Particle image velocity measurement method based on neural operator

Abstract

The invention discloses a neural operator-based particle image velocimetry (PIVNO) method. Efficient mapping of a particle image pair to a flow field is realized through an end-to-end learning framework. According to the method, a neural operator normal form is adopted, cost volume construction with large calculation amount in a traditional method is avoided, and calculation efficiency and estimation precision are improved. The method comprises the steps of obtaining a synthetic data set and carrying out multi-scale down-sampling, constructing a PIVNO model to realize high-precision flow field prediction, supervising pre-training on the synthetic data set, self-supervising a fine-tuning optimization model on real data, finally predicting a real flow field, and supporting estimation of any resolution. According to the method, global feature aggregation, iterative optimization and high-resolution reconstruction technologies are combined, the flow field estimation precision is improved, and the adaptability of the model on real data is enhanced. The method is efficient in calculation, high in generalization ability and suitable for the fields of aerodynamics, turbulence research, biological fluid and the like, and provides technical support for high-precision flow field measurement.

Description

Technical Field

[0001] The present invention relates to the cross - field of computer vision, fluid mechanics and deep learning, and particularly to a particle image velocimetry method based on neural operators. Background Art

[0002] Particle Image Velocimetry (PIV) is a high - precision fluid measurement technology based on computer vision, which is widely used in many fields such as physics, material science, life science and engine design. This technology puts tracer particles in the fluid, collects a sequence of particle images with the help of a high - speed imaging system and a laser light source, and calculates based on the displacement between particles, and then derives the corresponding velocity field information. The significant advantage of PIV technology is that it can provide high - resolution, spatio - temporally discrete flow field observation data, so it is of great significance in fluid dynamics research and engineering applications.

[0003] However, traditional PIV methods mostly rely on local cross - correlation matching algorithms. In scenarios where the particle distribution is uniform or the image texture information is weak, the matching accuracy drops significantly, thus affecting the accuracy of the overall velocity field estimation. At the same time, the size of the matching window directly affects the estimated spatial resolution, forming a trade - off limitation between resolution and computational efficiency. In recent years, deep learning technology has been gradually applied to the PIV field, modeling the particle image features with a high - dimensional latent space, effectively alleviating the matching ambiguity problem. Deep neural networks can extract richer global spatial information while maintaining a relatively high spatial resolution, improving the overall performance of velocity field estimation. Nevertheless, deep - learning - based methods still face many challenges, including the dependence on a large amount of labeled data, the domain difference between synthetic data and real experimental data, and the limited model generalization ability. In addition, existing methods mainly focus on global velocity field estimation and are difficult to meet the requirements of high - precision, fine - grained local area flow analysis. Summary of the Invention

[0004] To solve the above problems, the present invention proposes a PIV method based on neural operators. This method introduces the neural operator paradigm, which can directly map the particle image pair I1 and I2 to the velocity field U, thereby avoiding the construction of a cost volume with high computational cost in traditional methods and effectively improving the accuracy and computational efficiency of velocity field estimation. The method of the present invention combines a supervised pre-training and a self-supervised fine-tuning mechanism, which can be efficiently trained on a publicly available synthetic dataset and adapt to real experimental data without additional annotation, realizing the adaptive optimization of the model performance and significantly enhancing its generalization ability in different flow scenarios. In addition, the proposed neural operator model supports velocity field estimation at any resolution for any specified region, with high flexibility and adaptability, especially suitable for refined local flow field analysis scenarios. The implementation process of the method of the present invention includes the following steps:

[0005] Step S1: Obtain a publicly available synthetic PIV dataset for training, and randomly downsample the data by a factor of 1 to 4 to generate input image pairs of different resolutions. This enables the model to learn flow field features at different scales to support estimations at different resolutions;

[0006] Step S2: Construct a neural operator model PIVNO to realize the mapping between the particle image pair and the velocity field. The model is based on an end-to-end learning framework and can maintain high-precision velocity field estimation in different flow scenarios and resolutions;

[0007] Step S3: Use the synthetic dataset to perform supervised pre-training on PIVNO. Adopt the least square error and optimize the model parameters so that the model can accurately estimate the velocity field;

[0008] Step S4: Obtain the particle image data of the real flow field through an experimental device;

[0009] Step S5: Self-supervised fine-tune the pre-trained model on the real data and estimate the real velocity field.

[0010] Furthermore, in the step S1, first randomly generate a downsampling ratio s between 1 and 4 to determine the ratio of image reduction. Assuming the original image size is W×H, then calculate the size of the cropped area as W′ = 64×s and H′ = 64×s, ensuring that the final size of the image after downsampling is always fixed at 64×64 pixels. Subsequently, randomly select a sub-region of size W′×H′ in the original image for cropping to ensure data diversity and enhance the generalization ability of the model. The cropped sub-region is downsampled by bilinear interpolation and scaled to 64×64, thereby unifying the size of the input data, enabling the model to learn flow field features at different scales, and ensuring the stability and consistency of training.

[0011] Furthermore, the model PIVNO in step S2 includes the following sub-steps:

[0012] Step S21: Use a feature encoder to extract local features from the input particle image pairs and concatenate them into local context features;

[0013] Step S22: Use the RoPE Galerkin Attention module to perform position encoding on the local features and aggregate the global features using the Galerkin-type attention mechanism to obtain global context features;

[0014] Step S23: Iteratively optimize the context features through the Conv-GRU module to extract the flow field features from the context features;

[0015] Step S24: Use the continuous scale flow field velocity measurement module to map the flow field features to velocity field estimates at any scale.

[0016] Furthermore, in step S21, the feature encoder consists of six residual units, and each residual unit is composed of two convolutional kernel residual connections. Among them, the convolutional kernel extracts local features through convolution operations on the image, while the residual connection ensures the effective transmission of key information in the deep network, avoiding gradient disappearance or feature loss. The extracted features are concatenated to form context features to provide a richer information representation.

[0017] Furthermore, in step S22, the RoPE Galerkin Attention module includes the following steps:

[0018] Step S221: First, map the input features to queries (Q), keys (K), and values (V), which is the core step in the attention mechanism for judging the information correlation between different positions. And use Rotary Position Encoding (RoPE) to enhance K and V to introduce relative position information and improve the local feature expression ability. Subsequently, perform layer normalization (LN) on K and V to stabilize the training and optimize the calculation.

[0019] Step S222: Use the Galerkin-type attention integral operator to calculate the attention weights, without constructing a computationally expensive cost volume, thus achieving efficient global feature aggregation. Specifically, perform matrix multiplication of Q and K processed by RoPE to generate the attention distribution and multiply it by the value V to obtain the attention output;

[0020] Step S223: In the feed-forward network part, introduce a 3×3 depth convolution, which is embedded between the first fully connected layer and the GELU activation function, and can capture local absolute position information and enhance the model's local information perception ability;

[0021] Step S224: In the final feature mapping stage, a layer-by-layer calculation framework is formed by combining multi-layer fully connected transformation, non-linear activation (GELU), and hierarchical linear operators to improve the coherence of information processing and computational efficiency. The entire module combines the attention mechanism with local convolution to achieve the synergistic effect of global feature aggregation and local feature enhancement, thereby enhancing the expression ability of the model.

[0022] Furthermore, in the step S23, the GRU (Gated Recurrent Unit) takes the context features obtained through step S22 as the initial hidden state and the input at each moment, and performs iterative optimization through four steps: update gate, reset gate, candidate hidden state calculation, and final hidden state update, so as to extract the flow field features from the context features.

[0023] Furthermore, in the step S24, the continuous scale flow field velocity measurement module in PIVNO can achieve super-resolution reconstruction of the velocity field. This module combines methods such as random sampling, continuous interpolation, and graph convolution, and infers the high-resolution velocity field distribution based on the flow field features extracted in step S23. Through this process, PIVNO can generate velocity field estimation results with different resolutions in the specified area, which is especially suitable for high-precision analysis of local flow fields.

[0024] Furthermore, in the step S3, supervised pre-training aims to preliminarily train the PIVNO constructed in step 2 using the synthetic data set obtained in step 1 to ensure that the model has the ability to accurately estimate the velocity field. It specifically includes the following steps:

[0025] Step S31: Use the synthetic data set obtained in step S1 that covers various flow conditions, which includes different velocity distributions, turbulence characteristics, and boundary conditions, so as to ensure that the model can learn diverse flow field features.

[0026] Step S32: Use the least square error as the loss function, calculate the error between the velocity field estimated by the model and the true velocity field, and optimize the model parameters through gradient descent. During the training process, the model needs to maintain high precision under different flow scenarios and resolutions to ensure its generalization ability and improve its adaptability to real experimental data.

[0027] Furthermore, in the step S4, particle image data of the real flow field is obtained through the experimental device to provide experimental data for model optimization. It specifically includes the following steps:

[0028] Step S41: Release tracer particles into the fluid to be measured so that they move with the fluid for subsequent imaging analysis.

[0029] Step S42: Use a light source to irradiate the tracer particles and take consecutive particle image pairs by means of high-speed camera shooting.

[0030] Further, in the said step S5, it specifically includes the following steps:

[0031] Step S51: Construct a self-supervised loss function, adopting a combination of a data term, a smoothing term, and a divergence regularization term. The data term is used to model the similarity between particle image pairs, the smoothing term is used to ensure the spatial smoothness of the velocity field, and the divergence regularization term is used to maintain the incompressibility of the velocity field to ensure physical consistency;

[0032] Step S52: Self-supervised optimize the model parameters. By minimizing the combined loss term, make PIVNO gradually adjust the parameters on real experimental data, so as to match the statistical distribution of the real velocity field and improve the adaptability of the model;

[0033] Step S53: Velocity field estimation. Adopt the optimized PIVNO model to estimate the velocity field of real particle image pairs, and support the generation of velocity fields at any resolution. Combine the continuous-scale flow field velocity measurement module, and through random sampling, continuous interpolation, and graph convolution methods, realize the super-resolution reconstruction of the velocity field, so that PIVNO can provide high-precision local velocity field estimation within a specified area to meet the needs of fine flow analysis.

[0034] Compared with the prior art, the present invention adopts the above technical solutions and has the following beneficial effects:

[0035] By introducing the neural operator framework, the present invention has successfully realized the direct mapping from particle image pairs to the velocity field, avoiding the construction of computationally expensive cost volumes in traditional methods, thus significantly improving the computational efficiency. At the same time, the method enhances the accuracy of velocity field estimation through technical means such as global feature aggregation, iterative optimization, and high-resolution reconstruction, and can accurately capture complex flow structures. The self-supervised fine-tuning strategy further improves the adaptability of the model in real experimental data, enabling the method to maintain high stability and accuracy under different flow conditions and resolutions. Compared with existing particle image velocimetry methods, the present invention not only is more efficient in computing resources, but also has obvious advantages in velocity field estimation accuracy and generalization ability, and can be widely applied to scientific and engineering fields such as aerodynamics, turbulence research, biofluids, and ocean fluids, and has important application value in high-precision velocity field measurement and complex flow analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 is the overall flow schematic diagram of the method of the present invention;

[0037] Figure 2 is the schematic diagram of the neural operator network structure in the embodiment of the present invention;

[0038] Figure 3 It is a schematic structural diagram of the particle image acquisition device in the embodiment of the present invention;

[0039] Figure 4 It is a schematic diagram of the input and output results of the neural operator network in the embodiment of the present invention. Specific implementation manners

[0040] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0041] The present invention proposes a PIV method based on neural operators. This method introduces the neural operator paradigm, which can directly map the particle image pairs I1 and I2 to the velocity field U, avoiding the construction of cost volumes with high computational costs in traditional methods and significantly improving the accuracy of velocity field estimation. By combining supervised pre-training and self-supervised fine-tuning strategies, this method can be efficiently trained on synthetic datasets and adaptively optimized on real data, significantly enhancing the generalization ability of the model. In addition, the neural operator model proposed by the present invention supports outputting velocity field estimation results with arbitrary resolutions in any specified region, meeting the application requirements for high-precision analysis of local regions. Figure 1 It is a flowchart for implementing the method of the present invention, which includes the following steps:

[0042] Step S1: Obtain the publicly available synthetic PIV dataset for training, and perform random downsampling on the data by 1-4 times to generate input image pairs with different resolutions, enabling the model to learn flow field features at different scales to support estimations at different resolutions;

[0043] Step S2: Construct a neural operator model PIVNO to implement the mapping between particle image pairs and the velocity field. The model is based on an end-to-end learning framework and can maintain high-precision velocity field estimation in different flow scenarios and resolutions;

[0044] Step S3: Use the synthetic dataset to perform supervised pre-training on PIVNO, adopt the least square error, and optimize the model parameters to enable the model to accurately estimate the velocity field;

[0045] Step S4: Obtain the particle image data of the real flow field through the experimental device;

[0046] Step S5: Self-supervised fine-tune the pre-trained model on real data and estimate the real velocity field.

[0047] Furthermore, in the step S1, first obtain the publicly available PIV synthetic dataset for training, and perform random downsampling on the data by a factor of 1 to 4 to generate input images of different resolutions. This enables the model to learn flow field features at different scales and supports estimations at different resolutions. In the specific implementation process, first randomly generate a downsampling ratio s between 1 and 4 to determine the ratio of image reduction. Assuming the original image size is W×H, then calculate the size of the cropped area as W′ = 64×s and H′ = 64×s to ensure that the final size of the image after downsampling is always fixed at 64×64 pixels. Subsequently, randomly select a sub-region of size W′×H′ in the original image for cropping to ensure data diversity and enhance the generalization ability of the model. The cropped sub-region is downsampled by bilinear interpolation and scaled to 64×64, thus unifying the size of the input data, enabling the model to learn flow field features at different scales, and ensuring the stability and consistency of training.

[0048] Furthermore, the neural operator model PIVNO constructed in the step S2 is used to achieve an efficient mapping from particle image pairs to velocity fields. This model adopts an end-to-end learning framework and can maintain high-precision velocity field estimation in different flow scenarios and resolutions. Define PIVNO as a mapping operator G: A→U, where A represents the Hilbert space of particle image pairs: A = {a = [I1, I2] ∈ R 2 ∣I1(x) = I2(x - u(x))}, U represents the Hilbert space of velocity fields: U = {u: D→R 2}, where is the image domain. The network model is as Figure 2 shown. PIVNO enables G θ to learn the mapping from particle image pairs to velocity fields through end-to-end training parameters θ:

[0049]

[0050] PIVNO mainly consists of a feature encoder, a RoPE-Galerkin attention module, a GRU iterative optimization module, and a continuous scale flow velocity measurement module, and specifically includes the following:

[0051] Step S21: Use the feature encoder to extract local features from the input particle image pairs and concatenate them into local context features;

[0052] Step S22: Use the RoPE Galerkin Attention module to perform position encoding on the local features and use the Galerkin-type attention mechanism to aggregate global features to obtain global context features;

[0053] Step S23: Iteratively optimize the context features through the Conv-GRU module, so as to extract the flow field features from the context features;

[0054] Step S24: Use the continuous scale flow field velocity measurement module to map the flow field features to velocity field estimations of any scale.

[0055] Furthermore, in the step S21, the feature encoder is used to extract local features from the input pair of particle images and splice them into local context features. The feature encoder consists of six residual units to form the feature encoder, and each residual unit adopts a residual connection structure, so that key information is not lost in the deep network. The role of the feature encoder is to extract the local features of the input pair of particle images I1 and I2 and map them to a high-dimensional space. First, obtain the preliminary features P(I1) and P(I2) through convolution operations, and then splice them to obtain the context feature z0(x):

[0056] P(I)(c)=∫ N(c) κ(c - y)I(y)dy + I(c)

[0057] z0(x)=Concat(P(I1), P(I2))

[0058] where κ(c - y) is a 3×3 convolution kernel defined on the local neighborhood N(c) at position c, and I(c) is the residual connection term. Concat represents the splicing operation in the channel dimension, enabling the model to synthesize the information of two frames of images and learn a more accurate velocity field representation.

[0059] Furthermore, in the step S22, use the RoPE Galerkin Attention module to perform position encoding on the local features, and use the Galerkin-type attention mechanism to aggregate the global features to obtain the global context features. The RoPEGalerkin Attention module includes the following steps:

[0060] Step S221: First, map the input features to queries (Q), keys (K), and values (V), which is the core step in the attention mechanism and is used to judge the information correlation between different positions. And use the Rotary Position Encoding (RoPE) to enhance K and V to introduce relative position information and improve the local feature expression ability. Subsequently, perform layer normalization (LN) on K and V to stabilize the training and optimize the calculation;

[0061] Step S222: In the attention calculation stage, use the Galerkin-type attention integral operator to calculate the attention weights, without the need to construct a cost volume with a large amount of calculation, so as to achieve efficient global feature aggregation. Specifically, generate the attention distribution through the matrix multiplication of Q and K processed by RoPE and multiply it by the value V to obtain the attention output;

[0062] Step S223: In the feed-forward network part, introduce a 3×3 depth convolution, which is embedded between the first fully-connected layer and the GELU activation function, capable of capturing local absolute position information and enhancing the model's local information perception ability;

[0063] Step S224: In the final feature mapping stage, combine multi-layer fully-connected transformations, non-linear activation (GELU), and hierarchical linear operators to form a layer-by-layer calculation framework to improve the coherence of information processing and computational efficiency. The entire module combines the attention mechanism with local convolution to achieve the synergistic effect of global feature aggregation and local feature enhancement, thereby enhancing the model's expressive ability.

[0064] The calculation process of this module is as follows:

[0065] z t+1 (x) = σ(W t z t (x) + (K t (z t (x))b t ))

[0066]

[0067] First, perform a linear transformation on the input feature vector z0(x), where W t is the weight matrix and σ is the non-linear activation function. Then, calculate the key K and the value V through the attention mechanism, and apply the Rotary Position Encoding (RoPE) to them to obtain the encoded key K′ and value V′. Among them, represents the position of coordinate n on the x-axis, is the learnable frequency parameter. Finally, use the Galerkin linear attention mechanism to calculate the attention weights to enhance the feature expression ability and improve the accuracy of the velocity field estimation.

[0068] Furthermore, in the step S23, the GRU (Gated Recurrent Unit) uses the context feature z t (x) obtained through step S22 as the initial hidden state and the input at each moment, and performs iterative optimization through four steps: update gate, reset gate, candidate hidden state calculation, and final hidden state update, so as to extract the flow field feature from the context feature. Specifically, take z t (x) as the initial hidden state h0 and the input c t at each moment, and perform optimization through four steps: update gate, reset gate, candidate hidden state calculation, and final hidden state update. Its calculation process is as follows:

[0069] yt = σ(Conv 3×3 ([h t-1 , c t , W z )),

[0070] r t = σ(Conv 3×3 ([h t-1 , c t , W r )),

[0071]

[0072] First, update the gate y t from the previous hidden state h t-1 and the current input feature c t through 3×3 convolution and Sigmoid activation calculations to determine to what extent the current hidden state h t depends on historical information or new inputs. Second, the reset gate r t controls the influence of h t-1 on the candidate hidden state . A smaller r t value weakens historical information and makes the model focus more on the current input. The candidate hidden state combines the previous hidden state h t after the action of the reset gate r t-1 and the current input feature c t , and forms a new velocity field estimate through 3×3 convolution and Tanh activation. Finally, h t is weighted and fused by h t-1 and according to y t to balance historical information and new inputs. With multiple iterations, the velocity field estimate gradually converges, improving the accuracy and stability of the estimate.

[0073] Furthermore, in step S24, a continuous scale flow field velocity measurement module is used to map the flow field features to velocity field estimates at any scale. After the GRU iterative optimization module obtains the velocity field estimate h t , it is necessary to further generate the final high-resolution velocity field estimate through the continuous scale flow velocity measurement module. This module calculates the features at the target coordinates through random sampling and continuous interpolation methods to ensure that the velocity field estimates at different resolutions can maintain high accuracy. The calculation process is as follows:

[0074]

[0075] First, the module randomly samples the velocity field estimation output by the GRU and performs scale conversion in combination with the weights s of adjacent grid points to ensure the effective fusion of information at different resolutions, thereby enhancing the adaptability of the model. Among them, x i represents the target coordinates, and h l represents the hidden state h t of the velocity field estimation at different scales is the feature vector after weighted fusion of these scale information. Subsequently, the final velocity field is calculated using graph convolution operations where the convolution kernel κ(c - y) is a 3×3 graph convolution kernel, which is responsible for performing weighted summation on the local region R(c) to ensure the smoothness of the velocity field estimation and reduce the noise interference in the high-resolution estimation.

[0076] Furthermore, in step S3, the supervised pre-training aims to preliminarily train the PIVNO constructed in step 2 using the synthetic dataset obtained in step 1 to ensure that the model has the ability to accurately estimate the velocity field. It specifically includes the following steps:

[0077] Step S31: Use the synthetic dataset obtained in step S1 that covers various flow conditions, which includes different velocity distributions, turbulence characteristics, and boundary conditions, so as to ensure that the model can learn diverse flow field features.

[0078] Step S32: Adopt the least squares error as the loss function, calculate the error between the velocity field estimated by the model and the true velocity field, and optimize the model parameters through gradient descent. During the training process, the model needs to maintain high accuracy under different flow scenarios and resolutions to ensure its generalization ability and improve its adaptability to real experimental data.

[0079] The loss function of the supervised training is as follows:

[0080]

[0081] where is the velocity field estimated by the model.

[0082] Furthermore, in step S4, the particle image data of the true velocity field is obtained through the experimental device to provide experimental data for model optimization. It specifically includes the following steps:

[0083] Step S41: Release tracer particles into the fluid to be measured so that they move with the fluid for subsequent imaging analysis.

[0084] Step S42: Use a light source to irradiate the tracer particles and capture consecutive pairs of particle images through a high-speed camera.

[0085] Such asFigure 3 As shown, first, small tracer particles are introduced into the fluid medium to be measured, and these particles will move along with the surrounding fluid medium. Then, a high-power light source uniformly irradiates the tracer particles in the area to be measured, making them clearly visible during the imaging process. The illuminated tracer particles are displaced under the action of fluid motion, and the camera continuously captures images of this area, thereby obtaining a sequence of particle images containing fluid motion information. Finally, these consecutive particle image pairs can be used for subsequent velocity field calculation and analysis, providing real data support for fluid motion estimation.

[0086] Furthermore, in step S5, it specifically includes the following steps:

[0087] Step S51: Construct a self-supervised loss function, which adopts a combination of a data term, a smooth term, and a divergence regularization term. The data term is used to model the similarity between particle image pairs, the smooth term is used to ensure the spatial smoothness of the velocity field, and the divergence regularization term is used to maintain the incompressibility of the velocity field to ensure physical consistency;

[0088] Step S52: Self-supervised optimization of model parameters. By minimizing the combined loss term, PIVNO gradually adjusts the parameters on real experimental data, thereby matching the statistical distribution of the real velocity field and improving the adaptability of the model;

[0089] Step S53: Velocity field estimation. Using the optimized PIVNO model, estimate the velocity field for real particle image pairs and support the generation of velocity fields at any resolution. Combining with a continuous-scale flow field velocity measurement module, through random sampling, continuous interpolation, and graph convolution methods, achieve super-resolution velocity field reconstruction, enabling PIVNO to provide high-precision local velocity field estimates within a specified area to meet the needs of fine flow analysis.

[0090] Its loss function is defined as follows:

[0091] L P =L d +λ s L s +λ d L div

[0092]

[0093] L d (I1, I2) = σ(1 - E x (corr))

[0094]

[0095] During the self-supervised fine-tuning process, the model optimization is based on the total loss function L P, where the data item L d Calculate the correlation corr(x) of the images before and after the optical flow transformation to ensure that the estimated velocity field u(x) matches the original particle image pair. Here, I1(x) and I2(x) are the input particle image pair, I1(x) = I2(x - u(x)) is the image transformed according to the velocity field, and N(x) is the local neighborhood. is the dot product of the original image and the transformed image, E x represents the mean calculation over the entire image range, σ(z) = (z 2 +∈ 2 ) γ is the Charbonnier penalty function to improve the robustness of the calculation. The smoothing term L s constrains the spatial variation of the velocity field by calculating the Laplacian operator of the velocity field to ensure the smoothness of the velocity field estimation and reduce high-frequency noise. The divergence regularization term L div (u) constrains the mass conservation in hydrodynamics by calculating the divergence of the velocity field to make the estimated velocity field satisfy the physical properties of an incompressible fluid and prevent non-physical expansion or compression. This method first performs supervised pre-training on synthetic data and then performs self-supervised optimization on real data to make the model adapt to the statistical characteristics of the experimental data, improve the generalization ability, and ensure high-precision velocity field estimation even in an unsupervised situation.

[0096] Finally, as Figure 4 shown, use the fine-tuned PIVNO model to estimate the velocity field of real experimental data. Input the particle image pair I1, I2, and the model outputs the velocity field U(x):

[0097]

[0098] where θ * are the parameters of the fine-tuned model.

Claims

1. A particle image velocimetry method based on neural operators, characterized in that, This method constructs and trains a neural operator to achieve the mapping from particle image pairs to velocity fields. The method includes the following steps: Step S1: Obtain the publicly available synthetic PIV dataset for training, and perform random downsampling on the data by a factor of 1 to 4 to generate input image pairs with different resolutions. This enables the model to learn flow field features at different scales and enhances its adaptability to different image resolutions; Step S2: Construct the neural operator model PIVNO to achieve the mapping between particle image pairs and velocity fields under an end-to-end learning framework, ensuring high-precision estimation under various flow scenarios and image scales; Step S3: Use the synthetic dataset to perform supervised pre-training on PIVNO, and optimize the model parameters using the least squares error to improve its velocity field estimation accuracy; Step S4: Obtain particle image data under the real flow field through the experimental device as the data basis for subsequent model fine-tuning; Step S5: Perform self-supervised fine-tuning on the pre-trained model using the real data, and estimate the actual velocity field based on the optimized model.

2. The method according to claim 1, wherein In the step S1, first, a downsampling ratio s is randomly generated between 1 and 4 for the example of resizing the image. Let the size of the original image be W×H. Calculate the size of the cropped region as W ′ = 64×s (W ′ <= W) and H ′ = 64×s (H ′ <= H). Ensure that the final size of the downsampled image is 64×64. Subsequently, randomly select a sub-region of size W′×H′ in the original image for cropping to ensure data diversity and enhance the generalization ability of the model. The cropped sub-region is downsampled by the bilinear interpolation method to resize it to 64×64, so as to unify the input data size, enabling the model to learn flow field features at different scales while ensuring the stability and consistency of training.

3. The method according to claim 1, characterized in that In the above Step S2, the constructed neural operator model PIVNO includes: Step S21: Use a feature encoder to extract local features from the input particle image pairs and concatenate them into local context features; Step S22: Use the RoPE Galerkin Attention module to perform position encoding on the local features, and use the Galerkin-type attention mechanism to aggregate the global features to obtain the global context features; Step S23: Iteratively optimize the context features through the Conv-GRU module, and extract the flow field features from the context features; Step S24: Use the continuous scale flow field velocity measurement module to map the flow field features to velocity field estimates at any scale.

4. The method according to claim 3, wherein In the above Step S21, the feature encoder consists of six residual units, and each residual unit is composed of two convolutional kernel residual connections. Among them, the convolutional kernel extracts local features through convolution operations on the image, and the residual connection ensures that key information can be effectively transmitted in the deep network, avoiding gradient disappearance or feature loss. The extracted features are concatenated to form context features to provide a richer information representation.

5. The method according to claim 3, characterized in that, In the above Step S22, the RoPE GalerkinAttention module includes: Step S221: First, map the input features to queries (Q), keys (K), and values (V), which are the core steps in the attention mechanism for judging the information correlation between different positions. And use Rotary Position Encoding (RoPE) to enhance K and V to introduce relative position information and improve the local feature expression ability. Subsequently, perform layer normalization (LN) on K and V to stabilize the training and optimization calculations. Step S222: Calculate the attention weights using the Galerkin-type attention integral operator. There is no need to construct a cost volume with a relatively small computational amount, so as to achieve efficient global feature aggregation. Specifically, perform matrix multiplication on K processed by Q and RoPE to generate an attention distribution and multiply it by the value V to obtain the attention output; Step S223: In the feed-forward network part, introduce a 3×3 depth convolution, which is embedded between the first fully connected layer and the GELU activation function, and can capture local absolute position information and enhance the model's local information perception ability; Step S224: In the scale feature mapping stage, a layer-by-layer calculation framework is formed by combining multi-layer fully connected transformation, non-linear activation (GELU), and hierarchical linear operators to improve the coherence of information processing and computational efficiency. The entire module combines the attention mechanism with local convolution to achieve the synergistic effect of global feature aggregation and local feature enhancement, thereby enhancing the model's expression ability overall.

6. The method according to claim 3, characterized in that, In step S23, the GRU (Gated Recurrent Unit) uses the context features obtained in step S22 as the initial hidden state and the input at each moment, and performs iterative optimization through four steps: update gate, reset gate, candidate hidden state calculation, and final hidden state update, so as to extract the flow field features from the context features overall.

7. The method according to claim 3, wherein In step S24, the continuous scale flow field velocity measurement module in PIVNO can achieve super-resolution reconstruction of the velocity field. This module combines methods such as random sampling, continuous interpolation, and graph convolution, and infers the high-resolution velocity field distribution based on the flow field features extracted in step S23. Through this process, PIVNO can generate velocity field estimation results with different resolutions in the specified domain, especially suitable for high-precision analysis of local flow fields.

8. The method according to claim 1, characterized in that, In step S3, the supervised pre-training aims to preliminarily train the PIVNO constructed in step 2 using the synthetic dataset obtained in step 1 to ensure that the model has a certain velocity field estimation ability. It specifically includes the following steps: Step S31: Use the synthetic dataset obtained in step S1 that covers a variety of flow conditions, which includes different velocity distributions, turbulence characteristics, and boundary conditions, so as to ensure that the model can learn diverse flow field features overall. Step S32: Use the least square error as the loss function to calculate the error between the velocity field estimated by the model and the true velocity field, and optimize the model parameters through gradient descent. During the training process, the model needs to maintain high accuracy under different flow scenarios and resolutions to ensure its generalization ability and improve its adaptability to real experimental data.

9. The method according to claim 1, characterized in that, In step S4, particle image data of the real flow field is obtained through the experimental device to provide experimental data for model optimization. It specifically includes the following steps: Step S41: Release tracer particles into the fluid to be measured so that they move with the fluid for subsequent imaging analysis. Step S42: Use a light source to irradiate the tracer particles and capture consecutive pairs of particle images through a high-speed camera.

10. The method according to claim 1, wherein In step S5, it specifically includes the following steps: Step S51: Construct a self-supervised loss function, which adopts a combination of a data term, a smoothing term, and a divergence regularization term. The data term is used to model the similarity between pairs of particle images, the smoothing term is used to ensure the spatial smoothness of the velocity field, and the divergence regularization term is used to maintain the incompressibility of the velocity field to ensure physical consistency; Step S52: Self-supervised optimize the model parameters. By minimizing the combined loss term, PIVNO gradually adjusts the parameters on the real experimental data, so as to match the statistical distribution of the real flow field overall and improve the model's adaptability; Step S53: Velocity field estimation. Using the optimized PIVNO model, perform velocity field estimation on real particle image pairs, and support the generation of velocity fields at any resolution. Combining with the continuous scale flow field velocity measurement module, through random sampling, continuous interpolation, and graph convolution methods, achieve super-resolution velocity field reconstruction, enabling PIVNO to provide high-precision local velocity field estimation within the finger image domain to meet the needs of fine flow analysis.

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