps. .

[0024] The mask consistency constraint mechanism in step 2.3, such as Figure 3 As shown, the specific implementation method for maintaining data consistency in the known region during flow field completion is as follows:

[0025] Let the mask matrix be M, where the known regions have values ​​of 1 and the missing regions have values ​​of 0. The original complete latent variable z0 is calculated using the forward noise addition formula of the diffusion model to obtain the noisy latent variable z0 for the known regions at step t-1. t-1 In time step t of the reverse denoising process, the current sampled values ​​of the denoising network U-Net are processed to obtain the prediction generation result for the current step. Then, the mask matrix M is used to apply the mask matrix M to z. t-1 and The two parts are weighted and fused to update the final latent variable sample values. Its expression is:

[0026]

[0027] in, This indicates element-wise multiplication. The updated... This serves as the input for the reverse iteration at the next time step t-1, continuing until t=0. Through the above operations, the known region is always forced to remain the noisy version of the original observation data during the iteration process, while the missing region is filled with physically consistent data generated by the model, thus ensuring the data consistency of the completion result on the known boundaries.

[0028] The CAD mechanism in step 2.3 is used to constrain the structural consistency between the generated flow field and the known flow field. Its specific implementation is as follows:

[0029] First, an adaptive scaling function f based on feature space differences is constructed. The latent variable z used to generate the complete flow field from the completed model. t Latent variables in the known region The expression for the dynamic adjustment of constraint weights based on differences in the context feature space is as follows:

[0030]

[0031] in, Intensity adjustment coefficient; This represents the attenuation sensitivity parameter; It is a lightweight convolutional network consisting of 3 convolutional layers and 1 ReLU activation function, used to extract local structural features of the flow field. This represents the L2 norm. When there is a significant difference between the generated features and the known features, the adaptive weights, i.e., the adaptive scaling function f, are used. The output is increased, thereby strengthening the constraints on regions with inconsistent structures.

[0032] Secondly, a difference measurement function is constructed based on optimal transmission theory. Introducing the Brenier potential function The function consists of three multilayer sensing mechanisms, and the function is related to the current flow field latent variable z. t gradient The expression for the difference metric function, representing the optimal transport mapping direction from the generated flow field to the reference flow field, is as follows:

[0033]

[0034] in, Represents the square of the L2 norm; Let the Brenier potential function be the latent variable z of the current flow field. t The gradient; These are latent variables for a known region.

[0035] Finally, during the reverse diffusion process, to reduce the computational overhead of high-dimensional second-order differentiation, the core optimal transport mapping in the metric function is directly extracted. This is introduced as a structural constraint term into the inverse diffusion sampling update process.

[0036] The denoising network U-Net in step 2.3 adopts a U-shaped symmetrical architecture, including a downsampling encoding path, an intermediate bottleneck layer, and an upsampling decoding path. Its specific structure is as follows: The downsampling encoding path includes one 2D convolutional layer, five cascaded downsampling modules, and one transition module. Each downsampling module consists of one residual block, one self-attention layer, one cross-attention layer, and one downsampling layer. The transition module consists of one residual block, one self-attention layer, and one cross-attention layer. The overall structure of the upsampling decoding path is symmetrical to the encoding path, including one initial module and five cascaded upsampling modules. Before executing the upsampling modules, skip connections are used to concatenate and fuse the shallow spatial features of the corresponding layers in the encoding path with the current features. Finally, a normalization layer and a 2D convolutional layer are used to map back to the target number of channels, outputting the denoising result for the current time step.

[0037] Step 2.4: Construct the output layer, which uses the decoder of MFAE to output the noise-free latent variables from the reverse denoising process of the diffusion model module. Mapping back to physical space yields the complete flow field.

[0038] Step 3: Input the training set constructed in Step 1.2 into the completed model constructed in Step 2 for training. Update the model parameters using an iterative optimization method, setting the batch size to B and the initial learning rate to β, and use an optimizer to update the parameters. Continue until the model converges and the trained weighted model is obtained.

[0039] Step 4: Evaluate the effectiveness of the model in completing the missing flow field using the test set, and implement it according to the following specific steps:

[0040] Step 4.1: Input the test set data obtained in Step 1.2 into the model, use the trained model to complete the missing flow field, and use Mean Squared Error (MSE), Mean Absolute Error (MAE), Correlation Coefficient (CC), and Structural Similarity Index Measure (SSIM) as evaluation metrics.

[0041] Step 4.2: The results are compared with those of the Multi-Scale Autoencoder model (MS-AE), Enhanced Super-resolution Generative Adversarial Network (ESR-GAN), and U-shaped Attention-driven Deformable Generative Adversarial Network (UAD-GAN) to verify the performance of the proposed model.

[0042] The formula for the model evaluation index in step 4.1 is:

[0043]

[0044]

[0045]

[0046]

[0047] in, Mean square error; Mean absolute error; The correlation coefficient; is the structural similarity index; N is the number of samples; i represents the sample index; For real flow field data, Flow field data to complete the model, For the i-th sample value of the real flow field data, The i-th sample value of the flow field data used to complete the model. and These represent the mean values ​​of the actual flow field data and the model-completed flow field data, respectively. and These represent the standard deviations of the actual flow field data and the standard deviations of the flow field data completed by the model, respectively. Represents covariance; and These represent the mean of the actual flow field data and the mean of the flow field data completed by the model, respectively. and To avoid stability constants with a denominator of zero; For the summation operator, It is an absolute value operator.

[0048] The beneficial effects of this invention are as follows:

[0049] 1. This invention proposes a latent diffusion model incorporating multi-source constraints to effectively address the insufficient completion capability of existing deep learning techniques in handling large-scale continuous missing regions and complex turbulent boundary conditions in marine meteorological observation and environmental monitoring scenarios. By constructing a multi-source constraint guidance module, the physical information encoding unit, energy function guidance unit, and global information constraint unit are synergistically fused and applied to the inverse diffusion process of the diffusion model, thereby significantly improving the model's ability to represent the multi-scale structure and nonlinear characteristics of turbulence. Compared with existing generative methods, this invention significantly improves structural similarity and reconstruction accuracy in flow field completion tasks. The correlation coefficient (CC) between the completed turbulent velocity field and pressure field results is increased by approximately 4.5% compared to traditional generative models, and the mean square error (MSE) is reduced by approximately 4.1%.

[0050] 2. This invention introduces a context-adaptive difference constraint mechanism, which serves as a gradient correction strategy based on optimal transport theory. During the inverse diffusion process of the diffusion model, this mechanism dynamically corrects the generated results. By characterizing the local structural differences between the completed region and the known region, it adaptively adjusts the gradient direction during the generation process, thereby improving the structural consistency between the missing region and the known data. Compared to existing methods, this invention can effectively suppress over-smoothing in large-scale continuous missing regions and improve the recovery capability of small-scale vortex structures and high-frequency information. Experimental results show that, under the same test conditions, the structural similarity index (SSIM) of the generated results is improved by approximately 5.9%.

[0051] 3. This invention designs a mask consistency mechanism within the diffusion generation framework to explicitly constrain known regions, thereby avoiding boundary discontinuities and structural offsets present in existing methods during the generation process. This invention exhibits higher consistency in velocity field distribution, vortex structure characteristics, and energy spectrum, significantly improving the small-scale structural integrity of the reconstruction results, thus enhancing the integrity of observational data and strengthening the reliability of marine meteorological observation and environmental monitoring research. Attached Figure Description

[0052] Figure 1 This is a framework diagram of a conditional latent diffusion model based on multi-source constraints.

[0053] Figure 2 This is a diagram of the Fast Fourier Convolution module structure, where, Figure 2 (a) is the overall architecture diagram of the Fast Fourier Convolution module. Figure 2 (b) is a detailed diagram of the internal structure of the spectrum conversion module;

[0054] Figure 3 Diagram of mask consistency constraint mechanism;

[0055] Figure 4 To compare the errors and correlations between this invention and other methods in turbulent channel flow completion tasks and numerical simulation data, the present invention is presented. Figure 4 (a) is a graph comparing the mean absolute error under different missing rates. Figure 4 (b) is a graph comparing the mean squared error under different missing rates. Figure 4 (c) is a curve comparing the correlation coefficients under different missing rates. Figure 4 (d) is a comparison curve of structural similarity under different missing rates;

[0056] Figure 5 To compare the errors and correlations between this invention and other methods in the cylindrical wake completion task with numerical simulation data, wherein, Figure 5 (a) is a graph comparing the mean absolute error under different missing rates. Figure 5 (b) is a graph comparing the mean squared error under different missing rates. Figure 5 (c) is a curve comparing the correlation coefficients under different missing rates. Figure 5 (d) is a comparison curve of structural similarity under different missing rates; Detailed Implementation

[0057] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments.

[0058] The dataset used in Example 1 is turbulent channel flow, which is used to verify the effectiveness of the method of the present invention in the task of missing flow field data completion. Its processing flow is applicable to the scenario of missing turbulent data completion in marine meteorological observation, and includes the following steps:

[0059] Step 1: Acquire raw turbulence data, preprocess the data, and build a dataset. Follow these steps:

[0060] Step 1.1: First, access the public interface provided by the JHTDB website (https: / / turbulence.idies.jhu.edu) to download the friction velocity Reynolds number. A turbulent channel flow dataset with a computational domain size of 8πh×2h×3πh was generated. Then, instantaneous velocity and pressure field data in the two-dimensional yx plane at the z=0 section were extracted from this dataset, for a total of 1000 time steps. The flow field size is H×W=256×256. The learning time step Δt=0.065.

[0061] Step 1.2 involves handling the missing data in the aforementioned two-dimensional flow field data. The data from 1000 time steps is divided into training and testing sets in a 4:1 ratio, meaning the first 80% of time steps are used for model training, and the last 20% are used for model evaluation and prediction validation. In this embodiment, the missing percentages for all datasets are set to 12.5%, 25%, 35%, 45%, 55%, and 65%, resulting in training and testing sets for six different missing scenarios. Each missing scenario is constructed based on the aforementioned 1000 time steps of data.

[0062] Step 2: Design and build the flow field completion model, including the following sub-steps:

[0063] Step 2.1: Construct the input layer, which will contain the complete flow field data. and low-resolution flow field data obtained after downsampling. Organized as tensors, both of these tensors have dimensions (N, H, W, C), where: N represents the number of flow fields, H and W are the number of grids in the x and y spatial directions of the two-dimensional flow field at each time step, respectively, and C represents the number of channels, including both velocity components. and pressure .

[0064] Step 2.2: Construct a multi-source condition constraint guidance module, which includes a global information constraint unit, a physical information encoding unit, and an energy function guidance unit.

[0065] The global information constraint unit in step 2.2 includes the MFAE encoder and the Fast Fourier Convolution (FFC) module. Low-resolution data. First, the encoder module of MFAE extracts global features, and then the FFC module performs frequency domain feature enhancement to obtain the global variable z representing the global structure information of the flow field. g The FFC module adopts a dual-branch architecture, such as... Figure 2As shown, the left side includes two parallel convolutional layers and one ReLU activation function, while the right side includes one convolutional layer and one Spectral Transform Block (STB). The STB consists of one convolutional layer, one Fast Fourier Transform layer, one Inverse Fast Fourier Transform layer, and one residual connection. Subsequently, the global variable z... g The input is fed into the cross-attention layer of the denoising network U-Net, which is used to guide the global structure of the generated results at each stage of the backdiffusion.

[0066] The physical information encoding unit in step 2.2 consists of an input layer and a FiLM encoding module. First, the input Reynolds number scalar parameter Re is normalized, and then input into the FiLM encoding module, which consists of three multilayer perceptrons, for nonlinear mapping to obtain the corresponding physical feature vector g. The feature vector g is then input into the energy function guiding unit as a condition variable to participate in the reverse diffusion process of the diffusion model, thereby establishing the correlation between physical parameters and the multi-scale structure of the flow field.

[0067] The energy function guiding unit in step 2.2 is based on the energy function. Build, It consists of 3 convolutional layers and 1 multilayer perceptron, and the flow field latent variable z corresponds to each time step t of the diffusion sampling. t The concatenated vector is then concatenated with the physical feature vector g, and this concatenated vector is input into the energy function. Energy value obtained from The energy value is obtained by utilizing the automatic differentiation mechanism of the deep learning computing framework. Regarding its input item z t By taking the derivative, we obtain the expression in the spatial dimension that is related to z. t Same physical constraint gradient term This is introduced into the inverse sampling process of the diffusion model to guide the denoising process with physical conditions. The formula for the above process is:

[0068]

[0069] in, This is the guided mean after correction for the physical gradient of the energy function. The original mean value of the noise predicted by the denoising network U-Net at time step t is derived; z t Represents the noisy flow field latent variables at time step t; g represents the physical eigenvectors; z g Represents a global variable; These are weighting coefficients that vary over time. .

[0070] The MFAE module in step 2.2 consists of an encoder and a decoder. The encoder includes one position encoding layer, three cascaded downsampling modules, and one multilayer perceptron. The downsampling module comprises three parallel Fourier convolutional modules of different scales, one downsampling layer, and one residual connection. The Fourier convolutional module has two branches: the left branch consists of one 2D convolutional layer, one normalization layer, one ReLU activation function, and one residual connection; the right branch consists of one fast Fourier transform, one inverse Fourier transform, one residual connection, and one 2D convolution. The decoder consists of one multilayer perceptron, three sets of upsampling modules, and one shape transformation layer. The upsampling module consists of three parallel Fourier convolutional modules of different scales, one upsampling layer, and one residual connection.

[0071] Step 2.3: Construct the diffusion model module. This module is used to generate and complete the missing flow field in the latent feature space, including latent space mapping, forward diffusion process, and reverse diffusion process guided by the multi-source condition constraint guidance module. The reverse diffusion process includes mask consistency mechanism, CAD mechanism, and denoising network U-Net. Its specific implementation is as follows:

[0072] The input complete flow field data is processed by a pre-trained encoder of MFAE. Nonlinear feature extraction is performed to map the physical space to a latent feature space, obtaining an initial latent variable z0. A forward diffusion process is then performed in the latent space, gradually injecting Gaussian noise to add noise to the latent variable z0 until a noise latent variable z conforming to a standard Gaussian distribution is obtained at the T-th time step. T .

[0073] During the inference phase, this invention uses randomly initialized Gaussian noise z T The reverse diffusion process begins, gradually eliminating noise through the U-Net denoising network to recover the complete flow field latent variables. During each time step t, the generated trajectory is collaboratively guided by a multi-source condition constraint guidance module. First, the global variable z... g Injecting cross-attention layers into the denoising network U-Net to achieve global structure guidance; synchronously constructing the energy function. Using an automatic differentiation mechanism to determine the energy value Regarding its input item z t The gradient is obtained by taking the derivative. The gradient is then used as a correction term in the sampling equation to correct the sampling direction, ensuring that the generated results converge to the physically feasible region. A CAD mechanism based on optimal transport theory is constructed, which uses the features of the known observation area mapped to the latent space as a distribution reference to calculate the Brenier potential function and extract its gradient with respect to the latent variables. The consistency between the generated flow field and the known flow field in spatial distribution is dynamically corrected; finally, a hard data constraint is applied to the known region by combining a mask consistency operation, so that the sampling and update process of the multi-constraint-guided reverse diffusion process can be uniformly represented as:

[0074]

[0075]

[0076] in, This represents the intermediate latent variables generated at time step t-1 after the above multi-source constraint guidance, before the mask consistency operation is performed; This represents the updated latent variables that proceed to the next time step sampling operation after the mask consistency operation; μ θ (z t , t | z g The original mean value of the noise predicted by the denoising network U-Net at time step t is the result of the derivation of the noise. and These are the weighting coefficients of the context-adaptive difference constraint mechanism and the physical constraint term of the energy function, respectively. , ; To define the variance control parameter for the magnitude of random noise injected at time step t, ; Represents the true standard Gaussian noise of the sample; MaskConsist( The function represents the mask consistency operation; after T = 1000 time steps, the fully denoised flow field latent variables can be obtained. .

[0077] The mask consistency constraint mechanism in step 2.3, such as Figure 3 As shown, the specific implementation method for maintaining data consistency in the known region during flow field completion is as follows:

[0078] Let the mask matrix be M, where the known regions have values ​​of 1 and the missing regions have values ​​of 0. The original complete latent variable z0 is calculated using the forward noise addition formula of the diffusion model to obtain the noisy latent variable z0 for the known regions at step t-1. t-1 In time step t of the reverse denoising process, the current sampled values ​​of the denoising network U-Net are processed to obtain the prediction generation result for the current step. Then, the mask matrix M is used to apply the mask matrix M to z. t-1 and The two parts are weighted and fused to update the final latent variable sample values. Its expression is:

[0079]

[0080] in, This indicates element-wise multiplication. The updated... This serves as the input for the reverse iteration at the next time step t-1, continuing until t=0. Through the above operations, the known region is always forced to remain the noisy version of the original observation data during the iteration process, while the missing region is filled with physically consistent data generated by the model, thus ensuring the data consistency of the completion result on the known boundaries.

[0081] The CAD mechanism in step 2.3 is used to constrain the structural consistency between the generated flow field and the known flow field. Its specific implementation is as follows:

[0082] First, an adaptive scaling function f based on feature space differences is constructed. The latent variable z used to generate the complete flow field from the completed model. t Latent variables in the known region The expression for the dynamic adjustment of constraint weights based on differences in the context feature space is as follows:

[0083]

[0084] in, This represents the intensity adjustment coefficient, which is set in this case. =1.0; This represents the attenuation sensitivity parameter, which is set in this case. =2.0; It is a lightweight convolutional network consisting of 3 convolutional layers and 1 ReLU activation function; This represents the L2 norm. When there are significant differences between the generated features and known features, the adaptive weights increase, thereby strengthening the constraint on regions with inconsistent structures.

[0085] Secondly, a difference measurement function is constructed based on optimal transmission theory. Introducing the Brenier potential function The function consists of three multilayer sensing mechanisms, and the function is related to the current flow field latent variable z. t gradient The expression for the difference metric function, representing the optimal transport mapping direction from the generated flow field to the reference flow field, is as follows:

[0086]

[0087] in, Represents the square of the L2 norm; Let the Brenier potential function be the latent variable z of the current flow field. t The gradient; These are latent variables for a known region.

[0088] Finally, during the reverse diffusion process, to reduce the computational overhead of high-dimensional second-order differentiation, the core optimal transport mapping in the metric function is directly extracted. This is introduced as a structural constraint term into the inverse diffusion sampling update process.

[0089] The denoising network U-Net in step 2.3 adopts a U-shaped symmetrical architecture, including a downsampling encoding path, an intermediate bottleneck layer, and an upsampling decoding path. Its specific structure is as follows: The downsampling encoding path includes one 2D convolutional layer, five cascaded downsampling modules, and one transition module. Each downsampling module consists of one residual block, one self-attention layer, one cross-attention layer, and one downsampling layer. The transition module consists of one residual block, one self-attention layer, and one cross-attention layer. The overall structure of the upsampling decoding path is symmetrical to the encoding path, including one initial module and five cascaded upsampling modules. Before executing the upsampling modules, skip connections are used to concatenate and fuse the shallow spatial features of the corresponding layers in the encoding path with the current features. Finally, a normalization layer and a 2D convolutional layer are used to map back to the target number of channels, outputting the denoising result for the current time step.

[0090] Step 2.4: Construct the output layer, which uses the decoder of MFAE to output the noise-free latent variables from the reverse denoising process of the diffusion model module. Mapping back to physical space yields the complete flow field.

[0091] Step 3: Input the training set constructed in Step 1.2 into the completed model constructed in Step 2 for training. Update the model parameters using an iterative optimization method, setting the batch size to B=16 and the initial learning rate to β=10. -4 The parameters are updated using the Adam optimizer until the model converges and the trained weighted model is obtained.

[0092] Step 4: Evaluate the effectiveness of the conditional latent diffusion model based on multi-source constraints in completing the missing flow field using the test set, including the following sub-steps:

[0093] Step 4.1: Input the test set data obtained in Step 1.2 into the model, use the trained model to complete the missing flow field, and use mean squared error (MSE), mean absolute error (MAE), correlation coefficient (CC), and structural similarity index (SSIM) as evaluation indicators.

[0094] Step 4.2: Compare the results with the Multiscale Autoencoder (MS-AE), Enhanced Super-Resolution Generative Adversarial Network (ESR-GAN), and U-Shaped Self-Attention Driven Variable Generative Adversarial Network (UAD-GAN) to verify the performance of this model.

[0095] like Figure 4 As shown, the performance metrics of the MSCLDM proposed in this invention, compared with MS-AE, ESR-GAN, and UAD-GAN models, are illustrated on a channel stream dataset under different missing rates. Figure 4 (a) Mean Absolute Error (MAE) and Figure 4 (b) In the Mean Squared Error (MSE), the prediction error of all models increases with the increase of the missing percentage. However, MSCLDM significantly outperforms other models across all missing percentage gradients. Even under extreme conditions with a missing percentage as high as 65%, its MAE remains below 1.7%, and its MSE remains at an extremely low level, demonstrating strong robustness. Figure 4 In (c) CC and (d) SSIM indices, the model of this invention consistently maintains the highest value. In particular, in the SSIM index, when the missing rate reaches 65.0%, the performance of the traditional model degrades significantly, decreasing by approximately 0.65-0.70, while MSCLDM can still maintain above 0.73. This indicates that the model can effectively acquire the complex wall turbulence structure in the channel flow, reflecting its ability to process incomplete flow field data in practical scenarios such as marine meteorological observation.

[0096] This implementation case 2 uses cylindrical wake data to verify the effectiveness of the method of the present invention in the task of completing missing flow field data. Its processing flow is applicable to the scenario of completing missing turbulence data in marine meteorological observation, and includes the following steps:

[0097] Step 1: Turbulence data acquisition, including the following sub-steps:

[0098] Step 1.1: Numerical simulation of two-dimensional flow around a cylinder under different Reynolds numbers is performed based on the SST k-ω method. Fluent software is used for the numerical simulation. The simulation satisfies the incompressible viscous fluid assumption, solving only the continuity and momentum equations, neglecting the energy conservation equation. The computational domain is set as a two-dimensional rectangular region with the cylinder diameter D as the characteristic length, where D = 20 mm. The computational domain size is 40D × 20D. The distance from the cylinder center to the inlet and upper and lower boundaries is 10D, and the distance to the outlet is 30D. Structured meshing is used, with local refinement in the incoming and wake regions. The time step is 1 × 10⁻⁶. -6 Unsteady-state calculations were performed under the condition of a total simulation time of 0.001s, resulting in 5000 time steps of cylindrical wake data, from which two-dimensional velocity and pressure field data were extracted.

[0099] Step 1.2: For datasets obtained from numerical simulations under different Reynolds number conditions, construct corresponding sample sets. Each Reynolds number contains 1000 two-dimensional flow field slices. Construct training and testing sets at a 4:1 ratio. During the training phase, select the Reynolds number set {3.0 × 10⁻⁶}. 6 4.0×10 6 4.5×10 6 The model was trained using the training set of}, and the testing phase used a Reynolds number of 3.0 × 10⁻⁶. 6 The test set was used to verify the identically distributed performance, and the Reynolds number set that was not used in training was {3.5 × 10⁻⁶}. 6 5.0×10 6 The test set is used to evaluate the model's generalization ability across Reynolds numbers. In this embodiment, the missing percentages for all datasets are set to: 12.5%, 25%, 35%, 45%, 55%, and 65%.

[0100] Step 2: Design and construct a flow field completion model to verify the effectiveness of the method of the present invention in the task of completing missing flow field data. Its processing flow is applicable to the scenario of completing missing turbulence data in marine meteorological observation, including the following sub-steps:

[0101] Step 2.1: Construct the input layer, which will contain the complete flow field data. and low-resolution flow field data obtained after downsampling. Organized as tensors, both of these tensors have dimensions (N, H, W, C), where: N represents the number of flow fields, H and W are the number of grids in the x and y spatial directions of the two-dimensional flow field at each time step, respectively, and C represents the number of channels, including both velocity components. and pressure .

[0102] Step 2.2: Construct a multi-source condition constraint guidance module, which includes a global information constraint unit, a physical information encoding unit, and an energy function guidance unit.

[0103] The global information constraint unit in step 2.2 includes the MFAE encoder and the Fast Fourier Convolution (FFC) module. Low-resolution data. First, the encoder module of MFAE extracts global features, and then the FFC module performs frequency domain feature enhancement to obtain the global variable z representing the global structure information of the flow field. g The FFC module adopts a dual-branch architecture, such as... Figure 2As shown, the left side includes two parallel convolutional layers and one ReLU activation function, while the right side includes one convolutional layer and one Spectral Transform Block (STB). The STB consists of one convolutional layer, one Fast Fourier Transform layer, one Inverse Fast Fourier Transform layer, and one residual connection. Subsequently, the global variable z... g The input is fed into the cross-attention layer of the denoising network U-Net, which is used to guide the global structure of the generated results at each stage of the backdiffusion.

[0104] The physical information encoding unit in step 2.2 consists of an input layer and a FiLM encoding module. First, the input Reynolds number scalar parameter Re is normalized, and then input into the FiLM encoding module, which consists of three multilayer perceptrons, for nonlinear mapping to obtain the corresponding physical feature vector g. The feature vector g is then input into the energy function guiding unit as a condition variable to participate in the reverse diffusion process of the diffusion model, thereby establishing the correlation between physical parameters and the multi-scale structure of the flow field.

[0105] The energy function guiding unit in step 2.2 is based on the energy function. Build, It consists of 3 convolutional layers and 1 multilayer perceptron, and the flow field latent variable z corresponds to each time step t of the diffusion sampling. t The concatenated vector is then concatenated with the physical feature vector g, and this concatenated vector is input into the energy function. Energy value obtained from The energy value is obtained by utilizing the automatic differentiation mechanism of the deep learning computing framework. Regarding its input item z t By taking the derivative, we obtain the expression in the spatial dimension that is related to z. t Same physical constraint gradient term This is introduced into the inverse sampling process of the diffusion model to guide the denoising process with physical conditions. The formula for the above process is:

[0106]

[0107] in, This is the guided mean after correction for the physical gradient of the energy function. The original mean value of the noise predicted by the denoising network U-Net at time step t is derived; z t Represents the noisy flow field latent variables at time step t; g represents the physical eigenvectors; z g Represents a global variable; These are weighting coefficients that vary over time. .

[0108] Step 2.3: Construct the diffusion model module. This module is used to generate and complete the missing flow field in the latent feature space, including latent space mapping, forward diffusion process, and reverse diffusion process guided by the multi-source condition constraint guidance module. The reverse diffusion process includes mask consistency mechanism, CAD mechanism, and denoising network U-Net. Its specific implementation is as follows:

[0109] The input complete flow field data is processed by a pre-trained encoder of MFAE. Nonlinear feature extraction is performed to map the latent variable z0 from the physical space to the latent feature space, yielding an initial latent variable z0. Next, a forward diffusion process is executed in the latent space, progressively injecting Gaussian noise to add noise to the latent variable z0 until a noisy latent variable z0 conforming to a standard Gaussian distribution is obtained at the T-th time step. T .

[0110] During the inference phase, this invention uses randomly initialized Gaussian noise z T The reverse diffusion process begins, gradually eliminating noise through the U-Net denoising network to recover the complete flow field latent variables. During each time step t, the generated trajectory is collaboratively guided by a multi-source condition constraint guidance module. First, the global variable z... g Injecting cross-attention layers into the denoising network U-Net to achieve global structure guidance; synchronously constructing the energy function. Using an automatic differentiation mechanism to determine the energy value Regarding its input item z t The gradient is obtained by taking the derivative. The gradient is then used as a correction term in the sampling equation to correct the sampling direction, ensuring that the generated results converge to the physically feasible region. Simultaneously, a CAD mechanism constraint term based on optimal transport theory is constructed. By using the features of the known observation area mapped to the latent space as a distribution reference, the Brenier potential function is calculated and its gradient with respect to the latent variables is extracted. The consistency between the generated flow field and the known flow field in spatial distribution is dynamically corrected; finally, a hard data constraint is applied to the known region by combining a mask consistency operation, so that the sampling and update process of the multi-constraint-guided reverse diffusion process can be uniformly represented as:

[0111]

[0112]

[0113] in, This represents the intermediate latent variables generated at time step t-1 after the above multi-source constraint guidance, before the mask consistency operation is performed; This represents the updated latent variables that proceed to the next time step sampling operation after the mask consistency operation; μ θ(z t , t | z g The original mean value of the noise predicted by the denoising network U-Net at time step t is the result of the derivation of the noise. and These are the weighting coefficients of the context-adaptive difference constraint mechanism and the physical constraint term of the energy function, respectively. , ; To define the variance control parameter for the magnitude of random noise injected at time step t, ; Represents the true standard Gaussian noise of the sample; MaskConsist( The function represents the mask consistency operation; after T = 1000 time steps, the fully denoised flow field latent variables can be obtained. .

[0114] The mask consistency constraint mechanism in step 2.3, such as Figure 3 As shown, the specific implementation method for maintaining data consistency in the known region during flow field completion is as follows:

[0115] Let the mask matrix be M, where the known regions have values ​​of 1 and the missing regions have values ​​of 0. The original complete latent variable z0 is calculated using the forward noise addition formula of the diffusion model to obtain the noisy latent variable z0 for the known regions at step t-1. t-1 In time step t of the reverse denoising process, the current sampled values ​​of the denoising network U-Net are processed to obtain the prediction generation result for the current step. Then, the mask matrix M is used to apply the mask matrix M to z. t-1 and The two parts are weighted and fused to update the final latent variable sample values. Its expression is:

[0116]

[0117] in, This indicates element-wise multiplication. The updated... This serves as the input for the reverse iteration at the next time step t-1, continuing until t=0. Through the above operations, the known region is always forced to remain the noisy version of the original observation data during the iteration process, while the missing region is filled with physically consistent data generated by the model, thus ensuring the data consistency of the completion result on the known boundaries.

[0118] The CAD mechanism in step 2.3 is used to constrain the structural consistency between the generated flow field and the known flow field. Its specific implementation is as follows:

[0119] First, an adaptive scaling function f based on feature space differences is constructed. The latent variable z used to generate the complete flow field from the completed model. t Latent variables in the known region The expression for the dynamic adjustment of constraint weights based on differences in the context feature space is as follows:

[0120]

[0121] in, This represents the intensity adjustment coefficient, which is set in this case. =1.0; This represents the attenuation sensitivity parameter, which is set in this case. =2.0; It is a lightweight convolutional network consisting of 3 convolutional layers and 1 ReLU activation function; This represents the L2 norm. When there are significant differences between the generated features and known features, the adaptive weights increase, thereby strengthening the constraint on regions with inconsistent structures.

[0122] Secondly, a difference measurement function is constructed based on optimal transmission theory. Introducing the Brenier potential function The function consists of three multilayer sensing mechanisms, and the function is related to the current flow field latent variable z. t gradient The expression for the difference metric function, representing the optimal transport mapping direction from the generated flow field to the reference flow field, is as follows:

[0123]

[0124] in, Represents the square of the L2 norm; Let the Brenier potential function be the latent variable z of the current flow field. t The gradient; These are latent variables for a known region.

[0125] Finally, during the reverse diffusion process, to reduce the computational overhead of high-dimensional second-order differentiation, the core optimal transport mapping in the metric function is directly extracted. This is introduced as a structural constraint term into the inverse diffusion sampling update process.

[0126] The denoising network U-Net in step 2.3 adopts a U-shaped symmetrical architecture, including a downsampling encoding path, an intermediate bottleneck layer, and an upsampling decoding path. Its specific structure is as follows: The downsampling encoding path includes one 2D convolutional layer, five cascaded downsampling modules, and one transition module. Each downsampling module consists of one residual block, one self-attention layer, one cross-attention layer, and one downsampling layer. The transition module consists of one residual block, one self-attention layer, and one cross-attention layer. The overall structure of the upsampling decoding path is symmetrical to the encoding path, including one initial module and five cascaded upsampling modules. Before executing the upsampling modules, skip connections are used to concatenate and fuse the shallow spatial features of the corresponding layers in the encoding path with the current features. Finally, a normalization layer and a 2D convolutional layer are used to map back to the target number of channels, outputting the denoising result for the current time step.

[0127] Step 2.4: Construct the output layer, which uses the decoder of MFAE to output the noise-free latent variables from the reverse denoising process of the diffusion model module. Mapping back to physical space yields the complete flow field.

[0128] Step 3: Input the training set constructed in Step 1.2 into the completed model constructed in Step 2 for training. Update the model parameters using an iterative optimization method, setting the batch size to B=16 and the initial learning rate to β=10. -4 The parameters are updated using the Adam optimizer until the model converges and the trained weighted model is obtained.

[0129] Step 4: Evaluate the effectiveness of the conditional latent diffusion model based on multi-source constraints in completing the missing flow field using the cylindrical wake test set, including the following sub-steps:

[0130] Step 4.1: Input the test set data obtained in Step 1.2 into the model, use the trained model to complete the missing flow field, and use mean squared error (MSE), mean absolute error (MAE), correlation coefficient (CC), and structural similarity index (SSIM) as evaluation indicators.

[0131] Step 4.2: Compare the results with the Multiscale Autoencoder (MS-AE), Enhanced Super-Resolution Generative Adversarial Network (ESR-GAN), and U-Shaped Self-Attention Driven Variable Generative Adversarial Network (UAD-GAN) to verify the performance of this model.

[0132] like Figure 5 As shown, experiments with different missing rates were conducted to evaluate the model's adaptability to incomplete flow field data. The error curve of MSCLDM showed the most gradual increase as the missing rate increased from 12.5% ​​to 65.0%. Figure 5As can be seen from the MSE analysis in (b), when the missing rate exceeds 45.0%, the error between MS-AE and ESR-GAN increases significantly. However, the present invention effectively suppresses the spread of error through diffusion reconstruction with physical consistency. Figure 5 Figures (c) and (d) show that in complex flow field reconstruction, the CC and SSIM curves of the MSCLDM are consistently at the top. Even with a high missing rate of 65.0%, the CC index remains above 0.92, and the SSIM remains around 0.82. These results demonstrate that the MSCLDM proposed in this invention can accurately capture the spatiotemporal evolution structure of the detached vortex behind the cylinder, effectively reconstructing the dynamic evolution characteristics and physical details of the flow field even with limited known data.

[0133] To verify the applicability of the proposed method under different flow conditions, generalization tests were conducted on a cylindrical wake dataset under different Reynolds numbers. The training data included Reynolds numbers of 3.0 × 10⁻⁶. 6 4.0×10 6 and 4.5×10 6 The flow field sample was selected as 3.5 × 10⁻⁶. 6 and 5.0×10 6 The corresponding data were used as interpolation and extrapolation test conditions, respectively. Table 1 shows the test conditions for a Reynolds number of 3.5 × 10⁻⁶. 6 The interpolation test results at different missing percentages are shown in Table 1. As can be seen, this method can effectively reconstruct the flow field under various missing percentage conditions. Even when the missing percentage reaches 65%, the SSIM values ​​for the velocity and pressure fields still reach 0.8064 and 0.7634, respectively, indicating that the reconstructed results maintain a high degree of consistency with the reference data in terms of overall structure. Table 2 shows the interpolation test results for a Reynolds number of 5.0 × 10⁻⁶. 6 The extrapolation test results for this condition, which exceeds the distribution range of the training data, show in Table 2 that the proposed method still maintains relatively stable reconstruction performance. Although some evaluation indicators show a certain degree of decline, the main structural features of the flow field can still be effectively recovered. The results in Tables 1 and 2 demonstrate that the MSCLDM proposed in this invention has good reconstruction capabilities under both interpolation and extrapolation conditions, and is applicable to flow field completion tasks under different Reynolds numbers. This verifies the applicability and stability of the invention under complex flow conditions, reflecting its ability to process incomplete flow field data in practical scenarios such as marine meteorological observations.

[0134] Table 1 shows the values ​​at Re = 3.5 × 10⁻⁶. 6 Under the conditions, the quantitative evaluation of the extrapolation capability of MSCLDM proposed in this invention

[0135]

[0136] Table 2. When Re = 5.0 × 10 6Under the conditions, the quantitative evaluation of the extrapolation capability of MSCLDM proposed in this invention

[0137]

[0138] The specific examples described herein are merely illustrative of the invention. Those skilled in the art to which this invention pertains may make various modifications, additions, or similar substitutions to the described specific examples without departing from the scope defined by this invention.

Claims

1. A method for missing completion of a turbulent flow field based on a conditional latent diffusion model, characterized in that, Includes the following steps: Step 1: Obtain raw turbulence data, preprocess the data, and establish a dataset. The specific process is as follows: Step 1.

1. Obtain the two-dimensional turbulent velocity field and pressure field data of T time steps from the database, and map the data into two-dimensional flow field data in the form of HxW two-dimensional grid, each grid point containing two velocity components and pressure where represent the velocities in the x, y directions, respectively; Step 1.2: Perform missing processing on the two-dimensional flow field data described in Step 1.1, set different missing ratios to generate masks to construct a missing flow field dataset, and divide the missing flow field dataset into a training set and a test set in a 4:1 ratio; Step 2: Design and construct the flow field completion model, which includes: a multi-scale Fourier autoencoder module, a multi-source conditional constraint guidance module, and a diffusion model module. The specific process is as follows: Step 2.1, construct input layer, complete flow field data and low-resolution flow field data obtained by downsampling processing respectively organized as a tensor form, the dimensions of the above two tensors are (N, H, W, C), wherein: N represents the number of flow fields, H and W are the grid numbers of each time step in the x and y space directions respectively, C represents the channel number, including two velocity components and pressure ; Step 2.2: Construct a multi-source condition constraint guidance module, which includes: a global information constraint unit, a physical information encoding unit, and an energy function guidance unit; Step 2.3: Construct the diffusion model module, which includes latent space mapping, forward diffusion process, and reverse diffusion process guided by the multi-source condition constraint guidance module. The reverse diffusion process includes mask consistency mechanism, context adaptive difference constraint mechanism, and denoising network U-Net. Step 2.4: Construct the output layer, which uses a multi-scale Fourier autoencoder to output the noise-free latent variables from the inverse denoising process of the diffusion model module. Mapping back to physical space yields the completed flow field. Step 3: Input the training set constructed in Step 1.2 into the completed model constructed in Step 2 for training. Update the model parameters using an iterative optimization method until the model converges and the trained weight model is obtained. Step 4: Use the test set to evaluate the performance of the trained completion model, and use various error evaluation indicators to quantitatively analyze the completion effect of the missing flow field.

2. The method for completing missing turbulent flow fields based on a conditional latent diffusion model according to claim 1, characterized in that: The global information constraint unit in step 2.2 includes a multi-scale Fourier autoencoder and a fast Fourier convolution module. The fast Fourier convolution module adopts a dual-branch architecture, with two parallel convolutional layers and one ReLU activation function on the left, and one convolutional layer and one spectrum transform module on the right. The spectrum transform module includes one convolutional layer, one fast Fourier transform layer, one inverse fast Fourier transform layer, and one residual connection. Low-resolution data... First, global features are extracted using a multi-scale Fourier autoencoder, and then input into a fast Fourier convolution module to obtain the global variable z. g Subsequently, the global variable z g It is fed into the cross-attention layer of the denoising network U-Net; The physical information encoding unit in step 2.2 consists of an input layer and a FiLM encoding module. First, the input Reynolds number Re is normalized, and then it is input into the FiLM encoding module composed of three multilayer perceptrons for nonlinear mapping to obtain the corresponding physical feature vector g. The feature vector g is then input into the energy function guidance unit as a condition variable to guide the reverse diffusion process of the diffusion model. The energy function guiding unit in step 2.2 is based on the energy function. Build, It consists of 3 convolutional layers and 1 multilayer perceptron, and the flow field latent variable z corresponds to each time step t of the diffusion sampling. t The concatenated vector is obtained by concatenating the physical feature vector g with the concatenated vector, and then inputting the concatenated vector into the energy function. Energy value obtained from The energy value is obtained by utilizing the automatic differentiation mechanism of the deep learning computing framework. Regarding its input item z t By taking the derivative, we obtain the expression in the spatial dimension that is related to z. t Same physical constraint gradient term When this is introduced into the backsampling process of the diffusion model, the calculation formula for the above process is as follows: in, This is the guided mean after correction for the physical gradient of the energy function. The original mean value of the noise predicted by the denoising network U-Net at time step t is derived; z t Represents the noisy flow field latent variables at time step t; g represents the physical eigenvectors; z g Represents a global variable; These are weighting coefficients that change over time.

3. The method for completing missing turbulent flow fields based on a conditional latent diffusion model according to claim 2, characterized in that: The multi-scale Fourier autoencoder in step 2.2 consists of an encoder and a decoder. The encoder includes one position encoding layer, three cascaded downsampling modules, and one multilayer perceptron. The downsampling module consists of three parallel Fourier convolutional modules of different scales, one downsampling layer, and one residual connection. The Fourier convolutional module has two branches: the left branch consists of one 2D convolutional layer, one normalization layer, one ReLU activation function, and one residual connection; the right branch consists of one fast Fourier transform, one inverse Fourier transform, one residual connection, and one 2D convolution. The decoder consists of one multilayer perceptron, three sets of upsampling modules, and one shape transformation layer. The upsampling module consists of three parallel Fourier convolutional modules of different scales, one upsampling layer, and one residual connection.

4. The method for completing missing turbulent flow fields based on a conditional latent diffusion model according to claim 2, characterized in that: Step 2.3, construct the diffusion model module, the specific implementation method is as follows: The input complete flow field data is processed by a pre-trained multi-scale Fourier autoencoder. Nonlinear feature extraction is performed to obtain an initial latent variable z0. A forward diffusion process is then performed in the latent space, gradually injecting Gaussian noise to add noise to the latent variable z0 until a noise latent variable z conforming to a standard Gaussian distribution is obtained at the T-th time step. T ; During the inference phase, the randomized Gaussian noise z is used. T The reverse diffusion process begins, and noise is gradually eliminated through a denoising network U-Net to obtain the complete flow field latent variable z0. During the iteration at each time step t, the global variable z is first... g Inject into the cross-attention layer of the denoising network U-Net; simultaneously construct the energy function. Using an automatic differentiation mechanism to determine the energy value Regarding its input item z t The gradient is obtained by taking the derivative. The gradient is used as a correction term in the sampling equation to correct the sampling direction; a context-adaptive difference constraint mechanism based on optimal transport theory is constructed, using the features of the known observation area mapped to the latent space as the distribution reference, calculating the Brenier potential function and extracting its relationship with the latent variable z. t gradient Finally, a hard data constraint is applied to the known region using a mask consistency operation. The specific process is uniformly represented by the following formula: in, This represents the intermediate latent variables generated at time step t-1 after the above multi-source constraint guidance, before the mask consistency operation is performed; This represents the updated latent variables that proceed to the next time step sampling operation after the mask consistency operation; μ θ (z t , t | z g The original mean value of the noise predicted by the denoising network U-Net at time step t is the result of the derivation of the noise. and These are the weight coefficients of the context-adaptive difference constraint mechanism and the physical constraint term of the energy function, respectively. Variance control parameters for the magnitude of random noise injected at time step t; Represents the true standard Gaussian noise of the sample; MaskConsist( The function representing the mask consistency operation, after T time steps of reverse sampling, yields the fully denoised flow field latent variables. .

5. The method for completing missing turbulent flow fields based on a conditional latent diffusion model according to claim 2, characterized in that: The specific implementation of the mask consistency constraint mechanism in step 2.3 is as follows: Let the mask matrix be M, where the known regions have values ​​of 1 and the missing regions have values ​​of 0. The original complete latent variable z0 is calculated using the forward noise addition formula of the diffusion model to obtain the noisy latent variable z0 for the known regions at step t-1. t-1 In time step t of the reverse denoising process, the current sampled values ​​of the denoising network U-Net are processed to obtain the prediction generation result for the current step. Then, the mask matrix M is used to apply the mask matrix M to z. t-1 and The two parts are weighted and fused to update the final latent variable sample values. Its expression is: in, This indicates element-wise multiplication; the updated result will be... As the input for the reverse iteration at the next time step t-1, until iteration reaches t=0, the above operation ensures that the known region is always forced to remain the noisy version of the original observation data during the iteration process, while the missing region is filled by the physically consistent data generated by the model, thereby ensuring the data consistency of the completion result on the known boundary. The specific implementation of the context-adaptive difference constraint mechanism in step 2.3 is as follows: First, an adaptive scaling function f based on feature space differences is constructed. The latent variable z used to generate the complete flow field from the completed model. t Latent variables in the known region The expression for the dynamic adjustment of constraint weights based on differences in the context feature space is as follows: in, Indicates the intensity adjustment coefficient; This represents the attenuation sensitivity parameter; It is a lightweight convolutional network consisting of 3 convolutional layers and 1 ReLU activation function; Represents the L2 norm; Secondly, a difference measurement function is constructed based on optimal transmission theory. Introducing the Brenier potential function The function consists of three multilayer sensing mechanisms, and the function is related to the current flow field latent variable z. t gradient The optimal transport mapping direction from the generated flow field to the reference flow field is represented by the difference metric function, which is expressed as: in, Represents the square of the L2 norm; Let the Brenier potential function be the latent variable z of the current flow field. t The gradient; For known latent variables in the region; Finally, during the reverse diffusion process, the core optimal transport mapping in the metric function is extracted. This is introduced as a structural constraint term into the inverse diffusion sampling update process; The denoising network U-Net in step 2.3 adopts a U-shaped symmetrical architecture, including a downsampling encoding path, an intermediate bottleneck layer, and an upsampling decoding path. Its specific structure is as follows: The downsampling encoding path includes one two-dimensional convolutional layer, five cascaded downsampling modules, and one transition module. The downsampling module consists of one residual block, one self-attention layer, one cross-attention layer, and one downsampling layer. The transition module consists of one residual block, one self-attention layer, and one cross-attention layer. The overall structure of the upsampling decoding path is symmetrical to the encoding path, including one initial module and five cascaded upsampling modules. Before executing the upsampling modules, skip connections are used to concatenate and fuse the shallow spatial features of the corresponding level in the encoding path with the current features. Finally, a normalization layer and a two-dimensional convolutional layer are used to map back to the target number of channels, and the denoising result of the current time step is output.

6. The method for completing missing turbulent flow fields based on a conditional latent diffusion model according to claim 2, characterized in that, The model evaluation calculation formula in step 4 is as follows: in, Mean square error; Mean absolute error; The correlation coefficient; is the structural similarity index; N is the number of samples; i represents the sample index; For real flow field data, Flow field data to complete the model, For the i-th sample value of the real flow field data, The i-th sample value of the flow field data used to complete the model. and These represent the mean values ​​of the actual flow field data and the model-completed flow field data, respectively. and These represent the standard deviations of the actual flow field data and the standard deviations of the flow field data completed by the model, respectively. Represents covariance; and These represent the mean of the actual flow field data and the mean of the flow field data completed by the model, respectively. and To avoid stability constants with a denominator of zero; For the summation operator, It is an absolute value operator.

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