Flow field prediction method and system based on partial differential equation operator embedded convolutional network
By embedding partial differential equation operators and residual networks into flow field prediction, a multi-resolution prediction framework is constructed, which solves the problems of low computational efficiency and poor generalization ability of data-driven models in traditional methods, and achieves efficient and robust flow field prediction.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- HARBIN INST OF TECH
- Filing Date
- 2025-12-05
- Publication Date
- 2026-04-17
AI Technical Summary
Existing flow field prediction methods are computationally inefficient, traditional numerical solvers require large computational resources, data-driven models have poor generalization ability and are prone to divergence in long-term rolling predictions, and physical information neural networks have difficulty balancing data loss and physical loss.
We employ a method based on partial differential equation operator embedding convolutional networks. By constructing a residual network, we learn the residual characteristics between the physical prior solution and the true value of the flow field. We then combine this with discrete partial differential equations using fixed convolution kernels to construct a multi-resolution prediction framework. We use backpropagation of errors to update the network weights and maintain physical consistency.
It achieves efficient and robust flow field prediction, reduces computational load, improves the long-term stability and cross-condition generalization ability of the model, maintains high fidelity, and has a prediction error of less than 1.82% for two-dimensional pipe jet flow field. Its inference speed is faster than that of traditional CFD solvers.
Smart Images
Figure CN121880762A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of flow field prediction technology, specifically relating to a flow field prediction method based on partial differential equation operator embedded convolutional network. Background Technology
[0002] Computational modeling and simulation of physical processes are fundamental to understanding, predicting, and controlling various complex systems, such as turbulence, thermal-fluid coupling, and fluid-structure interaction. These physical phenomena are typically described by partial differential equations (PDEs), which can be solved using traditional numerical methods, such as the finite difference (FD), finite volume (FV), or finite element (FE) methods. These traditional numerical solvers are classic methods for flow field prediction. However, these methods rely on spatiotemporal discretization of the PDEs, a process that typically requires enormous computational resources and is time-consuming. Especially in applications requiring numerous repetitive model queries, such as optimization design, inverse problems, and uncertainty quantification, the computational cost of traditional methods makes them impractical. Furthermore, the stability and accuracy of traditional numerical methods are highly dependent on the quality of the computational mesh. In complex flow field problems, generating high-quality meshes is itself a challenge, while poor-quality or overly coarse meshes can easily lead to convergence difficulties in the numerical solution process. Therefore, how to significantly improve computational efficiency while ensuring prediction resolution and accuracy has become a key technical bottleneck in the current field of flow field prediction.
[0003] Defects and shortcomings of existing technology: To address the challenges of computational efficiency and application scenarios of traditional numerical solvers, the mainstream improvement approach is to build data-driven deep learning models. These methods aim to establish surrogate models capable of rapidly predicting the spatiotemporal dynamics of complex systems by learning from large amounts of simulation or experimental data. While these data-driven deep learning methods significantly improve the computational efficiency of flow field prediction, a series of problems still need to be addressed in practice. Specific challenges include: (1) The performance of a purely data-driven model is highly dependent on massive amounts of high-quality training data. The model has poor generalization ability, and its prediction accuracy will drop sharply for parameters and operating conditions outside the training data. Furthermore, when a purely data-driven model makes long-term time series predictions, the autoregressive pattern will produce a serious error accumulation effect, which will eventually lead to the failure of the prediction results.
[0004] (2) Physical information neural networks guide model training by calculating their residuals and adding them to the network's loss function. This can solve the problems of poor generalization ability and strong data dependence of pure data-driven models. However, the shortcoming of this method is that it is difficult to balance data loss and physical loss, which are loss terms that may have huge differences in magnitude and convergence speed. Furthermore, physical information neural networks only reflect physical information in the "target" of training (i.e., the loss function) and do not fundamentally change the "black box" architecture of the neural network itself. Summary of the Invention
[0005] This invention provides a flow field prediction method and system based on partial differential equation operator embedding convolutional networks. Its purpose is to solve the problems of huge numerical computation, lack of physical consistency of pure data-driven models and easy divergence in long-term flow field prediction.
[0006] Firstly, the purpose of this invention is to provide a flow field prediction method based on partial differential equation operator-embedded convolutional networks, comprising the following steps: S1, Data Acquisition and Preprocessing: Acquire fine-mesh flow field data and physical parameters, preprocess the fine-mesh flow field data to obtain the fine-mesh tensor, and record the physical parameter vector; S2, Physical Prior Calculation: The fine-grid flow field is downsampled to obtain the coarse-grid flow field. The partial differential equation operator with a fixed convolution kernel is discretized on the coarse-grid to advance one time step and obtain the coarse-grid physical solution. The coarse-grid physical solution is upsampled to align with the fine-grid resolution and obtain the physical prior solution. S3, Prediction framework construction: Construct a residual network, which uses a trainable convolutional neural network to learn the residual features between the physical prior solution and the true value of the flow field, and superimpose the residual features with the physical prior solution to form the predicted flow field at the next moment; S4, Model Training: Calculate the mean square error between the predicted residual and the true residual as the loss function, update the weights of the residual network through error backpropagation, wherein the parameters of the embedded partial differential equation operator convolution kernel are kept fixed, and the overall relative error is used as the evaluation index to verify the accuracy and physical consistency of the model in the long-term rolling prediction process, and obtain the trained residual network. S5, Result Inference: Based on the flow field at the current moment, the trained residual network is used to predict the flow field at the next moment.
[0007] Furthermore, a preferred solution is provided: In S1, the preprocessing of the fine-grid flow field data includes: performing dimensionless processing on the fine-grid flow field data, using an adaptive sliding window statistical strategy to detect and repair missing values, and rearranging the data into a standard tensor format of batch-time-channel-height-width.
[0008] Furthermore, a preferred solution is provided: in S2, downsampling is implemented using bilinear interpolation, and upsampling is implemented using bicubic interpolation.
[0009] Furthermore, a preferred embodiment is provided: in S2, the partial differential equation is the Navier-Stokes equation, and the fixed convolution kernel includes a convection operator, a diffusion operator, and a gradient operator.
[0010] Furthermore, a preferred option is provided: the partial differential equation operator in S2 is not trainable.
[0011] Furthermore, a preferred embodiment is provided: S3 includes: S31: Calculate the difference between the fine true value and the fine mesh object, and use it as a supervision label to generate residual labels; S32: Employs broadcast or learnable embedding to map physical parameter vectors to a high-dimensional space; S33: Combine the current flow field, coarse mesh material understanding, and physical parameters to obtain training and test data; S34: Construct multi-layer convolutional residual blocks and extract convolutional features; S35: Combine the convolutional features, physical prior solutions, and high-dimensional space, and input them into a residual CNN with jumpers to obtain the residual network.
[0012] Furthermore, a preferred solution is provided: in S5, a conservation superposition strategy is used to predict the process at the next moment.
[0013] Secondly, the purpose of this invention is to propose a flow field prediction system based on partial differential equation operator-embedded convolutional networks. This system is implemented based on a flow field prediction method based on partial differential equation operator-embedded convolutional networks as described in any one or more of the above-mentioned schemes. The system includes: Data acquisition and preprocessing module: used to acquire fine-mesh flow field data and physical parameters, preprocess the fine-mesh flow field data to obtain fine-mesh tensors, and record physical parameter vectors; Physics prior calculation module: used to downsample the fine-grid flow field to obtain the coarse-grid flow field, use a fixed convolution kernel to discretize the partial differential equation operator on the coarse grid, advance one time step to obtain the coarse-grid physics solution, and upsample the coarse-grid physics solution to align with the fine-grid resolution to obtain the physics prior solution; Prediction framework construction module: used to construct a residual network. The residual network uses a trainable convolutional neural network to learn the residual features between the physical prior solution and the true value of the flow field. The residual features are superimposed with the physical prior solution to form the predicted flow field at the next time step. Model training module: used to calculate the mean squared error between the predicted residual and the true residual as the loss function, update the weights of the residual network through error backpropagation, wherein the parameters of the embedded partial differential equation operator convolution kernel are kept fixed, and the overall relative error is used as the evaluation index to verify the accuracy and physical consistency of the model in the long-term rolling prediction process, and obtain the trained residual network. The result inference module is used to predict the flow field at the next time step based on the flow field at the current time step, using the trained residual network.
[0014] Thirdly, the present invention aims to provide a computer device, the computer device including a memory and a processor, the memory storing a computer program, and when the processor runs the computer program stored in the memory, the processor executes a flow field prediction method based on partial differential equation operator embedded convolutional network according to any one or more of the above-described schemes.
[0015] Fourthly, the present invention aims to provide a computer-readable storage medium for storing a computer program that executes a flow field prediction method based on partial differential equation operator embedded convolutional network as described in any one or more of the above-described schemes.
[0016] Compared with the prior art, the advantages of the present invention are: (1) This invention proposes an adaptive normalization repair data preprocessing module to address the scale differences and occasional missing data between CFD grid output and experimental measurements. The method first uses sliding window statistics to detect missing cells, then automatically adjusts the interpolation weights based on the local flow gradient to complete multi-directional linear reconstruction, and implements strict dimensionless transformation with reference velocity and reference length. This step effectively suppresses the interference of dimensional imbalance and random gaps in the source data on subsequent model training, providing numerically stable and scale-uniform input for physical operator convolution.
[0017] (2) This invention innovatively designs a physical constraint-trainable residual dual-branch convolutional framework. The left branch discretizes the Navier-Stokes equations with a fixed convolutional kernel and explicitly advances the time step by one time step in a coarse grid to ensure the conservation of mass and momentum; the right branch consists of multiple layers of convolutional-residual modules, which only fit the high-frequency residuals between the coarse solution and the true value. This dual-branch architecture maintains the physical consistency of hard constraints while improving the data-driven network's ability to represent complex nonlinear details, thus enhancing the convergence speed and long-term rolling stability.
[0018] (3) This invention innovatively proposes a fusion of explicit parameter embedding and multi-resolution evolution mechanism, significantly improving the generalization ability across working conditions. By using a downsampling-physical convolution-upsampling strategy to transfer information between coarse and fine grids, the computational cost of explicit PDE solving is reduced. Furthermore, by leveraging learnable embeddings, physical parameters such as Reynolds number and Mach number are extended into channel-level features and jointly modeled with the backbone feature field. Experiments show that this design enables the model to maintain accuracy and stability even without grid resolution, unknown parameter ranges, or even changing boundary conditions, reducing the dependence of traditional black-box networks on the training distribution.
[0019] Ultimately, through innovative designs involving physical operator embedding, residual compensation mechanisms, and multi-resolution co-evolution, this invention achieves fast, robust, and generalizable flow field prediction while maintaining high fidelity. It realizes high-precision prediction (Reynolds number) of two-dimensional pipe jet flow fields. The 500-step rolling average full-field relative error FRE is ≤1.82%, and the inference speed is 3.31 seconds (the explicit CFD solver takes about 370 seconds under the same grid conditions).
[0020] This invention is applicable to flow field prediction scenarios. Attached Figure Description
[0021] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.
[0022] Figure 1 The flowchart illustrates a flow field prediction method based on partial differential equation operator-embedded convolutional networks, as described in a specific embodiment of the present invention. Figure 2 This is a comparison chart of the flow field prediction results described in the specific embodiments of the present invention and the overall relative error of the pure data-driven black box network prediction. Figure 3 This is a comparison diagram of the method described in a specific embodiment of the present invention and the flow field prediction using a pure data-driven black-box network. Detailed Implementation
[0023] In the following description, specific details such as particular system architectures and techniques are set forth for illustrative purposes and not for limitation, in order to provide a thorough understanding of the embodiments of this application. However, those skilled in the art will understand that this application can also be implemented in other embodiments without these specific details. In other instances, detailed descriptions of well-known systems, apparatuses, circuits, and methods are omitted so as not to obscure the description of this application with unnecessary detail.
[0024] The technical solutions in the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.
[0025] Many specific details are set forth in the following description in order to provide a full understanding of this application. However, this application may also be implemented in other ways different from those described herein. Those skilled in the art can make similar extensions without departing from the spirit of this application. Therefore, this application is not limited to the specific embodiments disclosed below.
[0026] Implementation Method 1 This embodiment provides a flow field prediction method based on partial differential equation operator embedding convolutional networks, the process of which is as follows: Figure 1 As shown.
[0027] First, in the data preprocessing stage, this implementation addresses the inconsistency between the CFD computational grid and experimental sampling by designing a three-step process: normalization, recombination, and anomaly repair. An adaptive sliding window statistical strategy is used to perform multi-directional linear interpolation on occasional missing points, while simultaneously rearranging the multivariable tensors according to a fixed channel format to achieve dimensionless processing. Next, the multi-resolution physical prior construction stage begins: the fine-grid flow field is bilinearly downsampled, and the coarse-grid physical solution is derived using the Navier-Stokes equations (first-order convection and second-order diffusion templates) discretized with a fixed convolution kernel. Bicubic upsampling ensures resolution alignment. Based on this, this implementation constructs a "physics + trainable residual" dual-branch prediction framework: the left branch is an untrainable PDE physical operator, ensuring mass and momentum conservation; the right branch is a high-resolution convolutional feature branch, utilizing a multi-layer convolutional-residual network to learn the residual features between the coarse solution and the true value. The outputs of the two branches are superimposed using residuals to form the predicted flow field for the next time step. During the training phase, the difference between the predicted and actual residuals is measured by the overall relative error. Error backpropagation is used to update only the weights of the convolutional network, achieving fast and stable convergence. During the inference phase, each step first provides a coarse solution from the physics branch, and then the residual branch compensates for the loss, achieving high-fidelity and long-term stable rapid flow field prediction.
[0028] The method described in this embodiment specifically includes the following steps: (1) Obtain the flow field and parameters. Use the open-source computational fluid dynamics solver OpenFOAM to obtain the fine-grid instantaneous flow field and record the physical parameter vector at the same time. The instantaneous flow field of the fine mesh is expressed as: , in, Represents the Cartesian velocity x-direction component. Represents the y-axis component of the Cartesian velocity. Represents the z-direction component of the Cartesian velocity. Indicates pressure, Data dimension representation: number of channels × grid width × grid height; The physical parameter vector is represented as follows: , in, Represents the Reynolds number. Represents the Mach number. Indicates the angle of incidence.
[0029] (2) All variables are dimensionless, with reference speed as the reference value. With reference length The scaling formula for all field variables is shown in (1-1), which yields dimensionless variables.
[0030] , (1-1) in, The density is that of an ideal gas.
[0031] (3) In fine mesh tensors Adaptive sliding window is used in China The method detects occasional missing values or NaN values, and performs missing value repair using weighted linear interpolation as shown in formula (1-2), and replaces them with weighted linear interpolation to ensure data integrity.
[0032] (1-2) in, Represents the corresponding neighborhood point interpolation weights, Indicates an adaptive sliding window Dimensionless flow field values at effective neighboring points within the range.
[0033] (4) Rearrange the tensor dimensions as ,in For sample batches, For time step, The number of channels for flow field variables. For grid height, This is the grid width.
[0034] (5) For fine mesh tensors Proportional bilinear interpolation downsampling is performed as shown in Equation (1-3) to obtain the coarse-grid field. .
[0035] (1-3) (6) Based on this, perform discrete Navier-Stokes prediction, using a fixed convolution kernel on a coarse grid. , , The approximate Navier–Stokes equation is shown in formula (1-4), where This represents the first-order partial derivative convolution kernel in the x-direction, used to approximate the rate of change of flow field variables in the horizontal direction, corresponding to the x-components of the convection term and pressure gradient term in the equation; This represents the first-order partial derivative convolution kernel in the y-direction, used to approximate the rate of change of the flow field variables in the vertical direction, corresponding to the y-components of the convection term and pressure gradient term in the corresponding equation; This represents the Laplace operator (second-order differential) convolution kernel, used to approximate the second-order spatial derivative, corresponding to the diffusion term (or viscosity term) in the equation. Explicitly advancing one time step. To obtain coarse mesh material understanding .
[0036] (1-4) (7) Use bicubic interpolation for grid upsampling as shown in formula (1-5), and... By reverting to a finer mesh size, a physical prior solution can be obtained.
[0037] (1-5) (8) Calculate the difference between the fine ground truth and the physical prior solution as a supervision label to generate residual labels. The calculation formula is shown in (1-6). Here, "ground truth" is a standard term in the fields of machine learning and deep learning, referring to the target data used for supervised learning. "Fine / fine-grid" is a standard term in the field of computational fluid dynamics (CFD), referring to a high-resolution grid.
[0038] (1-6) (9) Scalar Through broadcast or learnable embedding maps This enables the network to explicitly perceive different operating conditions.
[0039] (10) Combine the current flow field, coarse mesh material analysis and physical parameters, and train the input structure as shown in formula (1-7) and combine them according to the same rules to obtain the result. However, it does not contain a label.
[0040] (1-7) (11) Construct multi-layer convolutional residual blocks in high-resolution branches, extract local high-frequency features to extract convolutional features, as shown in formula (1-8).
[0041] (1-8) in, This represents the trainable weight parameters of this part of the network. A defined high-resolution convolutional feature extraction network.
[0042] (12) will and , The input is merged into a residual CNN with jumps, forming the backbone network of NSPNN. The output prediction residual is shown in formula (1-9).
[0043] (1-9) in, This represents the trainable weight parameters of this part of the network. A defined residual prediction backbone network.
[0044] (13) The flow field prediction at the next moment is obtained by adopting the conservation superposition strategy, and the residual superposition is generated as shown in formula (1-10).
[0045] (1-10) (14) The loss function measures the difference between the prediction residual and the label using the mean squared error.
[0046] (1-11) in, For trainable weights of the network, This represents the total number of all grid points and channels in the current training batch. This represents the i-th data point involved in the calculation. It represents the difference between the fine true value calculated in step (8) and the physical prior solution corresponding to the true residual label of the i-th sample point.
[0047] (15) Based on this, the backpropagation of error is calculated, and the error is... Calculate gradient The Adam optimization algorithm is used to update the PDE convolution kernel gradient as shown in equations (1-12) to (1-16), ensuring that the gradient of the embedded PDE convolution kernel is always zero, thus ensuring hard physical constraints.
[0048] (1-12) (1-13) (1-14) (1-15) (1-16) in For the current batch gradient, It is the first-order momentum decay coefficient. It is the second-order momentum decay coefficient. For learning rate, To ensure numerical stability and prevent division by zero errors, For the current batch, The first-order momentum represents the average direction of the gradient over a past period.
[0049] (17) During the inference phase, repeat steps (5) to (15) to perform T-step time rolling to verify the long-term stability and physical consistency of the model as shown in formula (1-17).
[0050] (1-17) in For discrete residuals, The average relative error is taken as shown in equation (1-18).
[0051] (1-18) like If the value is much smaller than that of a pure black-box model and does not increase dramatically over time, it indicates that the model maintains physical conservation during the rolling process.
[0052] (18) Finally, the accuracy of the NSPNN flow field reconstruction model is evaluated using the full-field relative error (FRE), as shown in formula (1-19).
[0053] (1-19) in The total number of test parameters, In the first physical parameters The true flow field obtained from high-resolution numerical simulation (time) ), For the next step of predicting the flow field on the network, For the trainable part of NSPNN, given parameters and network weight The residual increment of the output at that time. It serves as a normalization mechanism, making the errors comparable across different samples and different physical quantities.
[0054] Implementation Method 2 This embodiment is a further illustrative example of the flow field prediction method based on partial differential equation operator embedding convolutional networks described in Embodiment 1.
[0055] The method steps described in this embodiment are as follows: (1) This embodiment takes the two-dimensional pipe jet flow field data generated by the open-source computational fluid dynamics software OpenFOAM as an example, and analyzes the flow field at different Reynolds numbers. The spatiotemporal evolution of the flow field under different conditions is predicted to verify the effectiveness of the proposed method in flow field prediction. Specifically, multiple operating conditions within a certain Reynolds number range are generated through numerical simulation, with each condition containing instantaneous flow field data at multiple time points. Simultaneously recorded physical parameter vector For a two-dimensional vector, the Reynolds number is used. and inlet jet position Composition, that is The training dataset covers a range of Reynolds numbers. Jet range .
[0056] (2) Select the characteristic velocity of the flow field problem according to formula (1-1). (e.g., inlet jet velocity) and characteristic length (e.g., pipe width) is used as a reference value for all field variables (Cartesian velocity). , ,pressure Dimensionless processing is performed to obtain input data with uniform scale. The computational domain of the two-dimensional pipe jet is... Therefore, the feature length The velocity at the inlet boundary is defined as a dynamically changing profile function. The maximum value of this function in the x-direction is 1.0. , The density is that of an ideal gas.
[0057] , (1-20) (3) In the acquired CFD fine mesh data In this process, the adaptive sliding window and weighted linear interpolation method described in formula (1-2) are used to detect and repair sporadic computational bad points (NaN) that may occur during the simulation, so as to ensure the integrity of the data tensor.
[0058] (4) Rearrange the dimensions of the preprocessed data tensors as follows: The standard format, in which In this embodiment, the batch size is... , In this embodiment, the time step is... , For the Navier-Stokes equations, a channel is defined as the velocity components u and v, and the pressure component p. , In this embodiment, the height and width of the grid are... , The preprocessed data tensor has a dimension of . .
[0059] (5) Based on formula (1-3), for the fine mesh tensor The proportion is Bilinear interpolation downsampling is used to construct a coarse-grid flow field using multi-resolution priors. From the downsampling ratio, it can be seen that Grid dimensions are That is, the dimension of the coarse grid data tensor after downsampling is .
[0060] (6) On the coarse grid, apply formula (1-4) through a set of non-trainable, fixed convolutional kernels: , , , To approximate the differential operators such as convection, diffusion, gradient, and divergence in the Navier-Stokes equations, we explicitly advance the time step by one step. This yields a rough solution based on the physical equations (physical priors). .
[0061] (7) Based on formula (1-5), bicubic interpolation is used as shown in formula (1-21) to obtain the physical prior solution on the coarse grid. Upsampling restores the mesh size to the same finer size as the original input, resulting in... .
[0062] (1-21) (8) Using formula (1-6), calculate the fine truth value (from the high-fidelity solution of OpenFOAM) and the upsampled physical prior solution at the next time step. The difference between them is used to generate residual labels for supervising neural network learning. .
[0063] (9) Convert the two-dimensional physical parameter vector Through a learnable embedding layer, it is mapped to a high-dimensional feature vector. This enables the network to explicitly perceive changes in parameters under different operating conditions.
[0064] (10) Following the rules of formula (1-7), the flow field at the current moment is... Upsampled physical prior solutions With parameter embedding vector Concatenate along the channel dimension to construct the input tensor for training trainable convolutional networks. .
[0065] (11) Based on formula (1-8), construct a high-resolution convolutional network consisting of multiple convolutional layers and residual connections, and process the input tensor. Feature extraction is performed, focusing on learning local high-frequency and nonlinear features that cannot be captured by physical priors.
[0066] (12) Input the extracted features into the residual CNN backbone network with jumpers, and finally output the predicted residual. As shown in formula (1-9).
[0067] (13) Using the conservation superposition strategy of formula (1-10), the residuals of the network prediction are... With physical prior solutions The data are superimposed to generate the final high-fidelity flow field prediction result for the next time step. .
[0068] (14) The loss function of the design model is the mean squared error (MSE) shown in formula (1-11), which is used to measure the prediction residual. True residual label The differences between them.
[0069] (15) The error backpropagation algorithm shown in formulas (1-12) to (1-16) is used to update the trainable weights of the network using the Adam optimizer. During this process, the gradient of the fixed convolution kernel representing the Navier-Stokes equation remains zero and does not participate in the update, thus achieving a hard constraint on the physical laws.
[0070] (16) During the reasoning stage, according to formula (1-17), the model output is... As input to the next time step, steps (5) to (13) are repeated in an autoregressive manner for rolling predictions up to 3 times the training time (e.g., training time is T, prediction time is 3T) to verify the long-term stability and physical consistency of the model.
[0071] (17) Finally, the full-field relative error (FRE) defined by formula (1-19) is... The actual calculations are as shown in formula (1-22), which evaluate the accuracy of the NSPNN flow field prediction model proposed in this embodiment. Figure 2 , Figure 3As shown, the results indicate that, under five sets of unseen test parameters, the prediction results of this method are in high agreement with the true values, while the pure data-driven black-box model suffers from severe error accumulation and eventually diverges over long-term rolling.
[0072] (1-22) Implementation Method 3 This embodiment proposes a flow field prediction system based on partial differential equation operator-embedded convolutional networks. The system is implemented based on a flow field prediction method based on partial differential equation operator-embedded convolutional networks as described in Embodiment 1. The system includes: Data acquisition and preprocessing module: used to acquire fine-mesh flow field data and physical parameters, preprocess the fine-mesh flow field data to obtain fine-mesh tensors, and record physical parameter vectors; Physics prior calculation module: used to downsample the fine-grid flow field to obtain the coarse-grid flow field, use a fixed convolution kernel to discretize the partial differential equation operator on the coarse grid, advance one time step to obtain the coarse-grid physics solution, and upsample the coarse-grid physics solution to align with the fine-grid resolution to obtain the physics prior solution; Prediction framework construction module: used to construct a residual network. The residual network uses a trainable convolutional neural network to learn the residual features between the physical prior solution and the true value of the flow field. The residual features are superimposed with the physical prior solution to form the predicted flow field at the next time step. Model training module: used to train the model by updating the weights of the residual network through error backpropagation, using the overall relative error as the loss function; The result inference module is used to predict the flow field at the next time step based on the flow field at the current time step, using the trained residual network.
[0073] It is understood that the present invention has been described through some embodiments, and those skilled in the art will recognize that various changes or equivalent substitutions can be made to these features and embodiments without departing from the spirit and scope of the invention. Furthermore, under the teachings of the present invention, these features and embodiments can be modified to adapt to specific situations and materials without departing from the spirit and scope of the invention. Therefore, the present invention is not limited to the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the protection scope of the present invention.
Claims
1. A flow field prediction method based on partial differential equation operator embedded convolutional network, characterized in that, Includes the following steps: S1, Data Acquisition and Preprocessing: Acquire fine-mesh flow field data and physical parameters, preprocess the fine-mesh flow field data to obtain the fine-mesh tensor, and record the physical parameter vector; S2, Physical Prior Calculation: The fine-grid flow field is downsampled to obtain the coarse-grid flow field. The partial differential equation operator with a fixed convolution kernel is discretized on the coarse-grid to advance one time step and obtain the coarse-grid physical solution. The coarse-grid physical solution is upsampled to align with the fine-grid resolution and obtain the physical prior solution. S3, Prediction Framework Construction: Construct a residual network, which uses a trainable convolutional neural network to learn the residual features between the physical prior solution and the true value of the flow field, and superimpose the residual features with the physical prior solution to form the predicted flow field at the next moment; S4, Model Training: Calculate the mean squared error between the predicted residual and the true residual as the loss function, update the weights of the residual network through error backpropagation, wherein the parameters of the embedded partial differential equation operator convolution kernel are kept fixed, and the overall relative error is used as the evaluation index to verify the accuracy and physical consistency of the model in the long-term rolling prediction process, and obtain the trained residual network. S5, Result Inference: Based on the flow field at the current moment, the trained residual network is used to predict the flow field at the next moment.
2. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, characterized in that, In step S1, the preprocessing of the fine-grid flow field data includes: performing dimensionless processing on the fine-grid flow field data, using an adaptive sliding window statistical strategy to detect and repair missing values, and rearranging the data into a standard tensor format of batch-time-channel-height-width.
3. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, characterized in that, In S2, downsampling is implemented using bilinear interpolation, and upsampling is implemented using bicubic interpolation.
4. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, characterized in that, In S2, the partial differential equation is the Navier-Stokes equation, and the fixed convolution kernel includes a convection operator, a diffusion operator, and a gradient operator.
5. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, wherein, The partial differential equation operators in S2 are not trainable.
6. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, characterized in that, S3 includes: S31: Calculate the difference between the fine true value and the fine mesh object, and use it as a supervision label to generate residual labels; S32: Employs broadcast or learnable embedding to map physical parameter vectors to a high-dimensional space; S33: Combine the current flow field, coarse mesh material understanding, and physical parameters to obtain training and test data; S34: Construct multi-layer convolutional residual blocks and extract convolutional features; S35: Combine the convolutional features, physical prior solutions, and high-dimensional space, and input them into a residual CNN with jumpers to obtain the residual network.
7. The flow field prediction method based on partial differential equation operator embedded convolutional network according to claim 1, wherein, In S5, a conservation superposition strategy is used to predict the process at the next moment.
8. A flow field prediction system based on partial differential equation operator embedded convolutional network, characterized in that, The system is implemented based on a flow field prediction method based on partial differential equation operator embedding convolutional networks as described in any one of claims 1-7, and the system includes: Data acquisition and preprocessing module: used to acquire fine-mesh flow field data and physical parameters, preprocess the fine-mesh flow field data to obtain fine-mesh tensors, and record physical parameter vectors; Physics prior calculation module: used to downsample the fine-grid flow field to obtain the coarse-grid flow field, use a fixed convolution kernel to discretize the partial differential equation operator on the coarse grid, advance one time step to obtain the coarse-grid physics solution, and upsample the coarse-grid physics solution to align with the fine-grid resolution to obtain the physics prior solution; Prediction framework construction module: used to construct a residual network. The residual network uses a trainable convolutional neural network to learn the residual features between the physical prior solution and the true value of the flow field. The residual features are superimposed with the physical prior solution to form the predicted flow field at the next time step. Model training module: used to calculate the mean squared error between the predicted residual and the true residual as the loss function, update the weights of the residual network through error backpropagation, wherein the parameters of the embedded partial differential equation operator convolution kernel are kept fixed, and the overall relative error is used as the evaluation index to verify the accuracy and physical consistency of the model in the long-term rolling prediction process, and obtain the trained residual network. The result inference module is used to predict the flow field at the next moment based on the flow field at the current moment, using the trained residual network.
9. A computer device, comprising: The computer device includes a memory and a processor. The memory stores a computer program. When the processor runs the computer program stored in the memory, the processor executes a flow field prediction method based on partial differential equation operator embedded convolutional network according to any one of claims 1-7.
10. A computer-readable storage medium, characterized in that, The computer-readable storage medium is used to store a computer program that executes a flow field prediction method based on partial differential equation operator embedded convolutional networks according to any one of claims 1-7.