A vortex flow field super-resolution reconstruction method and system based on residual U-Net

Abstract

This invention relates to the field of fluid dynamics data reconstruction and intelligent computing technology, specifically a method and system for super-resolution reconstruction of vortex flow fields based on residual U-Net. The invention first performs bicubic downsampling on the high-resolution flow field and interpolates it back to its original size to construct a degenerate input. This input is then fed into a network containing an encoder, bottleneck layer, decoder, skip connections, and global residual connections. The output residual field is added to the input to obtain the reconstructed flow field. During training, a combined loss consisting of mean square error and gradient loss is used for supervised optimization. This scheme can effectively recover small-scale vortices, shear layer boundaries, and high-gradient textures in low-resolution flow fields, reduce over-smoothing, and improve reconstruction accuracy and physical structure realism. It is suitable for recovering flow field data after compressed transmission, enhancing low-precision measurements, and reconstructing numerical simulation results.

Description

Technical Field

[0001] This invention relates to the field of fluid dynamics data reconstruction and intelligent computing technology, specifically a method and system for super-resolution reconstruction of vortex flow fields based on residual U-Net. Background Technology

[0002] In the field of fluid mechanics, high-resolution flow field data is crucial for revealing key physical mechanisms such as vortex evolution, shear layer entrainment, energy level stringing, boundary layer separation, and multi-scale turbulent interactions. Whether in direct numerical simulations, particle image velocimetry, laser-induced measurements, or engineering applications in oceanography, meteorology, aerodynamics, and combustion diagnostics, researchers strive to obtain flow field data with higher spatial resolution and fidelity to support subsequent mechanism analysis, parameter identification, and control optimization. However, the acquisition of actual flow field data is often limited by sensor sampling accuracy, experimental equipment cost, storage capacity, transmission bandwidth, and numerical computation resources. What can be directly obtained is often low-resolution, sparse, noisy, or even locally distorted flow field data. Especially for two-dimensional vortex flows or turbulent vorticity fields, low-resolution sampling can easily smooth or obscure details of small-scale vortex cores, thin shear layers, and high-gradient regions, leading to significant deviations in subsequent flow identification and physical analysis.

[0003] To alleviate the aforementioned problems, traditional techniques typically employ mathematical methods such as bilinear interpolation, bicubic interpolation, spline interpolation, and reconstruction filtering to amplify and restore low-resolution flow fields. These methods are simple to implement, computationally efficient, and can provide a certain degree of visual smoothing in regions with regular grids and gentle flow changes, thus they have long been widely used as low-cost data post-processing methods. However, these traditional interpolation methods essentially rely on local numerical relationships between neighboring pixels or grid points for estimation, failing to truly learn the complex nonlinear dynamic connections within the flow field and struggling to express the coupling relationships between vortex structures at different scales. Especially in regions with strong vortices, steep gradients, and fine-scale stripe structures, traditional interpolation often only generates relatively smooth transition results, failing to effectively recover the high-frequency physical information erased by the downsampling process, easily causing problems such as vortex core diffusion, boundary blurring, local extremum attenuation, and flow field texture distortion.

[0004] With the development of deep learning, image super-resolution methods using convolutional neural networks have been gradually introduced into flow field reconstruction. Existing research shows that end-to-end mapping based on deep networks can recover high-resolution details from low-resolution flow fields to a certain extent. The paper [Fukami K., Fukagata K., Taira K. Super-resolution reconstruction of turbulent flows with machine learning. Journal of FluidMechanics, 870 (2019): 106-120. DOI: 10.1017 / jfm.2019.238] proposes a machine learning-based super-resolution reconstruction method for turbulent flow fields. It employs convolutional neural networks and a hybrid downsampling skip-connected multi-scale model to reconstruct high-resolution two-dimensional cylindrical wakes and two-dimensional uniform turbulent fields. This paper points out that even with coarse-resolution flow field images as input, the proposed model can still accurately recover high-resolution flow field information, demonstrating the practical feasibility of data-driven super-resolution technology in turbulence reconstruction. This paper can be considered one of the earliest and most representative foundational works in the field of flow field super-resolution reconstruction. However, the technical approach represented by this institute mainly demonstrates the effectiveness of convolutional networks for flow field super-resolution, with its core focus on the mapping capability from low-resolution to high-resolution flow fields. For data objects such as two-dimensional vortex flow fields, which are characterized by high gradients, high vorticity, and sensitivity to local structures, relying solely on conventional convolutional networks or supervised training based on mean squared error can easily lead to models that are more inclined to approximate the overall numerical values, resulting in over-smoothing in local regions such as shear layers, vortex kernel edges, and fine-scale vortex chains. In other words, while existing methods can improve overall PSNR, MSE, and other metrics, they may not be able to fully maintain the structural consistency related to the first derivative in fluid physics, and are particularly difficult to stably recover the high-frequency boundary information that is crucial for vorticity field analysis.

[0005] Building upon this foundation, another type of research has begun to attempt to incorporate physical laws into the reconstruction process of deep learning. For example, the paper [Gao H., Sun L., Wang J.-X. Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels. Physics of Fluids, 33(7): 073603 (2021). DOI: 10.1063 / 5.0054312] proposes a physics-informed convolutional neural network super-resolution method under high-resolution label-free conditions. By integrating conservation laws and boundary conditions into the training process, the model can still perform flow field super-resolution and denoising even without high-resolution labels. The significance of this paper lies in demonstrating that flow field super-resolution should not be viewed merely as a general image magnification problem, but should be combined with the governing equations, boundary conditions, or physical constraints in fluid mechanics to improve the reliability and generalization ability of the results. This technique has high reference value for flow field recovery, noise suppression, and sparse data completion in high-dimensional parameter spaces. However, the main idea of ​​this technology is to constrain network training through hard or semi-hard physical information such as conservation laws and boundary conditions. Its focus is on weakening the dependence on high-resolution labels and achieving reconstruction from sparse, missing, and noisy data. For the single-frame super-resolution scene of two-dimensional vortex flow fields that this application focuses on, although this type of method has strong physical constraint capabilities, its network structure and training framework are usually more suitable for physically known, well-defined, or parameterized flow problems. When applied to situations with only supervised sample pairs and where the main objective is to recover the small-scale vortex texture and local gradient structure in the vortex field, existing physics-informed methods still have trade-offs between model complexity, training cost, and recovery accuracy for local vortex textures, and may not be able to balance implementation difficulty and reconstruction quality in engineering.

[0006] In addition, Chinese patent application CN118261796A discloses a method and system for super-resolution reconstruction of turbulent flow fields. This document also generally employs a deep learning framework, preprocessing and slicing turbulent data frames before inputting them into a reconstruction model for super-resolution restoration, thereby obtaining high-resolution turbulent data frames. This document explicitly points out that existing interpolation algorithms ignore the edge features of turbulent flow fields, making it difficult to reconstruct high-frequency details; therefore, its goal is also to improve clarity and enhance the recovery capability of small-scale structures. Thus, CN118261796A and this application are quite similar in application field and overall purpose, both belonging to high-resolution reconstruction techniques for turbulent or vortex flow fields. However, based on the published content, document [CN118261796A] leans more towards a general turbulent flow field super-resolution scheme involving slice preprocessing, deep feature extraction, and high-resolution fusion output. It emphasizes the overall reconstruction module and slice fusion strategy, without establishing a more targeted reconstruction constraint mechanism for the physical structural characteristics of two-dimensional single-channel vortex flow fields, especially the sensitivity to the first-order gradient of vorticity. In other words, for data objects like vorticity fields, which use local gradients, vortex boundaries, and fine-scale textures as core information carriers, general deep reconstruction networks still have two prominent problems: First, the input low-resolution field often loses a lot of high-frequency information, and if the network directly outputs high-resolution results, the training difficulty is high; Second, if the loss function mainly focuses on point value errors, it is easy to obtain reconstruction results that are numerically close but have blunted physical boundaries.

[0007] A review of existing technologies reveals that while current flow field super-resolution schemes have evolved from traditional interpolation methods to techniques based on convolutional neural networks, U-Net structures, multi-scale feature extraction, and even physics-informed constraints, they still suffer from the following shortcomings when dealing with two-dimensional vortex flow fields: First, existing methods often emphasize the fitting accuracy of the overall field values, while their ability to preserve small-scale vortex boundaries, shear layers, and local high-gradient regions in the vortex field remains insufficient. Second, existing methods often focus on directly predicting high-resolution outputs, failing to fully utilize the low-frequency fundamental information still retained in the low-resolution inputs. This results in the network needing to simultaneously learn low-frequency reconstruction and high-frequency compensation, leading to a heavy training burden. Third, while some physics-constrained methods enhance physical consistency, their implementation paths are relatively complex and may not be suitable for two-dimensional single-frame vortex flow field reconstruction scenarios with sufficient supervised samples. Fourth, existing patents and literature still lack more targeted solutions to the technical problem of restoring the gradient topological features of the vortex field and preventing edge smoothing while maintaining the feasibility of the network structure.

[0008] Therefore, it is still necessary to propose a new super-resolution reconstruction scheme for two-dimensional vortex flow fields. This scheme should maintain high operability in engineering implementation while making fuller use of the existing low-frequency information in the low-resolution input. By using an encoder-decoder network and residual learning mechanism suitable for flow field reconstruction tasks, the network learning should focus on the lost high-frequency details. Furthermore, by using constraints related to the local gradient of the flow field, the scheme can improve the recovery capability of vorticity field boundaries, vortex filaments, and shear structures, thereby obtaining high-resolution reconstruction results that balance numerical accuracy and physical structural realism. Summary of the Invention

[0009] The technical objective of this invention is to address the problems in existing vortex flow field super-resolution reconstruction methods, such as insufficient recovery of high-frequency vortex details, over-smoothing of local gradient structures, and poor physical consistency of reconstruction results. This invention provides a vortex flow field super-resolution reconstruction method, system, device, and medium based on the synergistic effect of residual U-Net and gradient constraint loss. This achieves high-fidelity recovery of small-scale vortex cores, shear layer boundaries, and complex texture structures in low-resolution flow fields, thereby improving reconstruction accuracy while enhancing the physical structural realism and engineering application value of the reconstructed flow field.

[0010] Firstly, in order to achieve the above-mentioned objectives, the present invention adopts the following technical solution:

[0011] In a first aspect, this invention discloses a super-resolution reconstruction method for vortex flow fields based on residual U-Net, comprising the following steps: S1: Obtain a two-dimensional high-resolution vortex flow field sample, and perform bicubic interpolation downsampling on the two-dimensional high-resolution vortex flow field sample according to a preset downsampling ratio. Then, interpolate the downsampled low-resolution flow field back to the original grid size through bicubic interpolation to obtain a degenerate input flow field with the same size as the two-dimensional high-resolution vortex flow field sample. S2: Construct a residual U-Net network for flow field reconstruction. The residual U-Net network includes an encoder, a bottleneck layer, a decoder, and global residual connections. The encoder is used to extract multi-scale features of the degraded input flow field, the decoder is used to restore spatial resolution and reconstruct high-frequency details, and a skip connection is set between the encoder and the decoder to fuse shallow spatial features and deep semantic features. The network output is a residual flow field of the same size as the input flow field. S3: Add the residual flow field to the degraded input flow field element by element to obtain the reconstructed high-resolution flow field; construct a combined loss function based on the pixel error and gradient error between the reconstructed high-resolution flow field and the corresponding real high-resolution flow field; S4: The residual U-Net network is trained under supervision using samples. During the training phase, local flow field blocks are obtained by random pruning for iterative training, and the entire flow field is used for verification during the validation phase. The trained flow field reconstruction model is obtained by optimizing the combined loss function. S5: The low-resolution two-dimensional vortex flow field to be reconstructed is back-interpolated to the target size using bicubic interpolation and then input into the trained flow field reconstruction model, outputting the corresponding high-resolution reconstructed flow field.

[0012] Preferably, in step S1, the two-dimensional high-resolution vortex flow field sample is a single-channel three-dimensional tensor defined on the real number domain, with 1 channel and spatial dimensions of the number of vertical grid points H and the number of horizontal grid points W. The degraded input flow field is obtained as follows: first, bicubic interpolation downsampling is performed on the two-dimensional high-resolution vortex flow field sample according to the downsampling factor, and then bicubic interpolation back-interpolation upsampling is performed on the downsampling result according to the same downsampling factor to obtain the degraded input flow field.

[0013] Preferably, in step S2, the output residual flow field of the residual U-Net network and the degraded input flow field satisfy the following relationship: the degraded input flow field and the residual flow field predicted by the residual U-Net network based on the input flow field are added element-by-element at the corresponding grid positions to obtain the reconstructed high-resolution flow field; And / or, in step S2, the residual U-Net network includes: The input convolutional layer is used to map the single-channel input to an initial feature map; A three-level downsampling coding layer is used to extract multi-scale features step by step; The first-level bottleneck layer is used to extract deep global semantic features; A three-stage upsampling decoding layer is used to restore spatial resolution step by step; The output convolutional layer is used to map the decoded feature map into a single-channel residual flow field; The output feature map sizes of each layer satisfy the following: The first layer encoded feature map is a three-dimensional feature map defined on the real number domain, with 64 channels and spatial dimensions of vertical and horizontal dimensions; the second layer encoded feature map has 128 channels, with spatial dimensions of half the vertical dimension and half the horizontal dimension, respectively; the third layer encoded feature map has 256 channels, with spatial dimensions of one-quarter of the vertical dimension and one-quarter of the horizontal dimension, respectively; and the bottleneck layer feature map has 512 channels, with spatial dimensions of one-eighth of the vertical dimension and one-eighth of the horizontal dimension, respectively.

[0014] Preferably, in step S2, each encoding layer and decoding layer adopts a dual convolution structure. The dual convolution structure is as follows: the input feature map is first subjected to a convolution operation with a kernel size of 3*3, then batch normalization operation and non-linear activation operation are performed, and then the result is subjected to a convolution operation with a kernel size of 3*3, batch normalization operation and non-linear activation operation again to obtain the output of the dual convolution operation unit. And / or, in step S2, the encoding layer uses max pooling for downsampling, specifically: the k-th level encoded input feature map is first subjected to max pooling downsampling, and then processed by the k-th level encoded convolution operation to obtain the k-th level encoded output feature map; the decoding layer uses transposed convolution for upsampling, specifically: the feature map after transposed convolution upsampling in the k-th level decoding stage is concatenated with the skip connection encoded feature map corresponding to the k-th level decoding layer according to the channel dimension, and then processed by the k-th level decoding convolution operation to obtain the k-th level decoded output feature map.

[0015] Preferably, in step S3, the combined loss function is obtained by weighting the mean squared error loss and the gradient loss according to the gradient loss weight coefficient; The mean square error loss is specifically calculated by squaring the numerical differences between the reconstructed high-resolution flow field and the real high-resolution flow field at corresponding positions of all longitudinal and transverse grid indices, summing the squared differences at all positions, and then dividing by the total number of grid points in the flow field. The gradient loss is specifically calculated as follows: the gradient difference between the reconstructed high-resolution flow field and the real high-resolution flow field in the x and y directions are calculated respectively; the absolute values ​​of the gradient difference in the x and y directions at each location are summed; and the summation of the results at all locations is then divided by the total number of grid points in the flow field.

[0016] Preferably, the gradients in the x and y directions are obtained using the Sobel operator. Specifically, the real high-resolution flow field is convolved with Sobel kernels in the x and y directions to obtain the gradients of the real high-resolution flow field in the x and y directions, respectively. The reconstructed high-resolution flow field is then convolved with Sobel kernels in the x and y directions, respectively, to obtain the gradients of the reconstructed high-resolution flow field in the x and y directions, respectively.

[0017] Preferably, in step S4, the training sample set, validation sample set, and test sample set are divided in a ratio of 8:1:1. And / or, in step S4, local training blocks are generated by random pruning during the training phase. The local training blocks are all single-channel three-dimensional tensors defined on the real number domain, and their spatial dimensions are all square regions with a side length of P. And / or, in step S4, the model training uses the AdamW optimizer, and the learning rate is updated using a cosine annealing strategy. Specifically, the learning rate corresponding to the t-th training round is obtained by adding the difference between the minimum learning rate and the initial learning rate after decaying by a cosine function to the minimum learning rate. And / or, in step S4, an early stopping mechanism is used to control the training termination condition. Training is stopped when the verification loss does not decrease within a preset number of rounds, where the preset number of rounds is 15.

[0018] Secondly, the present invention also provides a two-dimensional vortex flow field super-resolution reconstruction system, which is used to implement the method, including:

[0019] The data preprocessing module is used to perform step S1;

[0020] The network building module is used to execute step S2;

[0021] The loss construction module is used to execute step S3;

[0022] The model training module is used to perform step S4;

[0023] Rebuild the output module to perform step S5.

[0024] Thirdly, the present invention also provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implement the steps of the method.

[0025] Fourthly, the present invention also provides a computer program product, including a computer program or instructions that, when executed by a processor, implement the steps of the method.

[0026] Compared to existing flow field reconstruction schemes that use traditional interpolation or general convolutional networks, this invention can more effectively preserve small-scale vortex kernels, shear layer boundaries, and high-gradient texture structures during the process of restoring a low-resolution two-dimensional vortex flow field to a high-resolution flow field. On the one hand, by utilizing the encoder-decoder structure, skip connections, and global residual learning mechanism of the residual U-Net, the network can learn high-frequency details lost due to downsampling while inheriting the existing low-frequency information of the input flow field, thereby improving reconstruction accuracy and reducing the training difficulty caused by directly predicting the entire high-resolution flow field. On the other hand, by introducing gradient constraints into the loss function, the reconstruction result not only approximates the real flow field numerically, but also maintains a higher consistency with the real flow field at the spatial derivative level, thereby effectively suppressing the problems of excessive smoothing, edge blunting, and loss of local vortex details in the reconstruction result. Compared with existing flow field super-resolution methods that mainly focus on overall field value fitting or emphasize general turbulence reconstruction or reconstruction without high-resolution label physical constraints, this invention is more suitable for supervised high-fidelity reconstruction of two-dimensional single-channel vortex flow fields. It can improve the clarity, structural realism and physical consistency of flow field reconstruction while ensuring engineering feasibility. Therefore, it has higher practical value for applications such as fluid measurement data enhancement, high-quality recovery of compressed numerical simulation results, and subsequent vortex structure identification, flow mechanism analysis and intelligent diagnosis. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of the two-dimensional vortex flow field super-resolution reconstruction network structure based on residual U-Net in an embodiment of the present invention.

[0028] Figure 2 This is a schematic diagram of the training loss curve and learning rate change curve in an embodiment of the present invention.

[0029] Figure 3 This is a visual comparison of the flow field reconstruction results in the earlier stages of the training process in this embodiment of the invention.

[0030] Figure 4 This is a visual comparison of the flow field reconstruction results during the testing phase in an embodiment of the present invention. Detailed Implementation

[0031] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the protection scope of the present invention.

[0032] I. Terminology Explanation

[0033] To facilitate understanding of this invention, some terms used in this specification will be explained first, with the number of terms limited to no more than 10:

[0034] Two-dimensional vortex flow field: refers to the spatial distribution data of fluid represented by a two-dimensional regular grid. In this embodiment, it is preferably represented as a single-channel vortex scalar field.

[0035] High-resolution flow field: refers to flow field data with target grid size that can more completely express fine-scale vortices and boundary details.

[0036] Low-resolution flow field: refers to flow field data that has been downsampled, resulting in a reduction in the number of grid points and loss of detailed information.

[0037] Degraded input flow field: refers to the input flow field obtained by bicubic downsampling and bicubic back-interpolation of the high-resolution flow field to the original size. Its size is consistent with the true high-resolution flow field, but the high-frequency information has been weakened.

[0038] Residual field: refers to the information field predicted by the network, used to compensate for the difference between the degraded input flow field and the target high-resolution flow field.

[0039] Skip connection: refers to a connection method that directly transmits shallow features from a certain layer of the encoder to the corresponding layer of the decoder in order to preserve spatial location information and assist in detail reconstruction.

[0040] Global residual connection: refers to the connection method that directly adds the network input to the residual field of the network output to form the final reconstruction result.

[0041] Gradient loss: refers to the loss term that constrains the difference in spatial gradient between the reconstructed flow field and the real flow field, in order to enhance the recovery ability of edge and high gradient structures.

[0042] Random cropping: refers to randomly selecting local blocks from the entire flow field as training samples during training, in order to increase sample diversity and reduce memory usage.

[0043] Cosine annealing learning rate: refers to a scheduling strategy in which the learning rate gradually decreases from its initial value to a smaller value according to the trajectory of a cosine function.

[0044] II. System Structure of the Invention

[0045] Combination Figure 1 This invention provides a system for super-resolution reconstruction of two-dimensional vortex flow fields. The system can be deployed in servers, workstations, edge computing terminals, or electronic devices with GPU / CPU computing capabilities. The system includes at least:

[0046] The data preprocessing module is used to read two-dimensional high-resolution vortex flow field data and perform operations such as sample partitioning, degenerate input construction, normalization, and training block pruning.

[0047] The network building module is used to build a VortexUnet network with global residual connections, including an encoder, bottleneck layer, decoder, skip connections, and output layer.

[0048] The loss building module is used to calculate pixel-level errors and gradient errors on the network output and the real high-resolution flow field, and combine them to form the total loss;

[0049] The model training module is used to call the optimizer, learning rate scheduler, automatic mixed precision, and early stopping mechanism to complete the iterative update of parameters;

[0050] The reconstruction inference module is used to back-interpolate the low-resolution flow field to be reconstructed to the target size and input it into the trained model, and output the corresponding high-resolution flow field results.

[0051] The results evaluation and visualization module is used to output metrics such as MSE and PSNR, and generate [data / information]. Figure 2 , Figure 3 , Figure 4 Similar loss curves and flow field comparison diagrams.

[0052] The innovation of this invention does not lie in simply setting up the above modules, but in the fact that the above modules form a tightly coupled technical route around the specific technical object of two-dimensional vortex flow field: first, by constructing a fuzzy input of the same size, the model focuses on detail recovery, and then by combining residual learning, multi-scale feature fusion and gradient physical constraints, the problem that traditional interpolation or ordinary image super-resolution methods cannot balance numerical accuracy and physical structure authenticity is solved.

[0053] III. Technical Route of the Method of the Invention

[0054] The technical approach of this invention can be summarized as follows:

[0055] First, high-resolution two-dimensional vortex flow field data is acquired, and the low-resolution sampling process is simulated. Then, the degraded input is fed into a VortexUnet network with global residual connections to extract multi-scale features and predict the residual field. Next, numerical error and gradient error are constrained simultaneously by combining loss functions. Finally, the trained model is used to reconstruct the flow field at high resolution.

[0056] The following steps S1 to S5 are explained in detail.

[0057] (I) Step S1: Specific implementation of the data construction and degradation input generation steps

[0058] The purpose of step S1 is to construct supervised sample pairs suitable for training and inference in this invention.

[0059] 1. Acquisition and organization of raw flow field data

[0060] In this embodiment, the original data consists of 999 two-dimensional single-channel vortex flow field samples, preferably in .mat format. Each sample can be represented as a two-dimensional real matrix, reflecting the spatial distribution of two-dimensional vorticity at a certain moment or under a certain operating condition. To match the network input and output, each two-dimensional matrix can be expanded into a single-channel tensor.

[0061] High-resolution flow field samples can be represented as:

[0062] ;

[0063] in: Represents the true-valued flow field at high resolution; Represents the real number field; Indicates single channel; This indicates the number of grid points in the longitudinal direction of the flow field; This indicates the number of grid points in the transverse direction of the flow field.

[0064] In this embodiment, and The value can be on the order of 1024, but it is not limited to this, as long as the input and output grids are consistent.

[0065] 2. Dataset partitioning

[0066] To prevent data leakage between training, validation, and testing, this embodiment preferably divides all samples into training, validation, and testing sets in an 8:1:1 ratio. This ratio can be expressed as:

[0067] ;

[0068] in: Indicates the number of training samples; Indicates the number of validation samples; This indicates the number of test samples.

[0069] This step, through rigorous settling, ensures that the model faces unseen samples during the testing phase, thus more realistically reflecting its generalization ability.

[0070] 3. Construction method of degenerate input

[0071] Unlike some schemes that directly input low-resolution fields and then have the network simultaneously amplify and repair them, this embodiment preferably employs an input construction method that first downsamples and then interpolates back to the original size. Specifically, the high-resolution flow field is first magnified by... Bicubic interpolation downsampling is performed to obtain a low-resolution flow field, which is then back-interpolated to the original size through bicubic interpolation to form a fuzzy input with the same size as the high-resolution ground truth.

[0072] This process can be represented as:

[0073] ;

[0074] in: This represents a degenerate input flow field; Represents the true-valued flow field at high resolution; Indicates by multiplier Perform double triple downsampling; Indicates by multiplier Perform double triple insertion; This indicates the downsampling factor. In this preferred embodiment: .

[0075] The technical advantages of using the above construction method are:

[0076] First, the network input and output sizes remain consistent, eliminating the need for the network to simultaneously perform both size scaling and detail restoration tasks, thus simplifying the learning objective;

[0077] Secondly, the input still retains low-frequency contours and large-scale topology, allowing the network to concentrate its computational resources more effectively on compensating for fine-scale vortices, high-frequency boundaries, and shear layer textures.

[0078] Third, this degradation mode is similar to Figure 3 , Figure 4 The 8×Bicubic input shown in the upper left corner corresponds to the visual observation of the recovery effect.

[0079] 4. Normalization and random pruning

[0080] Since the dimensions and ranges of flow field values ​​may not be completely consistent, to improve training stability, each sample can be subjected to minimum-maximum normalization or linear normalization to ensure that the flow field values ​​fall within a uniform range. Normalization to a specific value is preferred. Interval.

[0081] During the training phase, to improve sample diversity and reduce GPU memory consumption, this embodiment employs a random pruning strategy, simultaneously pruning the entire flow field into local training blocks. A training block can be represented as:

[0082] ;

[0083] in: This represents a local block cropped from the high-resolution true-value flow field; This represents a local block obtained by clipping from the degenerate input flow field at the corresponding position; Indicates the side length of the training block; Indicates single channel; Represents the real number field.

[0084] During the verification and testing phase, in order to fully evaluate the model's ability to recover the topology of the entire field, this embodiment preferably uses direct inference of the entire flow field without further pruning.

[0085] (II) Step S2: Specific implementation of the residual U-Net network construction step

[0086] Combination Figure 1 The details of step S2 are as follows.

[0087] This invention does not allow the network to directly output a complete high-resolution field end-to-end from a fuzzy input. Instead, it allows the network to learn the missing portion of the degenerate input compared to the true high-resolution flow field, i.e., the residual field. Since the low-frequency background and large-scale flow contours are already preserved in the input, converting the learning objective into a residual helps reduce training difficulty, accelerate convergence, and highlight the recovery of high-frequency structures.

[0088] The final relationship to be rebuilt is as follows:

[0089] ;

[0090] in: This represents the reconstructed high-resolution flow field; This represents a degenerate input flow field; This indicates that the network is based on the input. The predicted residual field; This indicates that elements are added one by one according to grid points.

[0091] like Figure 1 As shown on the left, the input is first processed by an input convolutional module (inc) to obtain the first-layer feature map; then, it is progressively extracted into deeper multi-scale features through three levels of downsampling encoding layers. The preferred size of each feature map is:

[0092] ;

[0093] ;

[0094] ;

[0095] ;

[0096] in: , , , These represent the feature maps of each encoding layer and the bottleneck layer, respectively. , , , Indicates the number of feature channels; , These represent the height and width of the input flow field, respectively.

[0097] Each coding layer preferably employs a double convolutional structure followed by max pooling. The double convolutional structure can be represented as:

[0098] ;

[0099] in: Represents a double convolutional structure; Indicates the input feature map; Indicates the kernel size as Convolution operation; Indicates batch normalization; This represents a nonlinear activation function, with ReLU being the preferred choice.

[0100] Through the above structure, the encoder can gradually expand the receptive field, enabling the model to not only observe local small eddies, but also to comprehensively understand the larger-scale flow background and topological relationships.

[0101] Figure 1 The bottom layer is the bottleneck layer, and its preferred size is... The bottleneck layer serves to aggregate the deep semantic features obtained through multi-level encoding. For two-dimensional vortex flow fields, these deep features not only reflect local textures but also embody the spatial relationships between vortex structures at different scales. Therefore, the bottleneck layer provides a global context for reconstructing complex vortex morphologies in subsequent decoding stages.

[0102] like Figure 1 As shown on the right, the decoder recovers the spatial resolution step by step through three levels of transposed convolutions, and concatenates features with the corresponding layer of the encoder at each level. Its schematic relationship can be represented as follows:

[0103] ;

[0104] in: Indicates the first Level decoding output feature map; Indicates the first Level decoding convolution module; This represents the feature map obtained by upsampling the decoded result from the previous layer through transpose convolution; This represents the corresponding encoder feature map; This indicates splicing by channel dimension.

[0105] The advantage of skip connections lies in the fact that while deep features contain strong semantic information, their spatial localization accuracy decreases after multiple pooling operations; shallow features, on the other hand, retain richer edge positions and texture details. Fusing the two significantly enhances the ability to recover fine-scale vortices, shear stripes, and vortex boundaries.

[0106] Figure 1 outc is The convolutional output layer is used to compress the multi-channel features obtained from the last stage of decoding into a single-channel residual field. Subsequently, via... Figure 1 The global residual connection shown at the top will input... Adding it to the residual field yields the output. .

[0107] This design, on the one hand, enables the network to focus on learning missing information rather than repeatedly learning low-frequency content already present in the input; on the other hand, global residual connections can improve gradient propagation paths and alleviate the difficulty of training deep networks; furthermore, for detail-compensation tasks such as flow field reconstruction, residual learning is more in line with physical and numerical intuition than directly regressing the complete output.

[0108] (III) Step S3: Specific implementation of the reconstruction result generation and loss function construction steps

[0109] If only ordinary mean square error constraints are used, the model can easily obtain results that are numerically close to the true value, but with excessively smoothed local boundaries. To address the need for reconstructing high-gradient vortex cores, shear layers, and vortex filaments in two-dimensional vortex flow fields, this invention further introduces gradient consistency constraints to improve the realism of the physical structure of the reconstructed flow field.

[0110] 1. Pixel-level mean square error

[0111] Mean squared error loss is defined as:

[0112] ;

[0113] in: This represents the mean squared error loss; Indicates the number of longitudinal grid points in the flow field; This indicates the number of transverse grid points in the flow field; Indicates a vertical grid index; Indicates a horizontal grid index; This indicates that the reconstructed high-resolution flow field is in the first... line, number The value at the column position; This indicates that the true high-resolution flow field is in the first... line, number The value at the column position; this loss is used to ensure the overall accuracy of numerical reconstruction, so that the model output approximates the real high-resolution flow field at the field value level.

[0114] 2. Sobel gradient constraints

[0115] To explicitly focus on edges and high-gradient regions, this invention utilizes the Sobel operator to calculate separately. direction and Directional gradient. It is represented as:

[0116] ;

[0117] ;

[0118] ;

[0119] ;

[0120] in: , They represent the actual flow field at... , Gradient in direction; , They represent the reconstructed flow field at... , Gradient in direction; , These represent Sobel convolution kernels; This represents the convolution operation.

[0121] Preferably:

[0122] ,

[0123] ;

[0124] in: This represents the Sobel kernel used to compute the lateral gradient; This represents the Sobel kernel used to calculate the longitudinal gradient.

[0125] Based on the above gradient, the gradient loss is defined as:

[0126] ;

[0127] in: Indicates gradient loss; This indicates the reconstruction of the high-resolution flow field in Gradient in direction; This indicates that the true high-resolution flow field is in Gradient in direction; This indicates the reconstruction of the high-resolution flow field in Gradient in direction; This indicates that the true high-resolution flow field is in Gradient in direction; Indicates the corresponding matrix at the th line, number The value at the column position; This represents absolute value operations.

[0128] 3. Combination Loss Function

[0129] The final total loss is defined as:

[0130] ;

[0131] in: Indicates the total loss; This represents the mean squared error loss; Indicates gradient loss; These represent the weighting coefficients for the gradient loss. In this embodiment, the preferred method is: .

[0132] The technical effect of this combined loss is:

[0133] If only Models tend to produce smooth results because averaging makes it easier to reduce the overall error; while introducing Subsequently, the model will simultaneously focus on differences at the derivative level during the optimization process, thus being more inclined to recover vorticity boundaries, local extreme value transitions, and striped fine structures. Figure 4 The vorticity gradient plot in the lower right corner reflects this: the gradient texture output by the model still shows a continuous and clear vortex distribution, rather than being uniformly smoothed out.

[0134] Therefore, step S3 not only improves the visual quality of the reconstructed image, but also enhances the realism of the physical structure from the perspective of the first derivative.

[0135] (iv) Step S4: Specific implementation of the network training step

[0136] Step S4 is responsible for implementing the aforementioned network structure and loss function into a repeatable engineering training process. While its technical contribution is primarily reflected in ensuring stability at the implementation level, it is equally indispensable for achieving usable accuracy in the final invention.

[0137] 1. Optimizer and Learning Rate Strategy

[0138] This embodiment preferably uses the AdamW optimizer because it balances adaptive learning rate and weight decay control, making it suitable for stable training of deep networks. The learning rate uses cosine annealing scheduling, expressed as:

[0139] ;

[0140] in: Indicates the first The learning rate corresponding to each training round; Indicates the initial learning rate; This represents the minimum learning rate; Indicates the current training round; Indicates the total number of training rounds; Pi is a constant. This represents the cosine function.

[0141] In this embodiment, the preferred initial learning rate is: Minimum learning rate Total training rounds .

[0142] 2. Training batches and mixed precision

[0143] In this embodiment, the training batch size is preferably 64. To reduce memory usage and improve training speed, automatic mixed-precision training can be used. Mixed-precision training can significantly reduce memory overhead without significantly reducing numerical stability, making larger batch training possible.

[0144] 3. Early Termination Mechanism

[0145] To prevent ineffective iterations or overfitting in later training iterations, this embodiment implements an early stopping mechanism. Training stops when the validation loss no longer decreases over a consecutive number of training epochs. Its patience value can be denoted as: ,in: This indicates the number of patience rounds in the early stop mechanism.

[0146] 4. Training and Validation Process

[0147] During the training phase, the model reads randomly cropped local training blocks. Forward propagation yields the reconstructed block Calculate the portfolio loss Then, the parameters are updated through backpropagation. In the verification phase, the entire flow field is read, and the verification loss is calculated to monitor the generalization performance.

[0148] Figure 2 The results show that after training begins, both training loss and validation loss decrease rapidly, indicating that the network quickly learns the basic rules of compensating for high-resolution structures from fuzzy inputs; then they enter a phase of gradual decline, indicating that the model is making more detailed corrections to local details; the absence of a significant rebound in validation loss in the figure suggests that the training process is generally stable.

[0149] (V) Step S5: Specific implementation of the flow field reconstruction output step

[0150] After the model training is completed, the inference phase can begin. During inference, for any low-resolution two-dimensional vortex flow field to be reconstructed, bicubic interpolation, consistent with the training phase, is first used to back-interpolate it to the target mesh size. Then, the trained Vortex Unet network is input to obtain the residual field, which is added to the input to output a high-resolution reconstructed flow field.

[0151] This process is consistent with the training phase, which helps reduce training-inference bias. The inference results can be further output as a two-dimensional matrix, a pseudo-color flow field plot, or a data file for subsequent calculations.

[0152] IV. Application Examples

[0153] Application Example 1: 8x Super-Resolution Reconstruction of Two-Dimensional Vortex Flow Field

[0154] To verify the effectiveness of the proposed super-resolution reconstruction method for vortex flow fields based on the synergistic effect of residual U-Net and gradient constraint loss, the applicant conducted a supervised reconstruction experiment on two-dimensional vortex flow field data in a laboratory environment. This experiment used a two-dimensional single-channel vortex flow field as the research object. By constructing a low-resolution degenerate input and using the proposed VortexUnet model to perform high-resolution reconstruction, the technical effectiveness of the invention in terms of flow field detail recovery, physical structure preservation, and overall reconstruction accuracy was evaluated.

[0155] (I) Experimental Objective

[0156] The purpose of this application example is to verify the following technical effects:

[0157] Can this invention effectively restore the two-dimensional vortex flow field after 8x downsampling to its original high-resolution state?

[0158] Can this invention effectively recover small-scale vortex kernels, shear layer boundaries, and local high-gradient textures that are difficult to preserve using traditional bicubic interpolation methods?

[0159] In this invention, the residual learning mechanism and gradient loss work together to suppress the common over-smoothing phenomenon in the reconstruction results and improve the physical structure realism of the reconstructed flow field.

[0160] Does this invention have application value in terms of training convergence, generalization ability, and engineering feasibility?

[0161] (II) Data Sources and Sample Construction

[0162] This embodiment uses 999 two-dimensional vortex flow field samples in .mat file format. Each sample is a two-dimensional single-channel scalar field, preferably a two-dimensional vortex field. To avoid sample leakage and ensure the objectivity and reliability of the test results, all data are independently divided into training, validation, and test sets in a ratio of 8:1:1.

[0163] The data partitioning relationship is represented as follows:

[0164] ;

[0165] in: Indicates the number of samples in the training set; Indicates the number of samples in the validation set; This indicates the number of samples in the test set.

[0166] In this embodiment, to simulate the information degradation process caused by insufficient sampling resolution or data compression and transmission in actual flow fields, bicubic interpolation is used to downsample the high-resolution flow field by a factor of 8. The downsampling result is then interpolated back to the original size using bicubic interpolation, and this is used as the network input. The process can be represented as follows:

[0167] ;

[0168] in: This represents a degenerate input flow field; This represents the original high-resolution true flow field; Indicates by multiplier Bicubic downsampling operation; Indicates by multiplier Bicubic interpolation operation; This indicates the downsampling factor.

[0169] In this embodiment: ;in: This represents the space degradation factor, which is set to 8.

[0170] This processing method corresponds to the attached document. Figure 3 and attached Figure 4 The image shown on the left in the top row is an 8xBicubic input image. As can be seen from the attached image, although the input is restored to its original size after processing in this way, many fine-scale vortex textures have become significantly blurred, and the boundary transitions have become blunt, indicating that bicubic interpolation alone cannot recover the details of the high-frequency flow field.

[0171] (III) Model Structure and Training Conditions

[0172] This embodiment uses the Vortex Unet proposed in this invention for training and testing. (See attached diagram.) Figure 1 The network consists of an input convolutional layer, a three-level encoding downsampling layer, a bottleneck layer, a three-level decoding upsampling layer, skip connections, an output convolutional layer, and global residual connections.

[0173] The network output satisfies the following formula:

[0174] ;

[0175] in: This represents the reconstructed high-resolution flow field output by the network; This represents a degenerate input flow field; This represents the residual field predicted by the network.

[0176] The loss function used in this embodiment is a combination of pixel mean square error and gradient loss:

[0177] ;

[0178] in: Indicates the total loss; This represents the mean squared error loss; Indicates gradient loss; The weight coefficients represent the gradient loss.

[0179] In this embodiment, the gradient loss weights are taken as follows: ,in: This represents the loss weight that balances numerical accuracy and gradient consistency.

[0180] The training configuration uses the parameters you provided, summarized as follows.

[0181] Table 1 Training Configuration Parameters

[0182]

[0183] Among them, the learning rate changes with the number of training rounds, preferably satisfying the cosine annealing relationship:

[0184] ;

[0185] in: Indicates the first The learning rate corresponding to each round; Indicates the initial learning rate; This represents the minimum learning rate; Indicates the current training round; Indicates the total number of training rounds; Pi is a constant. This represents the cosine function.

[0186] The training conditions and Figure 2 The learning rate changes consistently show a trend: a higher learning rate in the early stages of training to accelerate convergence, followed by a gradual decrease to refine parameter search and improve the final reconstruction quality.

[0187] (iv) Training process and convergence analysis

[0188] Figure 2 The curves showing the changes in training loss and validation loss with Epoch and the learning rate are presented. Figure 2 It is evident that in the early stages of training, both training and validation losses decrease rapidly, indicating that the network can quickly learn the basic mapping relationship from the degraded flow field to the high-resolution flow field. Subsequently, the loss curve enters a stable decreasing phase, indicating that the model begins to further fit fine-scale texture and boundary information.

[0189] The optimal validation loss during the training process in this embodiment is: ;in: This represents the minimum loss value on the validation set.

[0190] from Figure 2 It can also be seen that the training set loss and the validation set loss follow a basically consistent trend, and there is no obvious divergence between them. This indicates that the training strategy, residual structure and gradient constraints adopted in this invention can maintain good generalization performance while ensuring training convergence, and no significant overfitting occurs.

[0191] Therefore, from the perspective of the training process, it can be proven that the present invention is not only feasible in terms of theoretical structure, but also has strong stability and feasibility in actual laboratory training.

[0192] (v) Quantitative assessment results

[0193] This embodiment quantitatively evaluates the trained model on the test set. The test results are shown in the table below.

[0194] Table 2 Quantitative Evaluation Results of the Model Test Set

[0195]

[0196] The mean square error can be expressed as:

[0197] ;

[0198] in: This represents the mean squared error loss; Indicates the number of longitudinal grid points in the flow field; This indicates the number of transverse grid points in the flow field; Indicates a vertical grid index; Indicates a horizontal grid index; This indicates that the reconstructed high-resolution flow field is in the first... line, number The value at the column position; This indicates that the true high-resolution flow field is in the first... line, number The value at the column position.

[0199] Peak signal-to-noise ratio can be expressed as:

[0200] ;

[0201] in: Indicates peak signal-to-noise ratio; This represents the maximum allowable amplitude after the flow field is normalized. Indicates mean square error; It represents a logarithmic operation with base 10.

[0202] As shown in Table 2, the average MSE of the method of this invention on the test set is only 0.000547, indicating that the reconstruction result is highly close to the real high-resolution flow field in terms of numerical value; the average PSNR reaches 32.66 dB, indicating that the present invention can still achieve a high-fidelity reconstruction effect under the challenging task of 8x super-resolution. For super-resolution reconstruction tasks, a PSNR greater than 30 dB is generally considered to be of high reconstruction quality, therefore, the result of 32.66 dB fully demonstrates the significant technical effectiveness of the present invention.

[0203] (vi) Qualitative visualization results analysis

[0204] 1. Analysis of Sample Reconstruction Results During Training

[0205] Appendix Figure 3 The reconstruction comparison results of a certain sample during the training process are presented. (Attached) Figure 3 The top row, from left to right, shows: 8x bicubic interpolation input plot, VortexUnet output plot, and actual flow field plot; the bottom row shows: local magnified plot, absolute error thermogram, and vorticity gradient plot.

[0206] From the appendix Figure 3 As seen in the top row, the bicubic interpolation input image on the left has lost a large number of tiny vortex structures. The vortex filaments, narrow shear layers, and local high gradient transitions that should have been present in the image are significantly blurred. In contrast, the output image of the middle model has been able to reconstruct more continuous vortex textures, and its overall flow field trend is quite close to the real image on the right. This indicates that the network of this invention already has a significant ability to recover details even before the training is completely finished.

[0207] From the appendix Figure 3 As can be seen from the enlarged view in the lower row, the model output can form relatively clear transition boundaries and texture directions in local areas, indicating that the encoder-decoder structure and skip connections used in this invention can effectively integrate local spatial features and deep semantic features.

[0208] From the appendix Figure 3 As can be seen from the absolute error heatmap in the middle of the bottom row, most areas are darker, with relatively high errors only in a few drastic locations. This indicates that the model has established a good fit for most areas of the field, and only in local extreme high-frequency areas does it still need further training and optimization.

[0209] From the appendix Figure 3 As can be seen from the gradient plot in the lower right corner, the model has been able to recover some continuous gradient stripes and vortex boundaries, which indirectly shows that gradient loss does indeed play a promoting role in the recovery of high gradient structures.

[0210] 2. Analysis of Test Sample Reconstruction Results

[0211] Appendix Figure 4 The reconstruction results on the test set samples are presented after training. (See attached...) Figure 3 In comparison, attached Figure 4 This better reflects the final technical effect of the invention.

[0212] From the appendix Figure 4 The three images in the top row clearly show that: the bicubic interpolation input image on the left is relatively smooth overall, but a lot of details are missing; the output image of the model in the middle is significantly better than the input on the left in terms of overall texture direction, local vortex structure and color transition, and is very close to the real high-resolution flow field on the right; especially in several local small-scale vortex concentration areas, the model output can recover the vortex core position and boundary direction close to the true value.

[0213] From the appendix Figure 4 As can be seen from the magnified view in the lower left corner, the model still maintains a high level of detail recovery in areas of strong local change. The boundary transition at the red-blue junction is clear, indicating that the method of this invention is not a simple smoothing or sharpening, but rather learns the nonlinear mapping relationship between low-resolution input and the real high-resolution flow field.

[0214] From the appendix Figure 4 As can be seen from the absolute error heatmap at the bottom center, the error is mainly distributed in a very few drastically changing local areas, while the large areas show a low error state. This phenomenon is consistent with the low MSE results in Table 2.

[0215] From the appendix Figure 4 As can be seen from the vorticity gradient plot in the lower right corner, the gradient structure output by the model is continuous, fine, and clearly directional, showing high consistency with the physical texture features corresponding to the real flow field. This result indicates that the gradient loss introduced in this invention not only improves visual clarity but, more importantly, enhances the ability to preserve the physical structure of the flow field at the first derivative level, thereby effectively avoiding the edge blunting and over-smoothing phenomena commonly seen when only MSE is targeted.

[0216] (vii) Experimental Results

[0217] Based on the above quantitative and qualitative results, it can be proven that the present invention has at least the following technical effects.

[0218] 1. It can significantly improve the reconstruction accuracy of degraded flow fields by 8 times.

[0219] In this embodiment, the model achieved an average MSE of 0.000547 and an average PSNR of 32.66 dB on the test set, indicating that the present invention can effectively recover high-resolution information of the two-dimensional vortex flow field under high-magnification degradation conditions, with high numerical accuracy and excellent overall reconstruction quality.

[0220] 2. It can effectively recover small-scale vortex and high-frequency boundary information.

[0221] Combined with appendix Figure 3 Appendix Figure 4 The visualization results show that this invention not only recovers the macroscopic flow field profile, but also has a strong ability to recover local small-scale vortices, shear boundaries, and complex textures. This indicates that the residual U-Net structure, skip connections, and multi-scale feature extraction mechanism in this invention can effectively compensate for the high-frequency information lost during downsampling.

[0222] 3. It can improve the realism of the physical structure of the reconstructed flow field.

[0223] Appendix Figure 4 The gradient plots show that the method of this invention, while recovering the field values, also maintains the vorticity gradient fringes and boundary orientation relatively well. This proves that by introducing gradient loss based on the Sobel operator, this invention enables the model to approximate the real flow field from the first derivative level, thereby improving the physical consistency of the reconstruction results.

[0224] 4. The training process is stable and feasible for engineering applications.

[0225] Figure 2 The training loss and validation loss curves shown decrease smoothly, and the learning rate decay process is controllable, indicating that the technical route of the present invention is not only theoretically sound, but also has the possibility of stable training and practical deployment. It is suitable for applications such as recovery of flow field data after compressed transmission, enhancement of low-precision measurement data, and rapid high-fidelity reconstruction of numerical simulation results.

[0226] In summary, this application example demonstrates that the proposed super-resolution reconstruction method for vortex flow fields based on residual U-Net and gradient constraint loss can achieve high-fidelity recovery of high-resolution flow fields under 8x degradation conditions. Experimental results show that the model achieves excellent performance on the test set with an average mean square error of 0.000547 and an average peak signal-to-noise ratio of 32.66 dB; combined with... Figure 2 To be continued Figure 4 It can be seen that the present invention not only has a stable training convergence process, but also can effectively recover the small-scale vortex structure, boundary transition and high gradient texture that have been blurred or even lost in the bicubic interpolation input. The reconstruction results show obvious advantages in both numerical accuracy and physical structure authenticity, thus proving that the present invention has good technical effect and engineering application value.

[0227] The foregoing description of embodiments of the present invention, through which those skilled in the art are able to implement or use the present invention, will be readily apparent to those skilled in the art. Various modifications to these embodiments will be readily apparent to those skilled in the art. The general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novelty disclosed herein.

[0228] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0229] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0230] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0231] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0232] In a typical configuration, a computing device includes one or more processors (CPU), input / output interfaces, network interfaces, and memory.

[0233] Memory may include non-persistent memory in computer-readable media, such as random access memory (RAM) and / or non-volatile memory, such as read-only memory (ROM) or flash RAM. Memory is an example of computer-readable media.

[0234] Computer-readable media includes both permanent and non-permanent, removable and non-removable media that can store information using any method or technology. Information can be computer-readable instructions, data structures, modules of programs, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, magnetic disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.

Claims

1. A super-resolution reconstruction method for vortex flow fields based on residual U-Net, characterized in that, Includes the following steps: S1: Obtain a two-dimensional high-resolution vortex flow field sample, and perform bicubic interpolation downsampling on the two-dimensional high-resolution vortex flow field sample according to a preset downsampling ratio. Then, interpolate the downsampled low-resolution flow field back to the original grid size through bicubic interpolation to obtain a degenerate input flow field with the same size as the two-dimensional high-resolution vortex flow field sample. S2: Construct a residual U-Net network for flow field reconstruction. The residual U-Net network includes an encoder, a bottleneck layer, a decoder, and global residual connections. The encoder is used to extract multi-scale features of the degraded input flow field, the decoder is used to restore spatial resolution and reconstruct high-frequency details, and a skip connection is set between the encoder and the decoder to fuse shallow spatial features and deep semantic features. The network output is a residual flow field of the same size as the input flow field. S3: Add the residual flow field to the degraded input flow field element by element to obtain the reconstructed high-resolution flow field; construct a combined loss function based on the pixel error and gradient error between the reconstructed high-resolution flow field and the corresponding real high-resolution flow field; S4: The residual U-Net network is trained under supervision using samples. During the training phase, local flow field blocks are obtained by random pruning for iterative training, and the entire flow field is used for verification during the validation phase. The trained flow field reconstruction model is obtained by optimizing the combined loss function. S5: The low-resolution two-dimensional vortex flow field to be reconstructed is back-interpolated to the target size using bicubic interpolation and then input into the trained flow field reconstruction model, outputting the corresponding high-resolution reconstructed flow field.

2. The method according to claim 1, characterized in that, In step S1, the two-dimensional high-resolution vortex flow field sample is a single-channel three-dimensional tensor defined on the real number domain, with 1 channel and spatial dimensions of H vertical grid points and W horizontal grid points. The degraded input flow field is obtained as follows: first, bicubic interpolation downsampling is performed on the two-dimensional high-resolution vortex flow field sample according to the downsampling factor, and then bicubic interpolation back-interpolation upsampling is performed on the downsampling result according to the same downsampling factor to obtain the degraded input flow field.

3. The method according to claim 1, characterized in that, In step S2, the output residual flow field of the residual U-Net network and the degraded input flow field satisfy the following relationship: the degraded input flow field and the residual flow field predicted by the residual U-Net network based on the input flow field are added element by element at the corresponding grid positions to obtain the reconstructed high-resolution flow field; And / or, in step S2, the residual U-Net network includes: The input convolutional layer is used to map the single-channel input to an initial feature map; A three-level downsampling coding layer is used to extract multi-scale features step by step; The first-level bottleneck layer is used to extract deep global semantic features; A three-stage upsampling decoding layer is used to restore spatial resolution step by step; The output convolutional layer is used to map the decoded feature map into a single-channel residual flow field; The output feature map sizes of each layer satisfy the following: The first layer encoded feature map is a three-dimensional feature map defined on the real number domain, with 64 channels and spatial dimensions of vertical and horizontal dimensions; the second layer encoded feature map has 128 channels, with spatial dimensions of half the vertical dimension and half the horizontal dimension, respectively; the third layer encoded feature map has 256 channels, with spatial dimensions of one-quarter of the vertical dimension and one-quarter of the horizontal dimension, respectively; and the bottleneck layer feature map has 512 channels, with spatial dimensions of one-eighth of the vertical dimension and one-eighth of the horizontal dimension, respectively.

4. The method according to claim 3, characterized in that, In step S2, each encoding layer and decoding layer adopts a dual convolution structure. The dual convolution structure is as follows: the input feature map is first subjected to a convolution operation with a kernel size of 3*3, then batch normalization operation and non-linear activation operation are performed, and then the result is subjected to a convolution operation with a kernel size of 3*3, batch normalization operation and non-linear activation operation again to obtain the output of the dual convolution operation unit. And / or, in step S2, the encoding layer uses max pooling for downsampling, specifically: the k-th level encoded input feature map is first subjected to max pooling downsampling, and then processed by the k-th level encoded convolution operation to obtain the k-th level encoded output feature map; the decoding layer uses transposed convolution for upsampling, specifically: the feature map after transposed convolution upsampling in the k-th level decoding stage is concatenated with the skip connection encoded feature map corresponding to the k-th level decoding layer according to the channel dimension, and then processed by the k-th level decoding convolution operation to obtain the k-th level decoded output feature map.

5. The method according to claim 1, characterized in that, In step S3, the combined loss function is obtained by weighting the mean squared error loss and the gradient loss according to the gradient loss weight coefficient; The mean square error loss is specifically calculated by squaring the numerical differences between the reconstructed high-resolution flow field and the real high-resolution flow field at corresponding positions of all longitudinal and transverse grid indices, summing the squared differences at all positions, and then dividing by the total number of grid points in the flow field. The gradient loss is specifically calculated as follows: the gradient difference between the reconstructed high-resolution flow field and the real high-resolution flow field in the x and y directions are calculated respectively; the absolute values ​​of the gradient difference in the x and y directions at each location are summed; and the summation of the results at all locations is then divided by the total number of grid points in the flow field.

6. The method according to claim 5, characterized in that, The x-direction gradient and y-direction gradient are obtained through the Sobel operator, specifically: the real high-resolution flow field is convolved with the Sobel convolution kernels in the x-direction and y-direction respectively to obtain the gradient of the real high-resolution flow field in the x-direction and the gradient in the y-direction; the reconstructed high-resolution flow field is convolved with the Sobel convolution kernels in the x-direction and y-direction respectively to obtain the gradient of the reconstructed high-resolution flow field in the x-direction and the gradient in the y-direction.

7. The method according to claim 1, characterized in that, In step S4, the training sample set, validation sample set, and test sample set are divided in a ratio of 8:1:1; And / or, in step S4, local training blocks are generated by random pruning during the training phase. The local training blocks are all single-channel three-dimensional tensors defined on the real number domain, and their spatial dimensions are all square regions with a side length of P. And / or, in step S4, the model training uses the AdamW optimizer, and the learning rate is updated using a cosine annealing strategy. Specifically, the learning rate corresponding to the t-th training round is obtained by adding the difference between the minimum learning rate and the initial learning rate after decaying by a cosine function to the minimum learning rate. And / or, in step S4, an early stopping mechanism is used to control the training termination condition. Training is stopped when the verification loss does not decrease within a preset number of rounds, where the preset number of rounds is 15.

8. A two-dimensional vortex flow field super-resolution reconstruction system, characterized in that, The system is used to implement the method according to any one of claims 1-7, comprising: The data preprocessing module is used to perform step S1; the network construction module is used to perform step S2; the loss construction module is used to perform step S3; the model training module is used to perform step S4; and the reconstruction output module is used to perform step S5.

9. A computer-readable storage medium having a computer program or instructions stored thereon, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method described in any one of claims 1-7.

10. A computer program product, comprising a computer program or instructions, characterized in that, When the computer program or instructions are executed by a processor, they implement the steps of the method described in any one of claims 1-7.

Images