Method for optimizing fluid mechanics simulation model based on deep parallel neural operator

Abstract

The invention relates to the technical field of fluid mechanics, in particular to a method for optimizing a fluid mechanics simulation model based on a deep parallel neural operator, and the deep parallel neural operator comprises an encoder and a decoder; the optimization process of the fluid mechanics simulation model comprises the steps that 1, physical quantities of fluid mechanics are expressed as input functions through an encoder in an original space, and the input functions are preprocessed; 2, mapping the data obtained in the step 1 into a plurality of potential spaces through an encoder to solve a fluid mechanics simulation model, and compensating high-frequency information loss caused by modal truncation; and step 3, recovering the solved data of each potential space into the physical quantity of the fluid mechanics in the step 1 through a decoder, and projecting the physical quantity to the original space. Through the technical scheme, the method has the advantage of solving the problem of error accumulation in the iteration process.

Description

Technical Field

[0001] The present invention relates to solving partial differential equations related to fluid mechanics, and in particular to a method for optimizing a fluid mechanics simulation model based on a deep parallel neural operator. Background Art

[0002] In science and engineering, partial differential equations (PDEs) are often used to simulate physical phenomena in various fields, such as fluid mechanics, material mechanics, quantum mechanics, and electricity. The vast majority of PDEs lack analytical solutions, such as the Navier-Stokes equations, which makes numerical methods the main means of solving them. Although these methods are effective, they are computationally very expensive and usually require a lot of computing resources. Traditional numerical methods often require some compromises in accuracy while pursuing computational efficiency. Therefore, there is an urgent need to find a class of efficient solutions.

[0003] In recent years, deep learning researchers have made significant progress, making it possible for artificial intelligence to replace traditional numerical solvers and numerical simulators. The concept of neural operators and their innovative architectures show considerable promise in approximating complex mappings, and extensive research has been conducted in this area.

[0004] Neural operators are a series of methods proposed in recent years that use neural networks to learn gridless, infinite-dimensional operators. In addition, this method can also directly learn the mapping relationship between different parameters. This method has shown excellent performance in fields such as fluid mechanics and material mechanics.

[0005] The Fourier neural operator proposes a new approach to solving partial differential equation problems. The Fourier transform is a mathematical transformation that converts data into the Fourier domain for learning. It analyzes the frequency components of a signal by decomposing it into a combination of sine and cosine waves of different frequencies and has a wide range of applications in many scientific and engineering fields.

[0006] In the Fourier domain, differentiation operations become much simpler because differentiation is equivalent to multiplication in the Fourier domain. This makes the Fourier transform a powerful tool for solving partial differential equations, especially when frequency components are involved.

[0007] Recently, some spectral methods for solving partial differential equations have been extended to neural networks. A new neural operator architecture has been proposed. In the Fourier neural operator (FNO), a convolution operator is introduced to replace the kernel integral operator in equation 1. Let Representation function Fourier transform, represents the inverse Fourier transform, then: , the inverse Fourier transform is given by: ; By specifying , and applying the convolution theorem we get the Fourier kernel integral operator: This architecture defines a kernel integration operator directly in Fourier space with quasi-linear time complexity and advanced approximation capabilities.

[0008] Partial differential equations in fluid mechanics are of great significance in the field of engineering. Solving them can help engineers predict and design the behavior of various fluid dynamic systems. For example, in the field of aerospace, PDE helps engineers optimize the aerodynamic design of aircraft to increase lift, reduce drag, and ensure flight stability and efficiency. In automotive engineering, these equations are used to simulate and improve the aerodynamic characteristics of vehicles, as well as fluid flow in engines and transmission systems, thereby improving fuel efficiency and performance. At the same time, they also play a key role in links such as heat exchange and fluid transmission in industrial processes, helping engineers optimize these processes and improve energy efficiency and production efficiency.

[0009] Fluid mechanics related equations usually describe physical quantities such as fluid velocity and pressure based on space and time. These physical quantities involved change with changes in spatial position and time. Current mainstream neural operators usually adopt an iterative update strategy, but this will lead to the problem of spatial information loss due to iteration. Summary of the invention

[0010] In order to solve the problems in the related art, the present application provides a method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator, which solves the problem of spatial information loss caused by the iterative update process in the prior art.

[0011] The technical solution is as follows: A method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator, wherein the deep parallel neural operator includes an encoder and a decoder; and the optimization process of the fluid dynamics simulation model includes the following steps: Step 1: In the original space, the physical quantity in the fluid mechanics simulation model is represented as an input function through the encoder , and the input function After pre-processing, ; Step 2: Use the encoder to obtain Mapping to multiple latent spaces Solve the fluid dynamics simulation model in each potential space When solving the fluid mechanics simulation model respectively, the high-frequency information loss caused by modal truncation is compensated, wherein the fluid mechanics simulation model is a partial differential equation about the physical quantity in fluid mechanics; Step 3: Each latent space in step 2 The solved data is restored to the physical quantity of fluid mechanics in step 1 through the decoder and projected into the original space; Among them, the compensation method for the high-frequency information loss caused by the modal stage is: Through the encoder When mapping to multiple latent spaces, a parallel dual Fourier layer is used to fit the complex mapping between input and output in the frequency domain, and a simple linear layer is used to compensate for high-frequency information to obtain compensation. : , in, and Represent the same latent space The two parallel operators in Represents the latent space number, each parallel operator has a corresponding learnable parameter and , Represents a simple linear network.

[0012] Through the above technical solution, the error accumulation problem in the iterative process is solved by projecting the data into different latent spaces and learning through multiple parallel operators.

[0013] Preferably, the process of solving the partial differential equation is: , in, Indicates a cycle; Represents the sequential combination of operations; Used to encode data and map the encoded data into different latent spaces; decoding Project the data in the latent space to the original space; Refers to the complex mapping of input and output in the latent space approximating in different subspaces.

[0014] Preferably, in step 1, the input function including fluid velocity, temperature and pressure; The input function is passed through the encoder The pre-processing process is: Step 11: Input function Perform dimensionality increase After getting , where the dimension-raising operation The process is: ,in, ; Step 12: Perform local transformation , local transformation After obtaining : .

[0015] Preferably, each latent space is obtained in step 1 Perform projection calculation and get : , in, .

[0016] Preferably, the multiple latent spaces include a first latent space and a second latent space, and the processes of solving the partial differential equations of fluid mechanics in the first latent space and the second latent space are consistent.

[0017] Preferably, different Fourier layers in the same latent space are extracted by different modalities, and the extracted Fourier layers are obtained , The extraction process is: Through a dimensional network, The latent space data is projected onto potential space, and obtain ; Will and Output of the space Merge, which merges adjacent latent space outputs in a sequential manner until the process returns to the original space .

[0018] In summary, the beneficial effect of a method based on deep parallel neural operators to optimize fluid dynamics simulation models is that the data is projected into different latent spaces and learned through multiple parallel operators, thereby solving the error accumulation problem in the iterative process.

[0019] It is to be understood that the foregoing general description and the following detailed description are exemplary only and are not restrictive of the invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0021] Figure 1 A schematic diagram of the overall architecture of the deep parallel neural operator in the present invention; Figure 2 for Figure 1 Detailed schematic diagram of the projection process; Figure 3 for Figure 1Detailed diagram of the parallel Fourier layer in Fig. Figure 4 Schematic diagram of the true value, prediction result and error of each model in the present invention at T∈[11,20] time steps; where (a) represents the DPNO model, (b) represents the FNO model, and (c) represents the LSM model; Figure 5 Schematic diagram of the results of the deep parallel neural operator model and two baseline models on the Darcy flow problem in the present invention: the first row is the true value and the predicted value of the model, and the second row is the legend and the prediction error; Figure 6 It is a schematic diagram of the results of the deep parallel neural operator model and two baseline models in the present invention on the airfoil design problem, wherein the first row is the true value and the predicted value of the model, and the second row is the legend and the prediction error; Figure 7 It is a schematic diagram of the results of the deep parallel neural operator model and two baseline models in the pipeline flow problem in the present invention, wherein the first row is the true value and the predicted value of the model, and the second row is the legend and the prediction error; Figure 8 Schematic diagram of the parallel operator architecture in this embodiment Fig. 9 Schematic diagram of the serial operator architecture in this embodiment; DETAILED DESCRIPTION

[0022] Here, exemplary embodiments will be described in detail, and examples thereof are shown in the accompanying drawings. When the following description refers to the drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with the present invention.

[0023] In a possible embodiment, Figure 1-Figure 9 As shown, a method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator, the deep parallel neural operator includes an encoder and a decoder, and the optimization process of the fluid dynamics simulation model includes the following steps: Step 1: In the original space, the physical quantity in the fluid mechanics simulation model is represented as an input function through the encoder , and the input function After pre-processing, ; Step 2: Use the encoder to obtain Mapping to multiple latent spaces Solve the fluid dynamics simulation model in each potential space When solving the fluid mechanics simulation model respectively, the high-frequency information loss caused by modal truncation is compensated, wherein the fluid mechanics simulation model is a partial differential equation about the physical quantity in fluid mechanics; Step 3: Each latent space in step 2 The solved data is restored to the physical quantity of fluid mechanics in step 1 through the decoder and projected into the original space; Among them, the compensation method for the high-frequency information loss caused by the modal stage is: Through the encoder When mapping to multiple latent spaces, a parallel dual Fourier layer is used to fit the complex mapping between input and output in the frequency domain, and a simple linear layer is used to compensate for high-frequency information to obtain compensation. : , in, and Represent the same latent space The two parallel operators in Represents the latent space number, each parallel operator has a corresponding learnable parameter and , Represents a simple linear network.

[0024] By projecting the data into different latent spaces and learning through multiple parallel operators, the error accumulation problem in the iterative process is solved.

[0025] The process of solving the partial differential equation is: , in, Indicates a cycle; Represents the sequential combination of operations; Used to encode data and map the encoded data into different latent spaces; decoding Project the data in the latent space to the original space; Refers to the complex mapping of input and output in the latent space approximating in different subspaces.

[0026] In step 1, the input function including fluid velocity, temperature and pressure; The input function is passed through the encoder The pre-processing process is: Step 11: Input function Perform dimensionality increase After getting , where the dimension-raising operation The process is: ,in, ; Step 12: Perform local transformation , local transformation After obtaining : .

[0027] Each latent space is obtained in step 1. Perform projection calculation and get : , in, .

[0028] The multiple latent spaces include a first latent space and a second latent space, and the processes of solving the partial differential equations of fluid mechanics in the first latent space and the second latent space are consistent.

[0029] Different Fourier layers in the same latent space are extracted by different modalities, and the extracted , The extraction process is: Through a dimensional network, The latent space data is projected onto potential space, and obtain ; Will and Output of the space Merge, which merges adjacent latent space outputs in a sequential manner until the process returns to the original space .

[0030] In order to comprehensively evaluate the performance of the model, four fluid mechanics-related partial differential equation benchmark datasets were selected for training and testing, and the relative error of the model was recorded. The selected benchmark datasets are typical partial differential equations in fluid physics. See Table 1 for details; Table 1 Benchmark datasets

[0031] To better demonstrate the performance of our proposed model (i.e., Deep Parallel Neural Operator, denoted as DPNO), we compare it with seven currently recognized baseline models, including U-Net, FNO, WMT, U-FNO, UNO, F-FNO, and LSM. Among them, LSM is currently the best overall performing model on six datasets. We used the same experimental settings for all models. The detailed settings are shown in Table 2.

[0032] Table 2 Settings

[0033] To ensure that all models can converge, the number of training rounds is set to 500. The training batch size is 20. Each parameter of the model will be updated using the Adam optimizer with a learning rate of 0.001 and a weight decay of 0.0001.

[0034] Based on the problem setting, the entire task is to solve one or more sets of outputs from a set of inputs. The numerical value itself is small, so in order to more objectively describe the model performance, the mean square error (MSE) is used as the evaluation indicator.

[0035] MSE is a commonly used indicator for evaluating the performance of prediction models and is used in various types of machine learning tasks. It reflects the magnitude of the model error by measuring the mean square difference between the model's predicted value and the actual value. The smaller the MSE value, the closer the model's predicted value is to the true value.

[0036] Given a containing A dataset of samples, where each sample has an actual value and predicted values , the MSE calculation formula is as follows: , This example will specifically introduce the four fluid mechanics-related data sets used, and show the results of the model proposed in this example and seven baseline models.

[0037] Navier-Stokes equations: The Navier-Stokes equations are the basic equations that describe fluid motion. Consider the two-dimensional Navier-Stokes equations for a viscous incompressible fluid on a unit ring surface, described in vorticity form as: ; ; ; In this problem, the velocity field Belong to space , for any , It represents the vorticity, which is used to describe the local rotation speed of the fluid. Belong to space The viscosity coefficient is greater than 0, forcing the function Also belongs to space The main purpose of a neural network is to learn a solution operator , the solver learns the vorticity changes of the first 10 time steps to predict the vorticity field changes greater than 10 time steps.

[0038] The equations are solved using stream function representation combined with pseudospectral methods. Initially, a Poisson equation is solved in the Fourier domain to determine the velocity field. Subsequently, the derivatives and nonlinear terms of the vorticity are calculated in the physical domain, and then a de-aliasing process is performed to remove high-frequency noise. The time integration uses the Crank-Nicolson scheme, and the linear terms are implicitly processed while the nonlinear terms remain explicit. The dataset is generated by the baseline paper, and we use a 64*64 grid resolution for training and testing. The results are shown in Table 3 and Figure 4 shown.

[0039] Table 3 MSE error of the model in Navier-Stokes equation

[0040] Henry Darcy's law describes the flow characteristics of fluids through porous media. This law has a wide range of applications in many natural sciences, including Fourier's law in heat conduction, Ohm's law in electric current, and Fick's law in diffusion theory. The basic form of Darcy's law is as follows: , in, Refers to the volume of fluid passing through the medium per unit time. is the permeability coefficient, is the cross-sectional area of ​​the flow, is the pressure gradient. We consider the two-dimensional steady-state Darcy flow equation in a unit rectangle, which is a second-order linear elliptic partial differential equation with Dirichlet boundary conditions, and the equation form is as follows: , ; in, is the diffusion coefficient, is the forcing function.

[0041] For the above equations, our main focus on training the neural network is how to obtain the solution operator through the diffusion coefficient. We used the dataset provided in the baseline paper and used a fixed resolution of 85*85 in this example. The results are shown in Table 4 and Figure 5 middle.

[0042] Table 4 MSE error of the model on the Darcy equation

[0043] Airfoil problems can be analyzed using the Euler equations, which describe the motion of an inviscid (non-viscous) fluid. The Euler equations are a simplified form of the Navier-Stokes equations that apply to the flow around an airfoil at high speeds, where viscous effects can be neglected. The Euler equations are described as follows: ; ; ; in, It is the mass per unit volume, reflecting the concentration of the fluid; is the distance traveled by a fluid particle per unit time, a vector describing the change in fluid momentum and flow characteristics; is the force per unit area; is the total energy of the system. In this case, the far-field boundary condition is , , and ,in, is the Mach number, is the angle of attack. In addition, a no-penetration condition is imposed on the airfoil. The neural network fits the mapping from the shape parameters of the wing to the flow field around the wing. The grid point positions and the Mach numbers at these grid points are used as input and output data. In this example, 1000 training grid points and 200 test grid points are used. The results are shown in Tables 5 and Figure 6 middle.

[0044] Table 5 MSE error of the model in the airfoil problem

[0045] In fluid dynamics, pipe problems usually refer to the study of the flow of fluid through a pipe, which is governed by the Navier-Stokes equations. In the context of pipe problems, the Navier-Stokes equations can be simplified under the assumptions of steady state, incompressibility, and laminar flow. The simplified form is as follows: , , in, is the velocity vector, It's pressure. is the viscosity. In this case, a maximum velocity is established at the inlet. Parabolic velocity profile of the outlet. Free boundary conditions are used at the outlet, while the pipe surface is constrained by no-slip boundary conditions. The length of the pipe is 10 and the diameter is 1. The centerline of the pipe is defined by four piecewise cubic polynomials derived from the vertical positions and slopes of five equally spaced control nodes. The vertical positions of these nodes follow a uniform distribution , the slopes of these nodes are also evenly distributed in For this problem, the neural network mainly focuses on how to simulate the velocity field of the fluid through geometric parameters. The grid point position parameters (129*129) and the horizontal velocity at these grid points are used as input and output data. In this embodiment, 1000 training grid points and 200 test grid points are used. The results are shown in Tables 6 and Figure 7 middle.

[0046] Table 6 MSE error of the model on the pipeline problem

[0047] DPNO achieves state-of-the-art performance on 3 benchmark datasets and performs second best on one of them. Compared with the best overall performer LSM, it is improved by 8% on average. This demonstrates the versatility and superior performance of the proposed DPNO in fluid dynamics related PDE tasks. Compared with the baseline model FNO, DPNO improves the average performance by 36.69% on the six datasets. As shown in Table 7: Table 7 MSE comparison of 7 benchmark models and DPNO on 4 benchmark datasets: the best performing model is represented by (best), and the second best model is represented by (second best):

[0048] In order to verify the effectiveness of the parallel strategy in the model, a set of ablation experiments is designed in this embodiment to change the parallel operator into a serial operator and observe the performance changes. Specifically, the parallel operator is merged into a serial operator, such as Figures 8 to 9 The experimental results are shown in Table 8. In Table 8, DPNO indicates that the model is a parallel operator, and DPNO / P indicates that the model is a serial operator. Through the ablation experiment, it can be found that parallelization has a certain impact on the overall model performance, and the average performance has improved by 14.64%. This is because the error accumulation caused by iteration is reduced through the parallelization module.

[0049] Table 8 MSE comparison between DPNO / P and DPNO on four benchmark datasets

[0050] Excellent performance was achieved on six partial differential equation benchmark datasets. Compared with the baseline model FNO, the average performance of the deep parallel neural operator model was improved by 34%.

[0051] Those skilled in the art will readily appreciate other embodiments of the present invention after considering the specification and practicing the invention herein. This application is intended to cover any variations, uses or adaptations of the present invention that follow the general principles of the present invention and include common knowledge or customary techniques in the art that are not invented by the present invention.

[0052] It will be appreciated that the present invention is not limited to the exact construction that has been described above and shown in the drawings, and that various modifications and changes may be made without departing from the scope thereof.

Claims

1. A method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator, characterized in that: The deeply parallel neural operators include encoder and decoder; The optimization process of the fluid dynamics simulation model includes the following steps: Step 1: In the original space, the physical quantity in the fluid mechanics simulation model is represented as an input function through the encoder , and the input function After pre-processing, ; Step 2: Use the encoder to obtain Mapping to multiple latent spaces Solve the fluid dynamics simulation model in each potential space When solving the fluid mechanics simulation model respectively, the high-frequency information loss caused by modal truncation is compensated, wherein the fluid mechanics simulation model is a partial differential equation about the physical quantity in fluid mechanics; Step 3: Each latent space in step 2 The solved data is restored to the physical quantity of fluid mechanics in step 1 through the decoder and projected into the original space; Among them, the compensation method for the high-frequency information loss caused by the modal stage is: Through the encoder When mapping to multiple latent spaces, a parallel dual Fourier layer is used to fit the complex mapping between input and output in the frequency domain, and a simple linear layer is used to compensate for high-frequency information to obtain compensation. : ， in, and Represent the same latent space The two parallel operators in Represents the latent space number, each parallel operator has a corresponding learnable parameter and , Represents a simple linear network.

2. The method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator according to claim 1, characterized in that: The process of solving the partial differential equation is: ， in, Indicates a cycle; Represents the sequential combination of operations; Used to encode data and map the encoded data into different latent spaces; decoding Project the data in the latent space to the original space; Refers to the complex mapping of input and output in the latent space approximating in different subspaces.

3. The method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator according to claim 1, characterized in that: In step 1, the input function including fluid velocity, temperature and pressure; The input function is passed through the encoder The pre-processing process is: Step 11: Input function Perform dimensionality increase After getting , where the dimension-raising operation The process is: ,in, ; Step 12: Perform local transformation , local transformation After obtaining : .

4. The method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator according to claim 1, characterized in that: Each latent space is obtained in step 1. Perform projection calculation and get : ， in, .

5. The method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator according to claim 1, characterized in that: The multiple latent spaces include a first latent space and a second latent space, and the processes of solving the partial differential equations of fluid mechanics in the first latent space and the second latent space are consistent.

6. The method for optimizing a fluid dynamics simulation model based on a deep parallel neural operator according to claim 1, characterized in that: Different Fourier layers in the same latent space are extracted by different modalities, and the extracted ,in, The extraction process is: Through a dimensional network, The latent space data is projected onto potential space, and obtain ; Will and Output of the space Merge, which merges adjacent latent space outputs in a sequential manner until the process returns to the original space .

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