Three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraint
The method of super-resolution reconstruction of three-dimensional flow field with Bayesian physical constraints utilizes residual feature distillation and physical guided attention layer to extract spatial features, and optimizes model parameters through Bayesian methods. This solves the problems of high computational cost and low accuracy of high-resolution flow field data, and achieves high-precision and physically reasonable flow field reconstruction.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- DALIAN MARITIME UNIVERSITY
- Filing Date
- 2026-01-23
- Publication Date
- 2026-05-15
AI Technical Summary
Existing technologies are computationally expensive, time-consuming, and have low accuracy when acquiring high-resolution three-dimensional flow field data. Data-driven methods lack generalization ability and the reconstruction results have poor physical interpretability.
A three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints is adopted. Through a spatial feature extraction module and a spatiotemporal collaborative reconstruction module, initial spatial features are extracted using a residual feature distillation component and a physically guided attention layer. The model parameters are then optimized through Bayesian physical constraints to achieve high-resolution data reconstruction.
While ensuring computational efficiency, it outputs flow field predictions with high physical reliability, improves reconstruction accuracy, and achieves a dual improvement in spatial and temporal resolution, ensuring that the flow field is continuous, smooth, and conforms to the laws of fluid dynamics in the time dimension.
Smart Images
Figure CN122048652A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of super-resolution processing technology for low-resolution flow field data, and in particular to a method for super-resolution reconstruction of three-dimensional flow fields based on Bayesian physical constraints. Background Technology
[0002] High-resolution three-dimensional flow field data has significant application value in aerodynamic structure analysis, fluid mechanism research, aircraft equipment performance evaluation, and atmospheric evolution prediction. Therefore, obtaining high-resolution flow fields is one of the core objectives of research in related fields. In existing technologies, computational fluid dynamics (CFD) methods are an important means of obtaining high-precision flow field information. However, to analyze small-scale fluid structures, extremely fine meshes and dense time sampling are often required, which places extremely high demands on computational costs. This is especially true in three-dimensional flow field simulations, where the algorithm complexity is significantly increased.
[0003] Therefore, recovering fine fluid structures from flow field data with limited spatial resolution has gradually become an important research direction. Among them, constructing super-resolution reconstruction models using data-driven methods is an effective way to recover fine fluid structures from flow field data with limited spatial resolution. However, the high dependence of this method on training data can lead to problems such as insufficient generalization and poor physical interpretability of reconstruction results, which greatly reduces the accuracy of reconstruction. Summary of the Invention
[0004] This invention provides a method for super-resolution reconstruction of three-dimensional flow fields based on Bayesian physical constraints, in order to overcome the technical problems of high computational cost, long time cycle and low accuracy of existing methods for obtaining high-resolution three-dimensional flow field data.
[0005] To achieve the above objectives, the technical solution of the present invention is as follows: A method for super-resolution reconstruction of three-dimensional flow fields based on Bayesian physical constraints, comprising the following steps: S1. Acquire multiple frames of high-resolution 3D data and corresponding low-resolution 3D data to construct a training set; S2. Establish a three-dimensional flow field super-resolution reconstruction model. Train the three-dimensional flow field super-resolution reconstruction model based on the training set. During the training process, optimize the parameters in the three-dimensional flow field super-resolution reconstruction model based on Bayesian physical constraints. When the set loss function converges, obtain the trained three-dimensional flow field super-resolution reconstruction model. The three-dimensional flow field super-resolution reconstruction model includes a spatial feature extraction module and a spatiotemporal collaborative reconstruction module; The spatial feature extraction module, by introducing a residual feature distillation component and a physical guidance attention layer, can extract several initial spatial features fused with physical attention weights from the training set and transmit the initial spatial features to the spatiotemporal collaborative reconstruction module. The spatiotemporal collaborative reconstruction module is used to perform temporal interpolation on several initial spatial features to obtain interpolated spatial features, and extract spatiotemporal features from the initial spatial features and interpolated spatial features. Finally, pixel rearrangement is performed based on the spatiotemporal features to output three-dimensional high-resolution data. S3. Based on the trained 3D flow field super-resolution reconstruction model, 3D flow field reconstruction is performed on real-world 3D low-resolution data, and 3D high-resolution data is output.
[0006] Furthermore, the spatial feature extraction module includes several residual feature distillation modules, each of which includes a shallow feature extraction layer, a first feature distillation module, a second feature distillation module, a third feature distillation module, a fourth feature distillation module, and a first convolutional layer connected in sequence. In the process of extracting several initial spatial features from the training set, the spatial feature extraction module performs the following specific operations on each frame of 3D low-resolution data in the training set: After shallow features are extracted from the three-dimensional low-resolution data through the shallow feature extraction layer, they are transmitted to the first feature distillation module. After performing feature distillation and physical attention weight fusion operations on the shallow features by the first feature distillation module, the first distilled feature is output and transmitted to the second feature distillation module. After performing feature distillation and physical attention weight fusion operations on the first distillation feature through the second feature distillation module, the second distillation feature is output and transmitted to the third feature distillation module. After performing feature distillation and physical attention weight fusion operations on the second distilled feature through the third feature distillation module, the third distilled feature is output and transmitted to the fourth feature distillation module. After performing feature distillation and physical attention weight fusion operations on the third distilled feature through the fourth feature distillation module, the fourth distilled feature is output. After concatenating the first distillation feature, the second distillation feature, the third distillation feature, and the fourth distillation feature according to the channel dimension, the fifth distillation feature is output and transmitted to the first convolutional layer. After channel adjustment of the fifth distillation feature through the first convolutional layer, the initial spatial features corresponding to a single frame of three-dimensional low-resolution data are output. Furthermore, the first feature distillation module, the second feature distillation module, the third feature distillation module, and the fourth feature distillation module have the same structure, each including a first distillation branch, a second distillation branch, a third distillation branch, a fourth distillation branch, a first residual branch, a second residual branch, a third residual branch, a physically guided attention layer, and a second convolutional layer; Furthermore, the specific steps for each feature distillation module to perform feature distillation and physical attention weight fusion operations on the input features include: After performing a convolution operation on the input features through the first distillation branch, the first convolutional feature is output. The input features are first convolved by the first residual branch, and then the convolution result and the input features are residually concatenated to output the first residual feature, which is then transmitted to the second distillation branch and the second residual branch respectively. After performing a convolution operation on the first residual feature through the second distillation branch, the second convolution feature is output. The first residual feature is first convolved by the second residual branch, and then the convolution result and the first residual feature are residually concatenated to output the second residual feature, which is then transmitted to the third distillation branch and the third residual branch respectively. After performing a convolution operation on the second residual feature through the third distillation branch, the third convolution feature is output. The second residual feature is first convolved by the second residual branch, and then the convolution result and the second residual feature are convolved by residual connection to output the third residual feature and transmit it to the fourth distillation branch. After performing a convolution operation on the third residual feature through the fourth distillation branch, the fourth convolution feature is output. The first, second, third, and fourth convolutional features are concatenated according to the channel dimension to output the fifth convolutional feature; After channel adjustment of the fifth convolutional feature through the second convolutional layer, the sixth convolutional feature is output. After calculating the 3D low-resolution data in the training set through the physical guidance attention layer, the physical attention weights are output. After performing element-wise multiplication on the sixth convolutional feature and the physical attention weight, the weighted feature is output. After performing a residual connection between the weighted features and the features input to the first distillation branch, the distillation features are output.
[0007] Furthermore, the specific steps for the physically guided attention layer to perform calculations on the 3D low-resolution data in the training set include: Calculate vorticity separately , , , means as follows:
[0008] in, These represent the velocities of the model's predicted output in the x, y, and z directions, respectively. Based on the physical constraints of the vortex transport equation, the velocity field and vortex field residuals in the x, y, and z directions can be obtained, as follows:
[0009] in, Indicates kinematic viscosity. Represents the Laplace operator symbol; This represents the set of predicted velocity fields output by this model. This represents the set of predicted vorticity fields output by this model; Attention weights The calculation formula is:
[0010] Where Sig represents the Sigmoid activation function. This represents the initial low-resolution input data. This represents the attention weights obtained after training.
[0011] Furthermore, the spatiotemporal collaborative reconstruction module includes a time interpolation unit, several first three-dimensional gated loop units, a second three-dimensional gated loop unit, a first pixel rearrangement unit, and a second pixel rearrangement unit; The first three-dimensional gated loop unit is used to extract a first spatiotemporal feature from the initial spatial feature, and the first pixel rearrangement unit is used to rearrange the pixels of the first spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: The first three-dimensional gated loop unit processes the initial spatial feature sequence sequentially in time order. At each integer time t, it takes the hidden state of the previous time (t-1) and the initial spatial feature of the current time as input, calculates and outputs the hidden state of the current time as the first spatiotemporal feature; wherein, the hidden state at the initial time t=1 is initialized to zero. The first spatiotemporal features output by the first three-dimensional gated loop unit are rearranged pixel by pixel by the first pixel rearrangement unit, and the corresponding three-dimensional high-resolution data is output. The time interpolation unit is used to perform frame interpolation operations at the time between t+1 and t, i.e., at t+0.5, and outputs the interpolation spatial characteristics, as shown below:
[0012] in, and These represent the outputs of the residual characteristic distillation module at time t and t+1, respectively. The second three-dimensional gated loop unit is used to obtain the second spatiotemporal feature based on the frame interpolation spatial feature, and the second pixel rearrangement unit is used to rearrange the pixels of the second spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: At time t+0.5, the third spatiotemporal feature is obtained by performing an element-wise weighted average of the two first spatiotemporal features output by adjacent first three-dimensional gated recurrent units, as follows:
[0013] The second three-dimensional gated loop unit captures the long-range temporal dependence of the input interpolated spatial features and the third spatiotemporal features in the flow field evolution process, and outputs the second spatiotemporal features corresponding to the current moment. The second pixel rearrangement unit rearranges the pixels of the second spatiotemporal features output by the second three-dimensional gated loop unit and outputs the corresponding three-dimensional high-resolution data.
[0014] Furthermore, the loss function of the three-dimensional flow field super-resolution reconstruction model is defined as follows:
[0015] in, These are represented as network parameters in this model. Represents the data loss function; Indicates residual loss; in:
[0016] in, Represents the high-resolution velocity field predicted by the neural network. The velocity field representing the true label, The physical residual representing the prediction of the 3D flow field super-resolution reconstruction model and the continuity equation is expressed as:
[0017] Furthermore, the optimization of parameters in the 3D flow field super-resolution reconstruction model based on Bayesian physical constraints includes: Probabilistic Bayesian physics constraints are used to reduce the residual between the predicted flow field velocity and the actual high-resolution flow field velocity, and the parameters of the 3D flow field super-resolution reconstruction model are optimized, including: use The residual represents the difference between the reconstructed velocity field data and the true high-resolution velocity field data, and is considered to be a normal distribution with zero mean and negligible variance. The normal distribution of the velocity field residual is defined as follows:
[0018] in, Represents the high-resolution velocity field predicted by the neural network. The velocity field representing the true label, ( ) represents the probability density function. The representative value is 0 and the covariance is The normal distribution The covariance matrix representing the physical residuals; use The residual of the vorticity field data represents the difference between the reconstructed vorticity field data and the true high-resolution vorticity field data. This residual is considered to be a normal distribution with zero mean and negligible variance. The normal distribution of the vorticity field residual is defined as follows:
[0019] in, Represents the high-resolution vorticity field predicted by the neural network. Eddy field representing the true label.
[0020] Beneficial Effects: This invention, through a spatial feature extraction module using a residual feature distillation component and a physically guided attention layer, extracts several initial spatial features from the training set, incorporating physical attention weights. A spatiotemporal collaborative reconstruction module performs temporal interpolation on these initial spatial features to obtain interpolated spatial features. Spatiotemporal features are then extracted from both the initial and interpolated spatial features. Finally, pixel rearrangement is performed based on these spatiotemporal features to output high-resolution 3D data. During training, parameters in the 3D flow field super-resolution reconstruction model are optimized based on Bayesian physical constraints. This method, utilizing real physical constraints, not only outputs flow field predictions with high physical reliability but also optimizes network parameters based on the uncertainty of the prediction results, further improving reconstruction accuracy. This invention achieves more accurate multi-scale flow field modeling while maintaining modeling speed. Simultaneously, the output flow field is continuous and smooth in the temporal dimension and conforms to the laws of fluid dynamics, achieving a dual improvement in spatial and temporal resolution, ensuring the output of spatiotemporally continuous and physically reasonable high-fidelity flow field sequences. Attached Figure Description
[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.
[0022] Figure 1 This is a flowchart of a three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints in this invention; Figure 2 This is a data processing flowchart of the three-dimensional flow field super-resolution reconstruction model in an embodiment of the present invention; Figure 3 This is a schematic diagram of the RFDM and 3D-RFDB modules and the physically guided attention layer in an embodiment of the present invention; Figure 4 These are comparison images of energy spectra reconstructed from the Isotropic dataset using different methods in embodiments of the present invention. Figure 5 This is a bar chart comparing the Channel dataset using different methods in this embodiment of the invention, with L2 norm as the indicator. Figure 6 This is a comparison chart of the visualization results of three-dimensional reconstructed turbulent vortex structures using different methods on the Isotropic dataset in the embodiments of the present invention; Figure 7 This is a comparison chart showing the visualization results of three-dimensional reconstruction of turbulent vortex structures using different methods in the Channel dataset in the embodiments of the present invention. Detailed Implementation
[0023] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.
[0024] This embodiment provides a method for super-resolution reconstruction of three-dimensional flow fields based on Bayesian physical constraints, such as... Figure 1 As shown, the specific steps include: S1. Acquire multiple frames of high-resolution 3D data and corresponding low-resolution 3D data to construct a training set; Specifically, this embodiment acquires three-dimensional high-resolution data at different times of N frames, and downsamples the multi-frame three-dimensional high-resolution data to obtain three-dimensional low-resolution data. The three-dimensional low-resolution data was used as input data for the three-dimensional flow field super-resolution reconstruction model, and the three-dimensional high-resolution data was used as labels. The three-dimensional low-resolution data... In This represents a set of low-resolution input velocity fields. This represents the low-resolution input vorticity field set. The low-resolution data input size is... D, H, and W represent the number of samples in the three-dimensional flow field data grid in the three-dimensional direction, respectively. The larger the number of samples, the higher the resolution.
[0025] S2, Establish as follows Figure 2 The illustrated 3D flow field super-resolution reconstruction model (3DGRU-RFDM) is trained based on the training set. During training, the parameters of the 3D flow field super-resolution reconstruction model are optimized based on Bayesian physical constraints. The trained 3D flow field super-resolution reconstruction model is obtained when the set loss function converges. The three-dimensional flow field super-resolution reconstruction model includes a spatial feature extraction module and a spatiotemporal collaborative reconstruction module; The spatial feature extraction module, by introducing a residual feature distillation component and a physical guidance attention layer, can extract several initial spatial features fused with physical attention weights from the training set and transmit the initial spatial features to the spatiotemporal collaborative reconstruction module. Specifically, at the spatial feature extraction level, this embodiment uses stacked residual feature distillation components, which achieves more accurate multi-scale flow field modeling while ensuring modeling speed. The spatial feature extraction module can accurately extract effective features from the input low-resolution data. The spatiotemporal collaborative reconstruction module is used to perform temporal interpolation on several initial spatial features to obtain interpolated spatial features, and extract spatiotemporal features from the initial spatial features and interpolated spatial features. Finally, pixel rearrangement is performed based on the spatiotemporal features to output three-dimensional high-resolution data. Specifically, in terms of spatiotemporal correlation modeling, the 3D Gated Cyclic (3DGRU) unit in the spatiotemporal collaborative reconstruction module can capture long-range temporal dependencies in the flow field evolution process by utilizing implicitly learned fluid motion evolution laws. Its recursive propagation mechanism ensures that the reconstruction at the current moment not only depends on instantaneous spatial features but also makes full use of the flow inertia and evolutionary trends of past moments, thereby significantly alleviating the temporal discontinuity of the reconstruction results at the details. At the same time, the spatiotemporal collaborative reconstruction module can predict the hidden intermediate states between two adjacent frames in the input sequence based on the learned physical evolution laws by performing highly physically reliable time interpolation. Compared with traditional linear interpolation, it can more accurately capture the nonlinear characteristics of the flow, making the output flow field continuous and smooth in the time dimension and in accordance with the laws of fluid dynamics. Ultimately, it achieves a dual improvement in spatial and temporal resolution, outputting a spatiotemporally continuous and physically reasonable high-fidelity flow field sequence.
[0026] Specifically, 3DGRU units are used to implicitly learn the evolution of fluid motion from a large amount of time-series flow field data and capture the dependencies between consecutive time frames. This temporal modeling capability enables the network to recover complex details in the velocity field that are difficult to infer from single-frame information.
[0027] S3. Based on the trained 3D flow field super-resolution reconstruction model, 3D flow field reconstruction is performed on real-world 3D low-resolution data, and 3D high-resolution data is output.
[0028] Specifically, the method proposed in this embodiment deeply integrates data-driven deep learning technology with the physical laws of fluid mechanics, ensuring computational efficiency while ensuring that the reconstruction results conform to the essential characteristics of fluid motion, thereby improving the reconstruction accuracy. In a specific embodiment, the spatial feature extraction module includes several residual feature distillation modules (RFDM), each of which includes a shallow feature extraction layer, a first feature distillation (3D-RFDB) module, a second feature distillation module, a third feature distillation module, a fourth feature distillation module, and a first convolutional layer connected in sequence. In the process of extracting several initial spatial features from the training set, the spatial feature extraction module performs the following specific operations on each frame of 3D low-resolution data in the training set: After shallow features are extracted from the three-dimensional low-resolution data through the shallow feature extraction layer, they are transmitted to the first feature distillation module. Specifically, the processing expression of the shallow feature extraction layer is as follows:
[0029] in, Represents shallow features, and , This represents a convolutional layer with a 3x3x3 kernel. This represents the Swish activation function.
[0030] After performing feature distillation and physical attention weight fusion operations on the shallow features by the first feature distillation module, the first distilled feature is output and transmitted to the second feature distillation module. After performing feature distillation and physical attention weight fusion operations on the first distillation feature through the second feature distillation module, the second distillation feature is output and transmitted to the third feature distillation module. After performing feature distillation and physical attention weight fusion operations on the second distilled feature through the third feature distillation module, the third distilled feature is output and transmitted to the fourth feature distillation module. After performing feature distillation and physical attention weight fusion operations on the third distilled feature through the fourth feature distillation module, the fourth distilled feature is output. After concatenating the first distillation feature, the second distillation feature, the third distillation feature, and the fourth distillation feature according to the channel dimension, the fifth distillation feature is output and transmitted to the first convolutional layer. After channel adjustment of the fifth distillation feature through the first convolutional layer, the initial spatial features corresponding to a single frame of three-dimensional low-resolution data are output.
[0031] like Figure 3 As shown, each RFDM contains four 3D-RFDB modules, which respectively produce four layers of distillation output. and , means as follows:
[0032] in, ; This represents the operation of feature distillation and physical attention weight fusion on the input features; Then to and Feature fusion is performed, as shown below:
[0033] Wherein, Concat represents the feature concatenation operation; Finally, channel adjustments are performed to obtain a single-frame output of RFDM. , represented as:
[0034] In a specific embodiment, the first feature distillation module, the second feature distillation module, the third feature distillation module, and the fourth feature distillation module have the same structure, each including a first distillation branch, a second distillation branch, a third distillation branch, a fourth distillation branch, a first residual branch, a second residual branch, a third residual branch, a physical guidance attention layer, and a second convolutional layer; Furthermore, the specific steps of each feature distillation module in performing feature distillation and physical attention weight fusion operations on the input features are as follows: Figure 3 As shown, it includes: After performing a convolution operation on the input features through the first distillation branch, the first convolutional feature is output. The input features are first convolved by the first residual branch, and then the convolution result and the input features are residually concatenated to output the first residual feature, which is then transmitted to the second distillation branch and the second residual branch respectively. After performing a convolution operation on the first residual feature through the second distillation branch, the second convolution feature is output. The first residual feature is first convolved by the second residual branch, and then the convolution result and the first residual feature are residually concatenated to output the second residual feature, which is then transmitted to the third distillation branch and the third residual branch respectively. After performing a convolution operation on the second residual feature through the third distillation branch, the third convolution feature is output. The second residual feature is first convolved by the second residual branch, and then the convolution result and the second residual feature are convolved by residual connection to output the third residual feature and transmit it to the fourth distillation branch. After performing a convolution operation on the third residual feature through the fourth distillation branch, the fourth convolution feature is output. The first, second, third, and fourth convolutional features are concatenated according to the channel dimension to output the fifth convolutional feature; After channel adjustment of the fifth convolutional feature through the second convolutional layer, the sixth convolutional feature is output. Specifically, each 3D-RFDB module contains four layers of distillation branches. (i=1,2,3,4) and three-level residual branch (i=1,2,3), the processing procedures for residual branch and distillation branch are as follows:
[0035] in, Represents the Swish activation function; Features representing the input 3D-RFDB module; This is represented as a convolutional layer with a 3x3x3 kernel; This indicates a convolutional layer with a 1x1x1 kernel; Representative input .
[0036] After calculating the 3D low-resolution data in the training set through the physical guidance attention layer, the physical attention weight Phy_att is output. In a specific embodiment, the specific steps for the physically guided attention layer to perform calculations on the three-dimensional low-resolution data in the training set include: Calculate vorticity separately , , , means as follows:
[0037] in, These represent the velocities of the model's predicted output in the x, y, and z directions, respectively. Based on the physical constraints of the vortex transport equation, the velocity field and vortex field residuals in the x, y, and z directions can be obtained, as follows:
[0038] in, Indicates kinematic viscosity. Represents the Laplace operator symbol; This represents the set of predicted velocity fields output by the model. The set of predicted vorticity fields represented by the model output; Attention weights The calculation formula is:
[0039] Where Sig represents the Sigmoid activation function. This represents the initial low-resolution data input into each layer of 3D-RFDM. This represents the attention weights obtained after training.
[0040] Specifically, this embodiment uses the absolute value of the physical residual as the attention weight, which not only makes the output more physically interpretable but also effectively concentrates computational resources on regions with strong physical constraints. Furthermore, employing a physical attention mechanism allows the model to ensure the accuracy of stepwise feature distillation processing through physical constraint equations. After performing element-wise multiplication on the sixth convolutional feature and the physical attention weight, the weighted feature is output. After performing a residual connection between the weighted features and the features input to the first distillation branch, the distillation features are output.
[0041] Specifically, such as Figure 3 As shown, physical attention weights are applied. and distillation results The process involves fusing the components and performing a final residual join operation to obtain the output of a single RFDM, as follows: The distillation results output from the first, second, third, and fourth distillation branches in the 3D-RFDB module are concatenated and output. The number of channels is adjusted using convolution operations to obtain a refined output. , means as follows:
[0042] Using element-wise multiplication to weight physical attention With refined output Multiplication is represented as follows:
[0043] Will With initial input Perform residual connection to obtain the output of a single characteristic distillation module. , means as follows:
[0044] In a specific embodiment, the spatiotemporal collaborative reconstruction module includes a time frame interpolation unit, several first three-dimensional gated loop units, a second three-dimensional gated loop unit, a first pixel rearrangement unit, and a second pixel rearrangement unit; The first three-dimensional gated loop unit is used to extract a first spatiotemporal feature from the initial spatial feature, and the first pixel rearrangement unit is used to rearrange the pixels of the first spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: The first three-dimensional gated loop unit processes the initial spatial feature sequence sequentially in time order. At each integer time t, it takes the hidden state of the previous time (t-1) and the initial spatial feature of the current time as input, calculates and outputs the hidden state of the current time as the first spatiotemporal feature; wherein, the hidden state at the initial time t=1 is initialized to zero. Specifically, at time t=1, the flow field data of the first frame processed by RFDM is then processed by the first three-dimensional gated loop unit and the first pixel rearrangement unit before it can be used as the spatiotemporal super-resolution output of the first frame, and the spatial size of the output flow field data is twice that of the input.
[0045] The first spatiotemporal features output by the first three-dimensional gated loop unit are pixel rearranged through the first pixel rearrangement (PixelShuffle) unit, and the corresponding three-dimensional high-resolution data is output. Specifically, the processing procedure of the first three-dimensional gated loop unit includes: Define reset gates separately Update door and candidate hidden state It is expressed as follows:
[0046] in, Represents the current frame The spatiotemporal features obtained after processing by the first three-dimensional gated loop unit; This represents element-wise multiplication. Represents the Sigmoid function; Conv3D represents a convolution kernel of 3. 3 3 learnable 3D convolutional layers; Specifically, the first pixel rearrangement unit performs a spatial upsampling of the input spatiotemporal features by a factor of two, that is, it doubles the number of grids in the three directions of the three-dimensional space, thereby obtaining high-resolution data for a single frame. This process is represented as follows:
[0047] Specifically, in this embodiment, the time interpolation unit is used to perform interpolation operations at the time between t+1 and t, i.e., at t+0.5, and outputs the interpolation spatial features, as follows:
[0048] in, and These represent the outputs of the RFDM at time t and t+1, respectively. The second three-dimensional gated loop unit is used to obtain the second spatiotemporal feature based on the frame interpolation spatial feature, and the second pixel rearrangement unit is used to rearrange the pixels of the second spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: At time t+0.5, the third spatiotemporal feature is obtained by performing an element-wise weighted average of the two first spatiotemporal features output by adjacent first three-dimensional gated recurrent units, as follows:
[0049] The second three-dimensional gated loop unit captures the long-range temporal dependence of the input interpolated spatial features and the third spatiotemporal features in the flow field evolution process, and outputs the second spatiotemporal features corresponding to the current moment. The second spatiotemporal features output by the second three-dimensional gated loop unit are pixel rearranged using the second pixel rearrangement (PixelShuffle) unit, and the corresponding three-dimensional high-resolution data is output.
[0050] Specifically, in this embodiment, the steps of capturing the long-range temporal dependence of the input interpolated spatial features and the third spatiotemporal features during the flow field evolution process through the second three-dimensional gated loop unit, and outputting the second spatiotemporal feature corresponding to the current moment, include: The interpolation space features are optimized to obtain the optimized interpolation space features, which are expressed as follows:
[0051] The optimized interpolation spatial features and the third spatiotemporal features are processed as follows:
[0052] Finally, the hidden features of the interpolated frames between every two frames in the input time series are obtained, and upsampling is performed to obtain the interpolated frame output. : Specifically, in this embodiment, the process by which the second pixel rearrangement unit rearranges the second spatiotemporal features output by the second three-dimensional gated loop unit is represented as follows:
[0053] In this embodiment, in order to obtain an output with improved spatiotemporal resolution, the first and second three-dimensional gated loop units can process a total of N low-resolution input features processed by RFDM. After time interpolation, the three-dimensional gated loop units are used to extract the temporal characteristics in the flow field data. The pixel rearrangement method is used to upsample and output the effective high-resolution data of the current frame, resulting in a total of 2N-1 high-resolution three-dimensional flow field data.
[0054] In a specific embodiment, to establish a unified constraint condition for practical applications, a Bayesian method is used to perform posterior inference of network parameters. This method, utilizing real physical constraints, not only outputs flow field predictions with high physical reliability but also optimizes network parameters based on the uncertainty of the prediction results, thereby further improving reconstruction accuracy. Specifically, a Bayesian learning framework based on physical constraints is used to optimize and train the 3D flow field super-resolution reconstruction model. Under conditions of sparse data, this enhances the physical consistency and reliability of the model output, giving it better accuracy and physical interpretability, including: In this embodiment, the loss function of the three-dimensional flow field super-resolution reconstruction model is defined as follows:
[0055] in, These are represented as network parameters in this model. Represents the data loss function; Indicates residual loss; Specifically, this embodiment establishes a hybrid loss function for multi-task collaborative optimization, including a physical constraint strategy based on the vortex transport equation to strictly constrain the reconstruction process of the 3D flow field super-resolution model. Simultaneously, both data fitting error and physical constraint residuals are modeled as normally distributed random variables, constructing a joint probability model within a Bayesian framework. The Stein variational gradient descent (SVGD) method is used to accelerate parameter optimization during training. Optimal parameters can be obtained by continuously reducing the loss function during the forward propagation of training. The optimal weights for the 3D super-resolution reconstruction model.
[0056] in:
[0057] in, The physical residual representing the prediction of the 3D flow field super-resolution reconstruction model and the continuity equation is expressed as:
[0058] In this embodiment, given prior physical knowledge, to ensure that the deep learning model's predictions are as close to the true values as possible, the model's parameters can be progressively optimized during the training of the neural network whose predictions closely approximate the true values. The process is represented as:
[0059] in, ·) represents the loss function; In this embodiment, probabilistic Bayesian physics constraints are used to reduce the residual between the predicted flow field velocity and the actual high-resolution flow field velocity, and to optimize the model parameters, including: use The residual represents the difference between the reconstructed velocity field data and the true high-resolution velocity field data. This velocity physical residual is the sum of many high-order infinitesimal quantities, and therefore can be regarded as a normal distribution with zero mean and negligible variance. The normal distribution of the velocity field residual is defined as follows:
[0060] in, Represents the high-resolution velocity field predicted by the neural network. The velocity field representing the true label, ( ) represents the probability density function. The representative value is 0 and the covariance is The normal distribution The covariance matrix representing the physical residuals, with parameters It can be updated in the data.
[0061] use The residual of the vorticity field data represents the difference between the reconstructed vorticity field data and the true high-resolution vorticity field data. This residual is considered to be a normal distribution with zero mean and negligible variance. The normal distribution of the vorticity field residual is defined as follows:
[0062] in, Represents the high-resolution vorticity field predicted by the neural network. Eddy field representing the true label.
[0063] In a specific embodiment, during the training process of the 3D flow field super-resolution reconstruction model, a likelihood function in a probabilistic physical constraint Bayesian learning framework is also constructed, including: First, the difference distribution between the physical residual of the high-resolution velocity field and the predicted value of the neural network velocity field is obtained, as shown below:
[0064] in, Represents a high-resolution velocity field. Represents the known parameters The predicted high-resolution velocity field, This represents the super-resolution vorticity field predicted by the neural network. Represents a true high-resolution vorticity field. R represents the optimized neural network parameters, and R(·) represents the physical residual of the vorticity-velocity equation. Secondly, a likelihood function influenced by physical principles is used to represent the probability of the data's correlation with the model, as follows:
[0065] in, Represents the given parameters Log-likelihood function for data under data item constraints Indicates the given parameters And the logarithmic physical residual likelihood function under physical information constraints, where D represents high-resolution real data and R represents physical residuals; This represents a combination of probability likelihood and equation likelihood, determined by judging the current parameters. The probability of predicting the data error and physical constraint residuals of super-resolution data is transformed from deterministic optimization to probabilistic inference optimization. Specifically, after constructing the likelihood function, the optimal likelihood distribution is inferred through the Bayesian posterior distribution, and Bayesian training is performed, including: The parameters of this model The prior distribution is defined as Data-driven constraints are set as follows: The posterior distribution is obtained using the Bayesian method, and is represented as follows:
[0066] in, This represents the posterior distribution obtained by the Bayesian method, i.e., the training objective; This represents the likelihood function given the parameters. In Bayesian learning, an easily computed distribution can be used. Approximate posterior distribution Furthermore, the Kroba-Leibler (KL) divergence is used as a measure of the difference between the variational distribution and the posterior distribution, i.e., by minimizing the KL divergence during training. The KL divergence method between the parameter and the posterior distribution is used for parameters. Optimize to obtain the optimal parameters. , means as follows:
[0067] Among them, This represents the selected approximate posterior distribution; Specifically, the above equation can redefine Bayesian inference as a deterministic optimization problem, namely, finding an approximate distribution. With This minimizes the KL divergence.
[0068] In this embodiment, the Stein variational gradient descent (SVGD) method is used to solve the above optimization problem, including: First, construct the Stein variational gradient descent update equation: Using a set of n particles Construct the posterior distribution from the previous step and minimize the KL divergence; these particles will move towards the target posterior distribution iteratively and gradually approach the high-probability region. That is, the SVGD method can effectively and monotonically reduce the KL divergence. The variational gradient descent update equation is as follows:
[0069] Among them, Represents the learning rate. Represents the parameters The prior function below The logarithmic probability, Represents the log probability of likelihood based on data-driven approaches. Representative parameters The gradient obtained from backpropagation, This represents using the kernel function value as the gradient. The weight.
[0070] In this embodiment, the parameters in the model are trained using the SVGD algorithm with physical constraints. : First, calculate the log-posterior distribution:
[0071] in, Represents the parameter The logarithmic prior probability, Indicates the parameter The logarithmic prior probability, Represents the given parameters Data likelihood Indicates the given parameters The physical constraint likelihood; Next, the backpropagation method is used to calculate... That is, for parameters The network model is used to obtain the backpropagation gradient, and then the kernel function is calculated. The kernel function represents the parameters. and the parameters input into the kernel function The similarity between them is calculated, and the operator is computed. This operator indicates the direction in which the particle should move; Finally, the optimal model parameters are updated using the Adam optimizer during training. Simultaneously, the model's performance is monitored using the initially constructed validation set during training. Training is stopped when performance no longer improves, and the optimal weights are saved. Ultimately, this model can be deployed for super-resolution of continuous low-resolution flow field data.
[0072] In this embodiment, to verify the ability of the three-dimensional flow field super-resolution reconstruction model to recover small-scale structures of turbulence, an ablation comparison experiment was carried out on the Isotropic dataset using the kinetic energy spectrum (E(K)) as a physical statistical index. Figure 4 The paper presents energy spectrum reconstruction results for different model configurations to evaluate the preservation and reconstruction effects of each model on small-scale energy distribution in the high wavenumber dissipation region. It can be seen that the 3D flow field super-resolution reconstruction model using Bayesian physical constraints and a physically guided attention layer more closely approximates the kinetic energy spectrum of the baseline model. This indicates that the 3D flow field super-resolution reconstruction model can better preserve small-scale vortex structures and possesses excellent physical interpretability. In addition, such as Figure 5 As shown, the experimental results, which use the root mean square error of the velocity field and vorticity field as indicators, show that the error of the three-dimensional flow field super-resolution reconstruction model, which includes physically constrained Baye learning and physically guided attention layers, is much smaller than that of the commonly used Tricubic interpolation method, achieving reductions of 37.9% and 52.6% in the velocity field and vorticity field, respectively.
[0073] Specifically, to further evaluate the physical accuracy of the reconstructed flow field, this embodiment conducts three-dimensional spatial super-resolution experiments on the Isotropic and Channel datasets. Figure 6 and Figure 7 As shown in the figures, the volumetric visualization of the vortex enclosed by the isosurfaces calculated using the Q criterion in both figures can fully demonstrate the reconstructed vortex structure and mass. Figure 6 Visualizing the vortex structure using an isosurface with Q=500 reveals that Bay-3DGRU-RFDN-NSATT demonstrates exceptional ability in reconstructing regular vortex structures, achieving a voxel reconstruction accuracy of 1.14%, which is very close to the baseline of 1.15%. This fully demonstrates that the combination of Bayesian learning and the physics-guided attention layer significantly improves the preservation of small-scale vortices.
[0074] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.
Claims
1. A method for super-resolution reconstruction of three-dimensional flow fields based on Bayesian physical constraints, characterized in that, The specific steps include: S1. Acquire multiple frames of high-resolution 3D data and corresponding low-resolution 3D data to construct a training set; S2. Establish a three-dimensional flow field super-resolution reconstruction model. Train the three-dimensional flow field super-resolution reconstruction model based on the training set. During the training process, optimize the parameters in the three-dimensional flow field super-resolution reconstruction model based on Bayesian physical constraints. When the set loss function converges, obtain the trained three-dimensional flow field super-resolution reconstruction model. The three-dimensional flow field super-resolution reconstruction model includes a spatial feature extraction module and a spatiotemporal collaborative reconstruction module; The spatial feature extraction module, by introducing a residual feature distillation component and a physical guidance attention layer, can extract several initial spatial features fused with physical attention weights from the training set and transmit the initial spatial features to the spatiotemporal collaborative reconstruction module. The spatiotemporal collaborative reconstruction module is used to perform temporal interpolation on several initial spatial features to obtain interpolated spatial features, and extract spatiotemporal features from the initial spatial features and interpolated spatial features. Finally, pixel rearrangement is performed based on the spatiotemporal features to output three-dimensional high-resolution data. S3. Based on the trained 3D flow field super-resolution reconstruction model, 3D flow field reconstruction is performed on real-world 3D low-resolution data, and 3D high-resolution data is output.
2. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 1, characterized in that, The spatial feature extraction module includes several residual feature distillation modules. Each residual feature distillation module includes a shallow feature extraction layer, a first feature distillation module, a second feature distillation module, a third feature distillation module, a fourth feature distillation module, and a first convolutional layer connected in sequence. In the process of extracting several initial spatial features from the training set, the spatial feature extraction module performs the following specific operations on each frame of 3D low-resolution data in the training set: After shallow features are extracted from the three-dimensional low-resolution data through the shallow feature extraction layer, they are transmitted to the first feature distillation module. After performing feature distillation and physical attention weight fusion operations on the shallow features by the first feature distillation module, the first distilled feature is output and transmitted to the second feature distillation module. After performing feature distillation and physical attention weight fusion operations on the first distillation feature through the second feature distillation module, the second distillation feature is output and transmitted to the third feature distillation module. After performing feature distillation and physical attention weight fusion operations on the second distilled feature through the third feature distillation module, the third distilled feature is output and transmitted to the fourth feature distillation module. After performing feature distillation and physical attention weight fusion operations on the third distilled feature through the fourth feature distillation module, the fourth distilled feature is output. After concatenating the first distillation feature, the second distillation feature, the third distillation feature, and the fourth distillation feature according to the channel dimension, the fifth distillation feature is output and transmitted to the first convolutional layer. After channel adjustment of the fifth distillation feature through the first convolutional layer, the initial spatial features corresponding to a single frame of three-dimensional low-resolution data are output.
3. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 2, characterized in that, The first feature distillation module, the second feature distillation module, the third feature distillation module, and the fourth feature distillation module have the same structure, each including a first distillation branch, a second distillation branch, a third distillation branch, a fourth distillation branch, a first residual branch, a second residual branch, a third residual branch, a physical guidance attention layer, and a second convolutional layer; Furthermore, the specific steps for each feature distillation module to perform feature distillation and physical attention weight fusion operations on the input features include: After performing a convolution operation on the input features through the first distillation branch, the first convolutional feature is output. The input features are first convolved by the first residual branch, and then the convolution result and the input features are residually concatenated to output the first residual feature, which is then transmitted to the second distillation branch and the second residual branch respectively. After performing a convolution operation on the first residual feature through the second distillation branch, the second convolution feature is output. The first residual feature is first convolved by the second residual branch, and then the convolution result and the first residual feature are residually concatenated to output the second residual feature, which is then transmitted to the third distillation branch and the third residual branch respectively. After performing a convolution operation on the second residual feature through the third distillation branch, the third convolution feature is output. The second residual feature is first convolved by the second residual branch, and then the convolution result and the second residual feature are convolved by residual connection to output the third residual feature and transmit it to the fourth distillation branch. After performing a convolution operation on the third residual feature through the fourth distillation branch, the fourth convolution feature is output. The first, second, third, and fourth convolutional features are concatenated according to the channel dimension to output the fifth convolutional feature; After channel adjustment of the fifth convolutional feature through the second convolutional layer, the sixth convolutional feature is output. After calculating the 3D low-resolution data in the training set through the physical guidance attention layer, the physical attention weights are output. After performing element-wise multiplication on the sixth convolutional feature and the physical attention weight, the weighted feature is output. After performing a residual connection between the weighted features and the features input to the first distillation branch, the distillation features are output.
4. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 3, characterized in that, The specific steps for the physical guidance attention layer to perform calculations on the 3D low-resolution data in the training set include: Calculate vorticity separately , , , means as follows: in, These represent the velocities of the model's predicted output in the x, y, and z directions, respectively. Based on the physical constraints of the vortex transport equation, the velocity field and vortex field residuals in the x, y, and z directions can be obtained, as follows: in, Indicates kinematic viscosity. Represents the Laplace operator symbol; This represents the set of predicted velocity fields output by this model. This represents the set of predicted vorticity fields output by this model; Attention weights The calculation formula is: Where Sig represents the Sigmoid activation function. This represents the initial low-resolution input data. This represents the attention weights obtained after training.
5. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 4, characterized in that, The spatiotemporal collaborative reconstruction module includes a time frame interpolation unit, several first three-dimensional gated loop units, a second three-dimensional gated loop unit, a first pixel rearrangement unit, and a second pixel rearrangement unit. The first three-dimensional gated loop unit is used to extract a first spatiotemporal feature from the initial spatial feature, and the first pixel rearrangement unit is used to rearrange the pixels of the first spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: The first three-dimensional gated loop unit processes the initial spatial feature sequence sequentially in time order. At each integer time t, it takes the hidden state of the previous time (t-1) and the initial spatial feature of the current time as input, calculates and outputs the hidden state of the current time as the first spatiotemporal feature; wherein, the hidden state at the initial time t=1 is initialized to zero. The first spatiotemporal features output by the first three-dimensional gated loop unit are rearranged pixel by pixel by the first pixel rearrangement unit, and the corresponding three-dimensional high-resolution data is output. The time interpolation unit is used to perform frame interpolation operations at the time between t+1 and t, i.e., at t+0.5, and outputs the interpolation spatial characteristics, as shown below: in, and These represent the outputs of the residual characteristic distillation module at time t and t+1, respectively. The second three-dimensional gated loop unit is used to obtain the second spatiotemporal feature based on the frame interpolation spatial feature, and the second pixel rearrangement unit is used to rearrange the pixels of the second spatiotemporal feature to output the corresponding three-dimensional high-resolution data. The specific steps include: At time t+0.5, the third spatiotemporal feature is obtained by performing an element-wise weighted average of the two first spatiotemporal features output by adjacent first three-dimensional gated recurrent units, as follows: The second three-dimensional gated loop unit captures the long-range temporal dependence of the input interpolated spatial features and the third spatiotemporal features in the flow field evolution process, and outputs the second spatiotemporal features corresponding to the current moment. The second pixel rearrangement unit rearranges the pixels of the second spatiotemporal features output by the second three-dimensional gated loop unit and outputs the corresponding three-dimensional high-resolution data.
6. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 5, characterized in that, The loss function of the 3D flow field super-resolution reconstruction model is defined as follows: in, These are represented as network parameters in this model. Represents the data loss function; Indicates residual loss; in: in, Represents the high-resolution velocity field predicted by the neural network. The velocity field representing the true label, The physical residual representing the prediction of the 3D flow field super-resolution reconstruction model and the continuity equation is expressed as: 。 7. The three-dimensional flow field super-resolution reconstruction method based on Bayesian physical constraints according to claim 6, characterized in that, Optimization of parameters in the 3D flow field super-resolution reconstruction model based on Bayesian physical constraints includes: Probabilistic Bayesian physics constraints are used to reduce the residual between the predicted flow field velocity and the actual high-resolution flow field velocity, and the parameters of the 3D flow field super-resolution reconstruction model are optimized, including: use The residual represents the difference between the reconstructed velocity field data and the true high-resolution velocity field data, and is considered to be a normal distribution with zero mean and negligible variance. The normal distribution of the velocity field residual is defined as follows: in, Represents the high-resolution velocity field predicted by the neural network. The velocity field representing the true label, ( ) represents the probability density function. The representative value is 0 and the covariance is The normal distribution The covariance matrix representing the physical residuals; use The residual of the vorticity field data represents the difference between the reconstructed vorticity field data and the true high-resolution vorticity field data. This residual is considered to be a normal distribution with zero mean and negligible variance. The normal distribution of the vorticity field residual is defined as follows: in, Represents the high-resolution vorticity field predicted by the neural network. Eddy field representing the true label.