Fluid solving method and system based on multi-step physical coding recursive convolutional neural network

Abstract

The invention discloses a fluid solving method and system based on a multi-step physical coding recursive convolutional neural network, and the method comprises the steps: constructing a partial differential equation, and initializing the construction of the physical coding recursive convolutional neural network. Performing forward iterative calculation on the partial differential equation in a physical driving mode to obtain multiple groups of fluid flow velocity solving results, and training and optimizing network parameters of the physical coding recursive convolutional neural network; initial condition data of a partial differential equation are obtained for the to-be-solved fluid flow velocity, and a fluid flow velocity solving result is obtained through the trained physical coding recursive convolutional neural network. The invention aims to solve the problems that a physical coding recursive convolutional neural network adopted during hydrodynamic analysis cannot converge frequently when solving a PDE with a relatively large time step, and a loss function is observed to have an imbalance phenomenon at different time steps. And the efficiency, the accuracy and the time extrapolation capability of solving the PDE by the physical coding recursive convolutional neural network during fluid dynamic analysis are optimized.

Description

Technical Field

[0001] The present invention relates to the field of fluid dynamics, and in particular to a fluid solving method and system based on a multi-step physical coding recursive convolutional neural network. Background Art

[0002] In the field of fluid dynamics, partial differential equations (PDEs) are a very important tool and are widely used to model and describe the behavior of various complex systems in fluid dynamics. In addition, PDEs play a key role in both the large-scale structure of the universe in astrophysics and neural networks in biology. However, for many practical problems, PDEs are often difficult to solve analytically, so numerical methods are needed to approximate the solution. Traditional numerical methods often face huge computational overhead when dealing with high-dimensional problems, which limits their application in large-scale and real-time simulations. However, with the continuous development of deep learning and automatic differentiation techniques, Physics-Informed Neural Networks (PINNs) have emerged as a new method for numerically solving PDEs. PINNs embed PDEs into the loss function of the neural network, use automatic differentiation techniques to calculate the differential operator, and optimize the loss function through a gradient-based optimization algorithm to obtain a numerical solution. This method not only simplifies the process of numerical solution, but also has higher flexibility and generalization ability.

[0003] Although most PINN-based models tend to use Fully Connected Neural Networks (FCNNs) to represent and obtain solution functions, this continuous learning model has some fundamental limitations. First, FCNN requires a large number of anchor points to optimize the loss function, which leads to a significant increase in training cost. Second, continuous networks need to use automatic differentiation techniques to perform differential operations in PDEs and calculate PDE residual losses, which leads to reduced computational efficiency. Finally, due to the presence of soft constraints in the loss function, physical constraints cannot be accurately enforced. To overcome these problems, Physics-Informed Convolutional Neural Networks (PICNNs) have been recently proposed. Compared with FCNN, Convolutional Neural Networks (CNNs) have the advantages of parameter sharing, fewer parameters, and better scalability. PeRCNN is a discrete learning model based on CNN that hardcodes initial conditions (ICs) and boundary conditions (BCs) into the network architecture. PeRCNN has been successfully used to model complex spatiotemporal dynamic systems (such as reaction-diffusion processes), showing superior performance compared to other methods. However, although PeRCNN has shown excellent performance in many cases, it often fails to converge when solving PDEs with large time steps, and an imbalance in the loss function is observed at different time steps. This problem seriously affects the effectiveness of PeRCNN in solving some equations and has become a key technical issue in the field of fluid dynamics that needs to be solved urgently. Summary of the invention

[0004] Technical problem to be solved by the present invention: In view of the above-mentioned problems in the prior art, a fluid solving method and system based on a multi-step physically encoded recursive convolutional neural network are provided. The present invention aims to solve the problem that the physically encoded recursive convolutional neural network used in fluid dynamics analysis often fails to converge when solving PDE with a large time step, and an imbalance in the loss function is observed at different time steps, and optimizes the efficiency, accuracy and time extrapolation capability of the physically encoded recursive convolutional neural network in solving PDE during fluid dynamics analysis.

[0005] In order to solve the above technical problems, the technical solution adopted by the present invention is: A fluid solution method based on a multi-step physical coding recursive convolutional neural network comprises the following steps: S1, construct the partial differential equation for solving the vorticity of the NS equations: , , , In the above formula, is the vorticity field, is the Laplace operator, is the velocity field, is the nabla operator, is the viscosity coefficient, For time, For the cycle, is the mandatory function, is the spatial coordinate, and is the initial vorticity; the initial condition data for solving the PDE for the fluid velocity is obtained by solving the partial differential equation for the vorticity of the NS equation; S2, initialize and construct a physical coding recursive convolutional neural network. Based on the initial condition data of the partial differential equation, the physical coding recursive convolutional neural network forward iterates the partial differential equation in a physically driven manner to obtain multiple sets of fluid flow rate solution results, where the expression of the iterative calculation is: , In the above formula, is the predicted flow velocity value of the k+1th iteration, is the predicted flow velocity value of the kth iteration, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the time step length; S3, combining the multiple sets of fluid velocity solution results obtained by the solution, using a multi-time step weight update algorithm to train and optimize the network parameters of the physical encoding recursive convolutional neural network; S4, for the fluid flow rate to be solved, the initial condition data of the partial differential equation corresponding to it is obtained based on numerical methods or observation means, the initial condition data of the partial differential equation is input into the trained physical coding recursive convolutional neural network, the partial differential equation is solved based on the specified number of time steps through the trained physical coding recursive convolutional neural network, and the final solution of the partial differential equation is output as the fluid flow rate solution result.

[0006] Optionally, obtaining initial condition data for solving the partial differential equation for fluid flow velocity in step S1 includes generating image data with a resolution of H×W from a Gaussian random field with periodic boundary conditions as initial condition data for solving the partial differential equation for fluid flow velocity, wherein H and W are the height and width of the image, respectively.

[0007] Optionally, the physical coding recursive convolutional neural network constructed in step S2 includes a multiplication module, the multiplication module includes multiple parallel convolutional layers, element product layers and a 1×1 convolutional layer, the velocity prediction value of the kth iteration is passed through multiple parallel convolutional layers to generate multiple feature maps, and the generated multiple feature maps are combined through the element product layer and then through the convolution layer with a convolution kernel size of 1×1 to merge multiple channels to obtain an output approximate estimate value, and the function expression of the multiplication module is: , In the above formula, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the predicted flow rate of the kth cycle, is the network parameter of the multiplication module, and Respectively represent the number of channels and the number of parallel convolutional layers, is the weight parameter corresponding to the cth channel in the convolution layer with a convolution kernel size of 1×1, Indicates The convolution filter weight parameters of the layer and the c-th channel, Indicates The bias parameters of the layer, Represents the element-wise product operation of an element-wise product layer.

[0008] Optionally, in step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the partial differential equation of initial condition data by the physically encoded recursive convolutional neural network, the boundary conditions are input into the physically encoded recursive convolutional neural network in a periodic filling manner as features, and the boundary conditions are input into the physically encoded recursive convolutional neural network in a periodic filling manner as features, including mapping the boundary conditions of the left boundary of the grid as features to the right boundary for input into the physically encoded recursive convolutional neural network, or mapping the boundary conditions of the right boundary of the grid as features to the left boundary for input into the physically encoded recursive convolutional neural network.

[0009] Optionally, in step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the partial differential equation of initial condition data by the physically coded recursive convolutional neural network, it includes filling the prediction results of the physically coded recursive convolutional neural network according to physical rules at each time step, and filling the prediction results of the physically coded recursive convolutional neural network in a periodic filling manner according to the physical rules.

[0010] Optionally, when the prediction result of the physical coding recursive convolutional neural network is filled in a periodic filling manner according to the physical rules, if the boundary condition of the fluid is a Dirichlet boundary condition, the filling function expression is: , If the boundary condition of the fluid is the Neumann boundary condition, the filling function expression is: , If the boundary condition of the fluid is the Robin boundary condition, the filling function expression is: , If the boundary condition of the fluid is a periodic condition, the filling function expression is: , in, is the predicted value of the equation to be solved, is the solution function corresponding to the Dirichlet boundary condition, is the point set corresponding to the Dirichlet boundary condition, is the mandatory function, is the point set corresponding to the Neumann boundary condition, and are the parameters of the Robin boundary condition, is the solution function corresponding to the Robin boundary condition, is the point set corresponding to the Robin boundary condition, is the predicted value of the equation to be solved at the left boundary, is the predicted value of the equation to be solved on the right boundary, and They represent the point set corresponding to the left boundary and the point set corresponding to the right boundary respectively.

[0011] Optionally, in step S3, when the network parameters of the physically encoded recursive convolutional neural network are trained and optimized using a multi-time step weight update algorithm, the solution results of the multiple groups of fluid flow rates are obtained based on the set local time step. , each time you take out The mean square error (MSE) of the predicted value output by the multi-step physically coded recursive convolutional neural network and the equation residual is calculated as the loss function for each group of fluid flow rate solution results, and the specified optimizer is used to optimize the network parameters of the physically coded recursive convolutional neural network based on the loss function until all fluid flow rate solution results are trained.

[0012] In addition, the present invention also provides a fluid solving system based on a multi-step physical coding recursive convolutional neural network, comprising a microprocessor and a memory connected to each other, wherein the microprocessor is programmed or configured to execute the fluid solving method based on the multi-step physical coding recursive convolutional neural network.

[0013] In addition, the present invention also provides a computer-readable storage medium, which stores a computer program or instruction, and the computer program or instruction is programmed or configured to execute the fluid solution method based on the multi-step physical encoding recursive convolutional neural network through a processor.

[0014] In addition, the present invention also provides a computer program product, including a computer program or instructions, which are programmed or configured to execute the fluid solution method based on the multi-step physical encoding recursive convolutional neural network through a processor.

[0015] Compared with the prior art, the present invention mainly has the following advantages: the present invention includes constructing a partial differential equation for solving the vorticity of the NS equation, initializing and constructing a physically encoded recursive convolutional neural network, forward iterating and calculating the partial differential equation in a physically driven manner based on the initial condition data of the partial differential equation, and obtaining multiple sets of fluid flow rate solution results, combining the multiple sets of fluid flow rate solution results obtained by the solution, and using a multi-time step weight update algorithm to train and optimize the network parameters of the physically encoded recursive convolutional neural network; obtaining the initial condition data of the partial differential equation corresponding to the fluid flow rate to be solved based on a numerical method or an observation method, and inputting the initial condition data of the partial differential equation into the training set; A good physical coding recursive convolutional neural network, through the trained physical coding recursive convolutional neural network, completes the solution of the partial differential equation based on the specified number of time steps, and outputs the final solution of the partial differential equation as the fluid flow rate solution result. The present invention uses a multi-time step weight update algorithm in the above manner to achieve very high accuracy when solving PDE, and its performance is better than that of a conventional physical coding recursive convolutional neural network, and the accuracy of equation solving is improved by two orders of magnitude; in addition, the multi-step physical coding recursive convolutional neural network can obtain better results in time extrapolation; in addition, the multi-step physical coding recursive convolutional neural network overcomes the problem that the model often fails to converge when solving PDE with a large time step. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 Schematic diagram of the basic flow of the method of the embodiment of the present invention.

[0017] Figure 2 Schematic diagram of the structure of a physical coding recursive convolutional neural network (MtS-PeRCNN) in an embodiment of the present invention.

[0018] Figure 3Schematic diagram of the application of the physical coding recursive convolutional neural network (MtS-PeRCNN) in an embodiment of the present invention. DETAILED DESCRIPTION

[0019] like Figure 1 As shown, the fluid solution method based on the multi-step physical coding recursive convolutional neural network in this embodiment includes the following steps: S1, construct the partial differential equation (PDE) for solving the vorticity of the NS equation: , , , In the above formula, is the vorticity field, is the Laplace operator, is the velocity field, is the nabla operator, is the viscosity coefficient, For time, For the cycle, is the mandatory function, is the spatial coordinate, and is the initial vorticity; the initial condition data for solving the PDE for the fluid velocity is obtained by solving the partial differential equation for the vorticity of the NS equation; S2, initialize and construct a physical coding recursive convolutional neural network. Based on the initial condition data of the partial differential equation, the physical coding recursive convolutional neural network forward iterates the partial differential equation in a physically driven manner to obtain multiple sets of fluid flow rate solution results, such as Figure 2 As shown, the expression for iterative calculation is: , In the above formula, is the predicted flow velocity value of the k+1th iteration, is the predicted flow velocity value of the kth iteration, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the time step length; in addition, other iterative calculations can be used as needed. For example, solving the two-dimensional Burgers equation is an example. The Burgers equation is a classic fluid dynamics problem, which is expressed as follows: ,in is the flow rate, is the viscosity coefficient, set to 0.005, is the Laplace operator, is the nabla operator; S3, combining the multiple sets of fluid velocity solution results obtained by the solution, using a multi-time step weight update algorithm to train and optimize the network parameters of the physical encoding recursive convolutional neural network; S4, for the fluid flow rate to be solved, the initial condition data of the partial differential equation corresponding to it is obtained based on numerical methods or observation means, the initial condition data of the partial differential equation is input into the trained physical coding recursive convolutional neural network, the partial differential equation is solved based on the specified number of time steps through the trained physical coding recursive convolutional neural network, and the final solution of the partial differential equation is output as the fluid flow rate solution result.

[0020] In this embodiment, obtaining the initial condition data for solving the partial differential equation for fluid flow velocity in step S1 includes generating image data with a resolution of H×W from a Gaussian random field with periodic boundary conditions as the initial condition data for solving the partial differential equation for fluid flow velocity, where H and W are the height and width of the image, respectively.

[0021] Physically encoded recursive convolutional neural networks are built to forward solve partial differential equations in a physically driven manner. They do not require labeled data for data-driven training. They only need to provide initial conditions, and boundary conditions are hard-coded into the model in the form of padding. In the PINN model, ICs and BCs are used as soft constraints to optimize the loss function. In contrast, the PeRCNN model improves the accuracy and efficiency of solving partial differential equations by hard-coding ICs and BCs into the model. At each time step, the model fills in the predicted results according to physical rules. For Dirichlet BCs, the predicted values ​​are filled with specified values, while for Neumann or Robin BCs, the filling values ​​are calculated based on boundary values ​​and gradient information. For example Figure 2 As shown, the multi-time stepping physics-encoded recurrent convolutional neural network (MtS-PeRCNN) constructed in step S2 of this embodiment includes a multiplication module (Π-block), which includes multiple parallel convolutional layers (Conv), element product layers and a 1×1 convolution layer. The velocity prediction value of the kth iteration generates multiple feature maps after passing through multiple parallel convolution layers, and the generated multiple feature maps are combined through the element product layer and then through the convolution layer (Conv1×1) with a convolution kernel size of 1×1 to merge multiple channels to obtain an output approximate estimate value, and the function expression of the multiplication module is: , In the above formula, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the predicted flow rate of the kth cycle, is the network parameter of the multiplication module, and Respectively represent the number of channels and the number of parallel convolutional layers, is the weight parameter corresponding to the cth channel in the convolution layer with a convolution kernel size of 1×1, Indicates The convolution filter weight parameters of the layer and the c-th channel, Indicates The bias parameters of the layer, Represents the element-wise product operation of an element-wise product layer. Figure 3 middle, ~ is the input for iterations 0 to 5, They represent the loss functions of the 0th to 5th iterations respectively, It represents the gradient of the loss function, which is used to optimize the network parameters of the physical coding recursive convolutional neural network after each iteration. Finally, MtS-PeRCNN is used to solve the partial differential equation and the total time can be output. (n time steps) corresponding to the solution set of the partial differential equation .

[0022] In this embodiment, in step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the initial condition data partial differential equation by the physically encoded recursive convolutional neural network, the boundary conditions are input as features into the physically encoded recursive convolutional neural network in a periodic padding manner, and the boundary conditions are input as features into the physically encoded recursive convolutional neural network in a periodic padding manner, including mapping the boundary conditions of the left boundary of the grid as features to the right boundary for input into the physically encoded recursive convolutional neural network, or mapping the boundary conditions of the right boundary of the grid as features to the left boundary for input into the physically encoded recursive convolutional neural network.

[0023] In this embodiment, in step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the partial differential equation of initial condition data by the physically coded recursive convolutional neural network, it includes filling the prediction results of the physically coded recursive convolutional neural network according to physical rules at each time step, and filling the prediction results of the physically coded recursive convolutional neural network in a periodic filling manner according to the physical rules.

[0024] In this embodiment, when the prediction result of the physical coding recursive convolutional neural network is filled in a periodic filling manner according to the physical rules, if the boundary condition of the fluid is a Dirichlet boundary condition, the filling function expression is: , If the boundary condition of the fluid is the Neumann boundary condition, the filling function expression is: , If the boundary condition of the fluid is the Robin boundary condition, the filling function expression is: , If the boundary condition of the fluid is a periodic condition, the filling function expression is: , in, is the predicted value of the equation to be solved, is the solution function corresponding to the Dirichlet boundary condition, is the point set corresponding to the Dirichlet boundary condition, is the mandatory function, is the point set corresponding to the Neumann boundary condition, and are the parameters of the Robin boundary condition, is the solution function corresponding to the Robin boundary condition, is the point set corresponding to the Robin boundary condition, is the predicted value of the equation to be solved at the left boundary, is the predicted value of the equation to be solved on the right boundary, and They represent the point set corresponding to the left boundary and the point set corresponding to the right boundary respectively.

[0025] The multi-time step weight update algorithm in this embodiment is described as follows: set a local time step number p, after iterating this local time step number in the network, perform back propagation to update the parameters, and then continue the iterative process until all n time steps are iterated. The multi-step physical coding recursive convolutional neural network uses previous predictions as input to predict the dynamics of the system through sequential forward propagation. This structure enables the multi-step physical coding recursive convolutional neural network to handle longer time step sequences while avoiding performance degradation caused by the gradient vanishing problem. By iteratively updating the weights every p steps, the model maintains stability and accuracy over a long time range, making it powerful in solving complex, time-dependent partial differential equation problems. In addition, the cyclic characteristics of the multi-step physical coding recursive convolutional neural network ensure the consistency of predictions over time, making full use of the sequential nature of the input data to improve the accuracy of model predictions. In each iteration of training the physical coding recursive convolutional neural network in step S2 of this embodiment, the loss function uses the mean square error (MSE) to calculate the mean square error between the predicted value output by the multi-step physical coding recursive convolutional neural network and the equation residual, and uses the Adam optimizer to optimize the network parameters.

[0026] In this embodiment, when the multi-time step weight update algorithm is used in step S3 to train and optimize the network parameters of the physical coding recursive convolutional neural network, the solution results of the multiple groups of fluid flow rates are obtained based on the set local time step. , each time you take out The mean square error (MSE) of the predicted value output by the multi-step physically coded recursive convolutional neural network and the equation residual is calculated as the loss function for each group of fluid flow rate solution results, and the specified optimizer is used to optimize the network parameters of the physically coded recursive convolutional neural network based on the loss function until all fluid flow rate solution results are trained.

[0027] Specifically, in this embodiment, when the model is initialized in step S1, it includes initializing MtS-PeRCNN, and the parameter set is ; Initial conditions Generated from a Gaussian random field with periodic boundary conditions, with a resolution of , as the input of MtS-PeRCNN. For this problem, the time step , a total of 400 time steps (total time 0.4 seconds) are used for training. Therefore, in each training cycle, MtS-PeRCNN will perform a total of 400 iterations based on the update method of the above network expression, starting from the input , calculated Next, step S2 of this embodiment is performed: model training: using a multi-time step weight update algorithm to update the parameters of the physical coding recursive convolutional neural network For this problem, the local time step is Therefore, MtS-PeRCNN consists of local initial conditions Iterate to Then, the four time snapshots calculated are used , calculate the physical information loss function described above, and use the Adam optimizer to update the network parameters . Then, update the local initial conditions for ,from Continue the subsequent iterative calculation process, and so on, until 400 iterative update processes are calculated and one cycle of training is completed. After completing the set total training cycle, the trained model parameters are obtained. Finally, step S3 of this embodiment is performed, model reasoning: for this problem, the total evolution time of the Burgers equation to be solved is 1 second, and the time step is , 1000 time steps (total time 1.0 second) are needed for inference. Set the number of iterations of MtS-PeRCNN to 1000 and enter the initial conditions , complete 1000 iterations, and output the solution set of the PDE corresponding to the total time of 1.0 second , each solution is a snapshot in time with the same resolution as the PDE being solved. , so far the solution to the Burgers equation is completed. For other PDEs, the three steps of this embodiment are also followed to solve them. Enter the initial conditions of the PDE to be solved After training the MtS-PeRCNN model, perform inference and the total model output time (n time steps) corresponding to the solution set of the PDE , each solution are all discrete-time snapshots of a PDE solution function.

[0028] In summary, the method of this embodiment discloses a multi-time stepping physics-encoded recurrent convolutional neural network (MtS-PeRCNN) for solving partial differential equations (PDEs). The method of this embodiment is based on a physics-encoded recurrent convolutional neural network, including obtaining initial condition data of the PDE to be solved based on a numerical method or an observation method; initializing the network parameters of the MtS-PeRCNN, inputting the initial conditions for forward solving the PDE; updating the parameters of the MtS-PeRCNN using a multi-time step weight update algorithm; completing model training after multiple iterations to obtain network parameters that can solve the PDE to be solved; based on the number of time steps required by a specific problem (the total evolution time of the equation), using the MtS-PeRCNN to solve the PDE, and outputting the solution of the PDE corresponding to the total time. The method of this embodiment can optimize the efficiency, accuracy and time extrapolation capability of the PeRCNN method for solving PDEs by using a multi-time step weight update algorithm.

[0029] In addition, this embodiment also provides a fluid solving system based on a multi-step physical coding recursive convolutional neural network, comprising a microprocessor and a memory connected to each other, wherein the microprocessor is programmed or configured to execute the fluid solving method based on the multi-step physical coding recursive convolutional neural network.

[0030] In addition, this embodiment also provides a computer-readable storage medium, which stores a computer program or instruction, and the computer program or instruction is programmed or configured to execute the fluid solution method based on the multi-step physical encoding recursive convolutional neural network through a processor.

[0031] In addition, this embodiment also provides a computer program product, including a computer program or instructions, which are programmed or configured to execute the fluid solution method based on the multi-step physical encoding recursive convolutional neural network through a processor.

[0032] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present application can take the form of a computer program product implemented on one or more computer-readable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The present application is described with reference to the flowcharts and / or block diagrams of the methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each process and / or box in the flowchart and / or block diagram, as well as the combination of the processes and / or boxes in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the process Figure 1 A process or multiple processes and / or boxes Figure 1 These computer program instructions can also be stored in a computer-readable memory that can guide a computer or other programmable data processing device to work in a specific way, so that the instructions stored in the computer-readable memory produce a product including an instruction device, which implements the functions specified in the process. Figure 1 A process or multiple processes and / or boxes Figure 1 These computer program instructions can also be loaded onto a computer or other programmable data processing device, so that a series of operation steps are executed on the computer or other programmable device to produce a computer-implemented process, so that the instructions executed on the computer or other programmable device provide for implementing the process in the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0033] The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above embodiments. All technical solutions under the concept of the present invention belong to the protection scope of the present invention. It should be pointed out that for ordinary technicians in this technical field, some improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

Claims

1. A fluid solution method based on a multi-step physical coding recursive convolutional neural network, characterized in that The steps include: S1, construct the partial differential equation for solving the vorticity of the NS equations: ， ， ， In the above formula, is the vorticity field, is the Laplace operator, is the velocity field, is the nabla operator, is the viscosity coefficient, For time, For the cycle, is the mandatory function, is the spatial coordinate, and is the initial vorticity; the initial condition data for solving the PDE for the fluid velocity is obtained by solving the partial differential equation for the vorticity of the NS equation; S2, initialize and construct a physical coding recursive convolutional neural network. Based on the initial condition data of the partial differential equation, the physical coding recursive convolutional neural network forward iterates the partial differential equation in a physically driven manner to obtain multiple sets of fluid flow rate solution results, where the expression of the iterative calculation is: ， In the above formula, is the predicted flow velocity value of the k+1th iteration, is the predicted flow velocity value of the kth iteration, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the time step length; S3, combining the multiple sets of fluid velocity solution results obtained by the solution, using a multi-time step weight update algorithm to train and optimize the network parameters of the physical encoding recursive convolutional neural network; S4, for the fluid flow rate to be solved, the initial condition data of the partial differential equation corresponding to it is obtained based on numerical methods or observation means, the initial condition data of the partial differential equation is input into the trained physical coding recursive convolutional neural network, the partial differential equation is solved based on the specified number of time steps through the trained physical coding recursive convolutional neural network, and the final solution of the partial differential equation is output as the fluid flow rate solution result.

2. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 1 is characterized in that: Acquiring the initial condition data for solving the partial differential equation for fluid flow velocity in step S1 includes generating image data with a resolution of H×W from a Gaussian random field with periodic boundary conditions as the initial condition data for solving the partial differential equation for fluid flow velocity, wherein H and W are the height and width of the image, respectively.

3. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 1 is characterized in that: The physical coding recursive convolutional neural network constructed in step S2 includes a multiplication module, which includes multiple parallel convolutional layers, element product layers and a 1×1 convolutional layer. The velocity prediction value of the kth iteration generates multiple feature maps after passing through multiple parallel convolutional layers, and the generated multiple feature maps are combined through the element product layer and then through the convolution layer with a convolution kernel size of 1×1 to merge multiple channels to obtain an approximate estimated value of the output, and the function expression of the multiplication module is: ， In the above formula, is the velocity prediction value and network parameters based on the kth iteration The approximate output value of is the predicted flow rate of the kth cycle, is the network parameter of the multiplication module, and Respectively represent the number of channels and the number of parallel convolutional layers, is the weight parameter corresponding to the cth channel in the convolution layer with a convolution kernel size of 1×1, Indicates The convolution filter weight parameters of the layer and the c-th channel, Indicates The bias parameters of the layer, Represents the element-wise product operation of an element-wise product layer.

4. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 3 is characterized in that: In step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the initial condition data partial differential equation by the physically encoded recursive convolutional neural network, the boundary conditions are input into the physically encoded recursive convolutional neural network in a periodic filling manner as features, and the boundary conditions are input into the physically encoded recursive convolutional neural network in a periodic filling manner as features, including mapping the boundary conditions of the left boundary of the grid as features to the right boundary for input into the physically encoded recursive convolutional neural network, or mapping the boundary conditions of the right boundary of the grid as features to the left boundary for input into the physically encoded recursive convolutional neural network.

5. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 4 is characterized in that: In step S2, when multiple sets of fluid flow rate solution results are obtained by forward iterative calculation in a physically driven manner based on the initial condition data partial differential equation by the physically coded recursive convolutional neural network, the prediction results of the physically coded recursive convolutional neural network are filled according to physical rules at each time step, and the prediction results of the physically coded recursive convolutional neural network are filled in a periodic filling manner according to the physical rules.

6. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 5 is characterized in that: When the prediction results of the physical coding recursive convolutional neural network are filled in a periodic filling manner according to the physical rules, if the boundary condition of the fluid is a Dirichlet boundary condition, the filling function expression is: ， If the boundary condition of the fluid is the Neumann boundary condition, the filling function expression is: ， If the boundary condition of the fluid is the Robin boundary condition, the filling function expression is: ， If the boundary condition of the fluid is a periodic condition, the filling function expression is: ， in, is the predicted value of the equation to be solved, is the solution function corresponding to the Dirichlet boundary condition, is the point set corresponding to the Dirichlet boundary condition, is the mandatory function, is the point set corresponding to the Neumann boundary condition, and are the parameters of the Robin boundary condition, is the solution function corresponding to the Robin boundary condition, is the point set corresponding to the Robin boundary condition, is the predicted value of the equation to be solved at the left boundary, is the predicted value of the equation to be solved on the right boundary, and They represent the point set corresponding to the left boundary and the point set corresponding to the right boundary respectively.

7. The fluid solving method based on multi-step physical coding recursive convolutional neural network according to claim 1 is characterized in that: When the multi-time step weight update algorithm is used in step S3 to train and optimize the network parameters of the physical encoding recursive convolutional neural network, including solving the multiple sets of fluid flow rate solutions, based on the set local time step , each time you take out The mean square error (MSE) of the predicted value output by the multi-step physically coded recursive convolutional neural network and the equation residual is calculated as the loss function for each group of fluid flow rate solution results, and the specified optimizer is used to optimize the network parameters of the physically coded recursive convolutional neural network based on the loss function until all fluid flow rate solution results are trained.

8. A fluid solving system based on a multi-step physical coding recursive convolutional neural network, comprising a microprocessor and a memory connected to each other, characterized in that: The microprocessor is programmed or configured to execute the fluid solution method based on a multi-step physical coding recursive convolutional neural network as described in any one of claims 1 to 7.

9. A computer-readable storage medium having a computer program or instruction stored therein, characterized in that: The computer program or instructions are programmed or configured to execute the fluid solution method based on a multi-step physical coding recursive convolutional neural network as described in any one of claims 1 to 7 through a processor.

10. A computer program product comprising a computer program or instructions, characterized in that The computer program or instructions are programmed or configured to execute the fluid solution method based on a multi-step physical coding recursive convolutional neural network as described in any one of claims 1 to 7 through a processor.

Images