A pinn and dg data fusion method for fluid strong discontinuity problem

Abstract

The application relates to the technical field of computational fluid dynamics based on computer data processing, and discloses a PINN and DG data fusion method for a fluid strong discontinuity problem, which comprises the following steps: describing the distribution of fluid states in a unit, solving numerical flux and fluid state quantities in the unit, and calculating the motion speed of a grid node; setting the calculation domain range, initial conditions and boundary conditions, defining the input and output of a neural network, and initially constructing a PINN model of the problem; calculating the partial differential equation loss term, the initial condition loss term and the boundary condition loss term of the loss function of the neural network; identifying a shock wave region and a smooth flow field region; introducing DG data in the shock wave region as a data loss term in the loss function, and constructing a total loss function based on the same; minimizing the total loss function, and finally outputting the density, speed and pressure of the full flow field space and each time step. In this way, the precise numerical solution can have both the shock wave capturing capability and the fitting capability of the smooth flow field region.

Description

Technical Field

[0001] This application relates to the field of computational fluid dynamics technology based on computer data processing, and for example to a PINN and DG data fusion method for fluid strongly discontinuous problems. Background Technology

[0002] Accurate solutions to strongly discontinuous flow fields are a core challenge in computational fluid dynamics. The mathematical descriptions of these problems typically involve hyperbolic conservation laws (such as the Euler equations), whose solutions exhibit large gradients in physical quantities within the shock wave region. While traditional numerical methods (such as the DG method and the finite volume method) have made significant progress in shock wave capture, they struggle to balance computational efficiency, numerical accuracy, and algorithmic robustness when solving strongly discontinuous problems, and they also find it difficult to solve inverse problems.

[0003] While the emerging PINN (Perimeter-Based Numerical Model) offers the advantage of being meshless, it faces significant challenges in handling strongly discontinuous problems. First, due to spectral bias, standard PINN tends to fit smooth flow field regions, leading to over-smoothing of the shock wavefront. Second, the sharp physical residual gradients near the shock wave easily cause an imbalance in the loss function weights, trapping the optimization process in local optima. Furthermore, the standard collocation method lacks strict volume integral conservation constraints, making it difficult to guarantee local conservation characteristics at discontinuities. Existing improvements such as weighted equations, artificial viscosity, or data fusion can alleviate some oscillations, but most remain confined to point-state collocation frameworks, resulting in low training efficiency in the shock region. Especially in fields involving strongly discontinuous computations, such as computational fluid dynamics, current techniques still struggle to provide accurate, stable, and efficient numerical simulation results.

[0004] It should be noted that the information disclosed in the background section above is only used to enhance the understanding of the background of this application, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0005] To provide a basic understanding of some aspects of the disclosed embodiments, a brief summary is given below. This summary is not intended as a general commentary, nor is it intended to identify key / important components or describe the scope of protection of these embodiments, but rather as a prelude to the detailed description that follows.

[0006] This disclosure provides a PINN and DG data fusion method for fluid strongly discontinuous problems, enabling accurate numerical solutions that combine shock wave capture capability with smooth flow field region fitting capability.

[0007] In some embodiments, the PINN and DG data fusion method for fluid strongly discontinuous problems includes: S10, using the DG method to spatially discretize the Euler equations and describe the distribution of fluid states within the cells; using the HLLC flux method to solve for numerical flux and fluid state variables within the cells; calculating the mesh node motion velocity based on the computational domain, the numerical flux, and the fluid state variables within the cells; S20, according to the Euler equations of the fluid in the problem, setting the computational domain range, initial conditions, and boundary conditions, clarifying the input and output of the neural network, and initially constructing the PINN model of the problem; S30, calculating the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network. Loss term; S40, analyze the gradient of the fluid state variables within the unit. When the gradient value is greater than a preset threshold, the computational domain of each problem is identified as a shock wave region; when the gradient value is less than or equal to the preset threshold, the computational domain of each problem is identified as a smooth flow field region; S50, introduce DG data into the shock wave region as a data loss term in the loss function; combine the partial differential equation loss term, the initial condition loss term, and the boundary condition loss term, and introduce a gradient weight factor to automatically balance the gradient contribution of each loss term to construct a total loss function; S60, minimize the total loss function, and finally output the density, velocity, and pressure of the entire flow field space and each time step.

[0008] In some embodiments, the PINN and DG data fusion device for fluid strongly discontinuous problems includes: a computation module configured to spatially discretize the Euler equations using the DG method and describe the distribution of fluid states within the cells; solve for numerical flux and fluid state variables within the cells using the HLLC flux method; calculate the movement velocity of grid nodes based on the computational domain, the numerical flux, and the fluid state variables within the cells; a condition setting and model building module configured to set the computational domain range, initial conditions, and boundary conditions according to the Euler equations of the fluid in the problem, define the input and output of the neural network, and initially construct the PINN model of the problem; and a loss term calculation module configured to calculate the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network. The system includes: a boundary condition loss term; a region identification module configured to analyze the gradient of fluid state variables within the unit; when the gradient value is greater than a preset threshold, the computational domain of each problem is identified as a shock wave region; when the gradient value is less than or equal to the preset threshold, the computational domain of each problem is identified as a smooth flow field region; a total loss construction module configured to introduce DG data into the shock wave region as a data loss term in the loss function; combining the partial differential equation loss term, the initial condition loss term, and the boundary condition loss term, and introducing a gradient weight factor to automatically balance the gradient contribution of each loss term to construct the total loss function; and an output module configured to minimize the total loss function and finally output the density, velocity, and pressure of the entire flow field space and each time step.

[0009] The PINN and DG data fusion method provided in this disclosure for fluid strongly discontinuous problems can achieve the following technical effects: 1. By introducing DG data into the standard PINN method to solve strongly discontinuous fluid problems, and by introducing the numerical solution of DG into the shock wave region as a data loss term, a total loss function that integrates physical constraints and DG data is constructed. Combined with an adaptive weight factor adjustment strategy, a high-precision and high-efficiency solution for strongly discontinuous flow field problems is achieved. 2. By integrating PINN and DG, the propagation characteristics and physical quantity mutations of shock waves in strongly discontinuous flow fields can be accurately simulated. This effectively solves the problems of numerical diffusion and over-smoothing caused by standard PINN when dealing with shock wave problems. At the same time, the fitting ability of neural networks is used to suppress the non-physical oscillations near the discontinuity of the traditional DG method. This overcomes the key technical bottlenecks of numerical oscillations in the simulation of strongly discontinuous flow using traditional numerical methods and numerical smoothing in PINN. 3. The PINN-DG method retains the advantages of the standard PINN method, such as being meshless, easily handling complex boundaries, and having strong generalization performance. It overcomes the non-physical oscillations generated by the traditional DG method, and provides a more accurate and efficient numerical method for underwater explosion dynamics and supersonic flow simulation in the field of computational fluid dynamics involving strong discontinuous shock wave calculations. It also provides theoretical basis and design reference. 4. The DG method based on PINN can accurately simulate the propagation characteristics and physical quantity changes of shock waves in strong discontinuous flow fields. It effectively solves the common numerical diffusion and oversmoothing problems of standard PINN when dealing with shock wave problems. At the same time, it uses the continuous representation characteristics of neural networks to suppress the non-physical oscillations near the discontinuity of the traditional DG method.

[0010] The above general description and the description below are exemplary and illustrative only and are not intended to limit this application. Attached Figure Description

[0011] One or more embodiments are illustrated by way of example with reference to the accompanying drawings. These illustrations and drawings do not constitute a limitation on the embodiments. Elements having the same reference numerals in the drawings are shown as similar elements. The drawings are not to be scaled. And wherein: Figure 1 This is a schematic diagram of the PINN and DG data fusion method for fluid strong discontinuity problems provided in the embodiments of this disclosure; Figure 2 This is a schematic diagram of the PINN-DG method calculation model provided in the embodiments of this disclosure; Figure 3 This is a schematic diagram of the results of solving smooth flow field problems using the PINN-DG method provided in the embodiments of this disclosure; Figure 4 This is a schematic diagram of the analytical solution to the smooth flow field problem provided in the embodiments of this disclosure; Figure 5 This is a schematic diagram showing the absolute error of the PINN-DG method for solving smooth flow field problems according to the embodiments of this disclosure; Figure 6 This is a schematic diagram illustrating the PINN-DG method for solving the pressure problem of the Sod shock tube provided in the embodiments of this disclosure; Figure 7 This is a schematic diagram illustrating the DG method for solving the pressure problem of the Sod shock tube provided in the embodiments of this disclosure; Figure 8 This is a schematic diagram illustrating the pressure results of solving the Sod shock tube problem provided in the embodiments of this disclosure; Figure 9 This is a schematic diagram illustrating the velocity of the Sod shock tube problem solved by the PINN-DG method provided in this embodiment of the disclosure; Figure 10This is a schematic diagram illustrating the DG method for solving the Sod shock tube problem using the embodiments of this disclosure; Figure 11 This is a schematic diagram illustrating the velocity results of solving the Sod shock tube problem provided in the embodiments of this disclosure; Figure 12 This is a schematic diagram illustrating the PINN-DG method for solving the density problem in a two-dimensional explosion, as provided in this embodiment of the disclosure. Figure 13 This is a schematic diagram illustrating the DG method for solving the density problem in a two-dimensional explosion, as provided in the embodiments of this disclosure. Detailed Implementation

[0012] To provide a more detailed understanding of the features and technical content of the embodiments of this disclosure, the implementation of the embodiments of this disclosure will be described in detail below with reference to the accompanying drawings. The accompanying drawings are for illustrative purposes only and are not intended to limit the embodiments of this disclosure. In the following technical description, for ease of explanation, several details are used to provide a full understanding of the disclosed embodiments. However, one or more embodiments may still be implemented without these details. In other cases, well-known structures and devices may be simplified in their depiction to simplify the drawings.

[0013] The terms "first," "second," etc., used in the specification and accompanying drawings of this disclosure are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate for the embodiments of this disclosure described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion.

[0014] Unless otherwise stated, the term "multiple" means two or more.

[0015] In this embodiment of the disclosure, the character " / " indicates that the objects before and after it are in an "or" relationship. For example, A / B means: A or B.

[0016] The term "and / or" describes an association between objects, indicating that three relationships can exist. For example, A and / or B means: A or B, or A and B.

[0017] The term "correspondence" can refer to an association or binding relationship. The correspondence between A and B means that there is an association or binding relationship between A and B.

[0018] Combination Figure 1 As shown, this disclosure provides a PINN and DG data fusion method for fluid strong discontinuity problems, including: S10 uses the DG method to spatially discretize the Euler equations and describe the distribution of fluid states within the element; the HLLC flux method is used to solve for the numerical flux and fluid state variables within the element; based on the computational domain, numerical flux, and fluid state variables within the element, the motion velocity of the mesh nodes is calculated. Specifically, the steps include: S11, the DG method is used to divide the entire computational domain into several non-overlapping quadrilateral units, each of which is an independent computational subdomain. S12. Inside each quadrilateral element, the Taylor basis functions at the centroid of the element are selected as a set of bases for the approximate space, and the distribution of fluid state variables within the element is described by combining the ideal gas law.

[0019] Fluid state quantities include: density ρ , x Directional velocity u , y Directional velocity v Always E and pressure p The pressure is calculated from density and energy using the ideal gas law.

[0020] The ideal gas law states: , in, Specific heat ratio; Internal energy per unit mass.

[0021] Taylor basis functions used at the element centroid, element W i Approximate numerical solutions within The expansion factor can be used Written as: , in: , in, and These are the Taylor expansion coefficients and normalized Taylor basis functions at the centroid, respectively; x c , y c () represents the centroid coordinates of the element; , , and These represent the nodes in x and y Maximum and minimum coordinate values ​​in the direction.

[0022] S13, at the common interface of adjacent elements, the numerical flux is calculated using the HLLC Riemann solver.

[0023] S14. Multiply the conserved form of the Euler equation by the test function and perform a volume integral over each quadrilateral element. Simultaneously, perform a boundary integral on the numerical flux to obtain the weak form corresponding to the DG method, as follows: , In the formula: For the test function, Representation unit W i The boundary, ( () is the vector representing the direction of the outward normal to the boundary. and as follows: , .

[0024] S15. The volume integral term and boundary integral term are calculated using the Gaussian numerical integration method. Combined with the local mass matrix (composed of the inner product of basis functions), the degree of freedom coefficients of the fluid state variables in each element are solved, thereby obtaining the flow field state variables at any point in the quadrilateral element: density, velocity, energy, and pressure.

[0025] S16, using a nodal solver, the velocity of each mesh node is calculated based on the momentum conservation variables of adjacent quadrilateral elements, allowing the mesh nodes to follow the local fluid motion and achieving Lagrange mesh evolution. During the stage where the mesh nodes follow the local fluid motion, the nodal solver introduces four pressures on each edge and two pressures on each side of the edge. The surface flux on each edge can be determined by the following formula: , .

[0026] Numerical flux of edges and Represented as: , , In the formula, and This represents the pressure on one half of the unit surface; Indicates the velocity at the node; and These are single-valued functions defined on each side of the interface; r and r+1 represent the two endpoints on one edge of the quadrilateral grid; r+ represents the midpoint between nodes r and r+1; i represents one side of the mesh edge (such as the left or right element side); * represents the interface analytical state obtained after calculation by the Riemann solver.

[0027] S17 uses the second-order Runge-Kutta method to recursively update the fluid state variables and grid node coordinates of each quadrilateral element in the entire computational domain until the preset termination time is reached.

[0028] In this way, the distribution of fluid state within the cell is obtained through S10, providing reliable reference data of strong discontinuous flow field for subsequent PINN training.

[0029] S20, based on the Euler equations for the fluid in the problem, define the computational domain, initial conditions, and boundary conditions, clarify the input and output of the neural network, and initially construct the PINN model for the problem. This includes the following steps: S21, First, we define the two-dimensional spatiotemporal computational domain Ω×[0, ... for solving the Euler equations. T ], where Ω is the spatial region, [0, T [ ] represents the time interval.

[0030] S22, specify the initial conditions: in t Density when =0 ρ 0( x , y ), x Directional velocity u 0( x , y ), y direction v 0( x , y ),pressure p 0( x , y The distribution of ) and boundary conditions (such as solid wall boundaries, periodic boundaries, etc.).

[0031] In this paper, the fluid motion governing equations are derived from the fact that inertial effects dominate in strongly discontinuous problems, while viscous effects of the fluid can be neglected. Therefore, the fluid motion can be described by the Euler equations, as follows: , , , , In the formula, Indicates fluid density; u、v For velocity components; p For fluid pressure; E This represents the total energy per unit volume.

[0032] Total energy and specific internal energy e The relationship is: .

[0033] To close the above set of equations and accurately describe the thermodynamic properties of the gas, we introduce the ideal gas law as mentioned above.

[0034] S23, see also Figure 2 A multi-layer fully connected neural network is used as the basic framework of the PINN model. The input layer of the neural network receives spatiotemporal coordinates (…). x , y , t ),in, x The horizontal position on a two-dimensional plane; y The position in the vertical direction on a two-dimensional plane; t This represents the time elapsed since the problem was studied. The output layer also outputs a conserved variable: density. ρ , x Directional velocity u , y Directional velocity v and pressure p The neural network contains six hidden layers, each with sixty-four neurons, and employs the nonlinear activation function tanh. Through training, it learns the mapping relationship from spatiotemporal coordinates to the flow field state. To comprehensively simulate the flow field characteristics, this step employs differentiated sampling strategies for PDE sampling points, initial condition points, and boundary condition points. Different sampling strategies, such as gridded sampling or random sampling, are used depending on the specific problem to ensure that the training data covers the entire flow field.

[0035] The physical governing equations solved by the PINN model include: , , , In the formula, Let be the physical field variables to be solved; For unknown solutions to time t The derivative; Represents the differential operator; and The computational domain and its boundaries; These are the initial conditions; For boundary operators.

[0036] S30, calculate the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network. Specifically, this includes the following steps: S31, using automatic differentiation technology, the output of the neural network ( ρ , u , v ,p ) Perform continuous differentiation with respect to the input spatiotemporal coordinates to obtain the first-order partial derivatives of each physical quantity (e.g. (etc.); Substituting all first-order partial derivatives into the left-hand side of the Euler equation, the loss term of the partial differential equation is calculated. The loss term of the partial differential equation is expressed as: .

[0037] S32, in the network prediction results, extract the output values ​​at all spatiotemporal points at the initial time points and compare them with the given initial condition distribution. ρ 0( x , y ), u 0( x , y ), v 0( x , y )as well as p 0( x , y By comparison, the initial condition loss term is calculated. The initial condition loss term is expressed as: .

[0038] S33, select boundary points on the spatial boundary of the computational domain, and construct corresponding constraints according to the preset boundary type; calculate the residual between the output of the neural network at the boundary points and the required boundary condition value to obtain the boundary condition loss term. The boundary condition loss term is expressed as: .

[0039] Thus, the general constraint terms of the PINN model loss function can be obtained through S30.

[0040] S40 analyzes the gradient of fluid state variables within the analysis unit to separate the shock wave region from the smooth flow field region. Based on the gradients of physical quantities such as density and velocity in the preliminary calculated flow field, a preset threshold is established. When the gradient value is greater than the preset threshold, the computational domain for each problem is identified as the shock wave region. Conversely, when the gradient value is less than or equal to the preset threshold, the computational domain for each problem is identified as the smooth flow field region.

[0041] In this way, the S40 adaptive partitioning mechanism provides a basis for the subsequent construction of differentiated loss functions, ensuring accurate simulation of the shock wave region.

[0042] S50 introduces DG data into the shock region as a data loss term in the loss function; it combines partial differential equation loss terms, initial condition loss terms, and boundary condition loss terms, and introduces a gradient weight factor to automatically balance the gradient contributions of each loss term, thus constructing the total loss function. Specifically, it includes the following steps: S51, at the selected spatiotemporal points, the residual between the neural network's predicted values ​​and the DG's exact solution is used as the data loss term to compensate for PINN's insufficient ability to capture strong discontinuities. The data loss term is expressed as: .

[0043] S52, the total loss function of the shock wave region Defined as: .

[0044] S53, the total loss function of the smoothed flow field region Defined as: ; in, , In the formula, These are the partial differential equation loss term, boundary condition loss term, initial condition loss term, and data loss term, respectively. , , , Adaptive weighting coefficients to balance the contributions of various gradients; These represent the number of sampling points for the partial differential equation, the number of sampling points for boundary conditions, the number of sampling points for initial conditions, and the number of data sampling points, respectively; i represents the i-th data point currently being calculated; x represents the spatial coordinate vector; and t represents the physical time variable. These represent the spatial locations of the i-th sampling points used to calculate the partial differential equation loss, boundary condition loss, initial condition loss, and data loss, respectively. These represent the specific times corresponding to the sampling point of the i-th partial differential equation, the boundary condition sampling point, the initial condition sampling point, and the data sampling point, respectively. This refers to the gradient weight factor in the adaptive weighting method. These are the weighting coefficients; This is the velocity vector.

[0045] In this way, the data loss term can be introduced into the neural network through S50 to construct the total loss function of the PINN-DG model.

[0046] S60 minimizes the total loss function, ultimately outputting the density, velocity, and pressure of the entire flow field space at each time step. Specifically, it includes the following steps: S61 employs a two-stage measurement process during training. First, the Adam optimizer is used to initially optimize the total loss function, causing it to decrease rapidly.

[0047] S62, switch to the L-BFGS optimizer to perform refined second-order derivative optimization of the total loss function to achieve higher-precision convergence.

[0048] S63, by minimizing the total loss function, the PINN model finally outputs high-precision calculation results of density, velocity and pressure in the entire flow field space and at each time step.

[0049] In this way, the density, velocity, and pressure of the entire flow field space and each time step can be obtained through S60, and the problem can be solved in the end.

[0050] In summary, the PINN and DG data fusion method for fluid strongly discontinuous problems provided in this disclosure can achieve the following technical effects: 1. By introducing DG data into the standard PINN method to solve strongly discontinuous fluid problems, and by introducing the numerical solution of DG into the shock wave region as a data loss term, a total loss function that integrates physical constraints and DG data is constructed. Combined with an adaptive weight factor adjustment strategy, a high-precision and high-efficiency solution for strongly discontinuous flow field problems is achieved. 2. By integrating PINN and DG, the propagation characteristics and physical quantity mutations of shock waves in strongly discontinuous flow fields can be accurately simulated. This effectively solves the problems of numerical diffusion and over-smoothing caused by standard PINN when dealing with shock wave problems. At the same time, the fitting ability of neural networks is used to suppress the non-physical oscillations near the discontinuity of the traditional DG method. This overcomes the key technical bottlenecks of numerical oscillations in the simulation of strongly discontinuous flow using traditional numerical methods and numerical smoothing in PINN. 3. The PINN-DG method retains the advantages of the standard PINN method, such as being meshless, easily handling complex boundaries, and having strong generalization performance. It overcomes the non-physical oscillations generated by the traditional DG method, and provides a more accurate and efficient numerical method for underwater explosion dynamics and supersonic flow simulation in the field of computational fluid dynamics involving strong discontinuous shock wave calculations. It also provides theoretical basis and design reference.

[0051] 4. The DG method based on PINN can accurately simulate the propagation characteristics and physical quantity changes of shock waves in strong discontinuous flow fields. It effectively solves the common numerical diffusion and oversmoothing problems of standard PINN when dealing with shock wave problems. At the same time, it uses the continuous representation characteristics of neural networks to suppress the non-physical oscillations near the discontinuity of the traditional DG method.

[0052] Comprehensive performance verification To verify the effectiveness and accuracy of the method of this invention in simulating strongly discontinuous flow fields, the calculation results of the method used in this invention are compared with those of the standard PINN method and the traditional DG method. A smooth flow field problem was selected for simulation, such as... Figures 3 to 5 As shown, Figure 3The PINN-DG method is given to predict the solution. Figure 4 An analytical solution to the problem is given. Figure 5 The absolute error represents the difference between the two results. The results show that the smooth flow field obtained by the PINN-DG calculation method used in this invention agrees well with the analytical solution, and the absolute error is low. This indicates that the PINN-DG method of this invention can accurately and effectively reflect the calculation results in a smooth flow field, providing reliable computational support for solving smooth flow fields.

[0053] Figures 6 to 11 This paper demonstrates the application results of the computational method of this invention in strongly discontinuous flow fields, using the classic Sod shock tube problem in computational fluid dynamics as a test. This problem is a typical Riemann problem, characterized by density and pressure jumps on both sides of the initial discontinuity, which can simultaneously generate shock waves, contact discontinuities, and expansion waves. The problem is solved in a two-dimensional computational domain of size [0,1]. [0,0.04], the boundary conditions are set to wall boundary conditions. The initial conditions are as follows: , The fluid specific heat ratio is set to =1.4. The calculation ends at t=0.2s. Figure 6 , Figure 7 and Figure 8 A comparison of the pressures obtained by solving the Sod shock tube problem using the PINN-DG method and the DG method is presented. Figure 9 , Figure 10 and Figure 11 A comparison of the velocities obtained by solving the Sod shock tube problem using the PINN-DG method and the DG method is presented. As can be seen from the figure, the results calculated by PINN-DG agree well with those calculated by DG, and PINN-DG overcomes the problem of numerical oscillations that easily occur in the DG method.

[0054] Figures 12 to 13 This paper demonstrates the application results of the computational method of this invention in a two-dimensional strongly discontinuous flow field, using the classic two-dimensional explosion problem in computational fluid dynamics as a test example. This example involves outwardly expanding circular shock waves, contact discontinuities, and inwardly propagating rarefaction waves. Compared to the one-dimensional example, the two-dimensional explosion problem places higher demands on the model's ability to capture discontinuities, effectively simulating the evolution characteristics of explosion shock waves in computational fluid dynamics.

[0055] The computation domain is set to [0,2]. The square region [0,2] is initially... t =0s, a high-pressure, high-density region with a radius of 0.5 exists at the center of the region (1,1). The initial conditions are as follows: .

[0056] Figure 12 and Figure 13 The calculation results of a strongly discontinuous flow field calculated using the method of this invention are presented. The results of the PINN-DG method agree well with the DG method, and compared with the traditional DG method, it can eliminate non-physical oscillations near the discontinuity. This fully demonstrates that this invention has the ability to accurately solve the entire strongly discontinuous flow field.

[0057] This embodiment also provides a PINN and DG data fusion device for addressing the problem of strong fluid discontinuity, including: The computation module is configured to spatially discretize the Euler equations using the DG method and describe the distribution of fluid states within the cells; solve for numerical flux and fluid state variables within the cells using the HLLC flux method; and calculate the mesh node motion velocity based on the computational domain, numerical flux, and fluid state variables within the cells. The condition setting and model building module is configured to set the computational domain, initial conditions and boundary conditions based on the Euler equations of the fluid in the problem, clarify the input and output of the neural network, and initially build the PINN model of the problem. The loss term calculation module is configured to calculate the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network; The region identification module is configured to analyze the gradient of fluid state variables within the analysis unit. When the gradient value is greater than a preset threshold, the computational domain of each problem is identified as a shock wave region; when the gradient value is less than or equal to the preset threshold, the computational domain of each problem is identified as a smooth flow field region. The total loss construction module is configured to introduce DG data in the shock region as the data loss term in the loss function; it combines the partial differential equation loss term, the initial condition loss term, and the boundary condition loss term, and introduces a gradient weight factor to automatically balance the gradient contribution of each loss term to construct the total loss function. The output module is configured to minimize the total loss function, and finally outputs the density, velocity and pressure of the entire flow field space and each time step.

[0058] The specific implementation process of the device can be found in the description of the above method embodiments, and will not be repeated here.

[0059] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0060] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A PINN and DG data fusion method for fluid strongly discontinuous problems, characterized in that, include: S10: The Euler equations are spatially discretized using the DG method, and the distribution of fluid states within the cell is described; the numerical flux and fluid state variables within the cell are solved using the HLLC flux method; based on the computational domain, the numerical flux, and the fluid state variables within the cell, the motion velocity of the grid nodes is calculated. S20. Based on the Euler equations for the fluid in the problem, the computational domain, initial conditions, and boundary conditions are defined, the input and output of the neural network are clarified, and the PINN model of the problem is initially constructed. S30, calculate the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network; S40, Analyze the gradient of the fluid state variables within the unit. When the gradient value is greater than a preset threshold, identify the computational domain of each problem as a shock wave region. When the gradient value is less than or equal to the preset threshold, the computational domain of each corresponding problem is identified as a smooth flow field region; S50, DG data is introduced into the shock wave region as the data loss term in the loss function; the partial differential equation loss term, the initial condition loss term, and the boundary condition loss term are combined, and a gradient weight factor is introduced to automatically balance the gradient contribution of each loss term to construct the total loss function. S60, minimize the total loss function, and finally output the density, velocity and pressure of the entire flow field space and each time step.

2. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 1, characterized in that, In step S10, the spatial discretization of the Euler equations using the DG method and the description of the fluid state distribution within the cell include: The DG method is used to divide the entire computational domain into several non-overlapping quadrilateral units, with each quadrilateral unit serving as an independent computational subdomain. Within each quadrilateral element, Taylor basis functions at the element's centroid are selected as a basis for approximate space, and the distribution of fluid state variables within the element is described in conjunction with the ideal gas law.

3. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 2, characterized in that, In step S10, the step of using the HLLC flux method to solve for numerical flux and intra-cell fluid state variables includes: Numerical flux is calculated at the common interface between adjacent elements using the HLLC Riemann solver. Multiply the conserved form of the Euler equation by the test function, perform a volume integral over each quadrilateral cell, and simultaneously perform a boundary integral on the numerical flux to obtain the weak form corresponding to the DG method. The volume integral and boundary integral terms are calculated using the Gaussian numerical integration method to obtain the flow field state variables at any point within the quadrilateral element.

4. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 3, characterized in that, In step S10, calculating the mesh node motion velocity based on the computational domain, the numerical flux, and the fluid state variables within the cell includes: Using the nodal solver, the motion velocity of each grid node is calculated based on the momentum conservation variables of adjacent quadrilateral elements; The second-order Runge-Kutta method is used to recursively update the fluid state variables and grid node coordinates of each quadrilateral element in the entire computational domain.

5. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 1, characterized in that, In step S20, the step of setting the computational domain, initial conditions, and boundary conditions based on the Euler equations for the fluid in the problem, and defining the input and output of the neural network, includes: The two-dimensional spatiotemporal computational domain for solving the Euler equations is defined as: Ω×[0, T ], where Ω is the spatial region, [0, T [This represents the time interval;] The initial condition is: in t Density when =0 ρ 0( x , y ), x Directional velocity u 0( x , y ), y direction v 0( x , y ),pressure p 0( x , y The distribution of ) and boundary conditions; The input to the neural network is ( x, y, t ),in, x The horizontal position on a two-dimensional plane; y The position in the vertical direction on a two-dimensional plane; t The time that the problem under study has been ongoing; The output of the neural network is ( ρ, u, v, p );in, ρ Density; u for x Directional velocity; v for y Directional velocity; p For pressure.

6. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 1, characterized in that, In step S20, the physical governing equations solved by the PINN model include: ， ， ， In the formula, Let be the physical field variables to be solved; For unknown solution pair time t The derivative; Represents the differential operator; and The computational domain and its boundaries; These are the initial conditions; For boundary operators.

7. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 5, characterized in that, S30 includes: Using automatic differentiation technology, the output of the neural network is continuously differentiated with respect to the input spatiotemporal coordinates to obtain the first-order partial derivatives of each physical quantity; the first-order partial derivatives are substituted into the left side of the Euler equation to calculate the loss term of the partial differential equation. In the network prediction results, the output values ​​at all spatiotemporal points at the initial time are extracted and compared with the given initial condition distribution to calculate the initial condition loss term. Boundary points are selected on the spatial boundary of the computational domain, and corresponding constraints are constructed according to the preset boundary type. The residual between the output of the neural network at the boundary points and the boundary condition requirement value is calculated to obtain the boundary condition loss term.

8. The PINN and DG data fusion method for fluid strong discontinuity problems according to claim 6, characterized in that, The S50 includes: The total loss function of the shock wave region Defined as: ， The total loss function of the smoothed flow field region Defined as: ， in, ， ， ， ， ， In the formula, These are, respectively, the partial differential equation loss term, the boundary condition loss term, the initial condition loss term, and the data loss term; , , , Adaptive weighting coefficients to balance the contributions of various gradients; These represent the number of sampling points for the partial differential equation, the number of sampling points for boundary conditions, the number of sampling points for initial conditions, and the number of data sampling points, respectively; i represents the i-th data point currently being calculated; x represents the spatial coordinate vector; and t represents the physical time variable. These represent the spatial locations of the i-th sampling points used to calculate the partial differential equation loss, boundary condition loss, initial condition loss, and data loss, respectively. These represent the specific times corresponding to the sampling point of the i-th partial differential equation, the boundary condition sampling point, the initial condition sampling point, and the data sampling point, respectively. This refers to the gradient weight factor in the adaptive weighting method. These are the weighting coefficients; This is the velocity vector.

9. A PINN and DG data fusion method for fluid strongly discontinuous problems according to any one of claims 1 to 7, characterized in that, In step S60, minimizing the total loss function includes: The Adam optimizer is used to initially optimize the total loss function, causing the total loss function to decrease rapidly; Switch to the L-BFGS optimizer to perform refined second-order derivative optimization on the total loss function.

10. A PINN and DG data fusion device for fluid strong discontinuity problems, characterized in that, include: The computation module is configured to spatially discretize the Euler equations using the DG method and describe the distribution of the fluid state within the cell; The numerical flux and intra-cell fluid state variables are solved using the HLLC flux method; the mesh node motion velocity is calculated based on the computational domain, the numerical flux, and the intra-cell fluid state variables. The condition setting and model building module is configured to set the computational domain, initial conditions and boundary conditions based on the Euler equations of the fluid in the problem, clarify the input and output of the neural network, and initially build the PINN model of the problem. The loss term calculation module is configured to calculate the partial differential equation loss term, initial condition loss term, and boundary condition loss term of the loss function of the neural network; The region identification module is configured to analyze the gradient of the fluid state variables within the unit, and when the gradient value is greater than a preset threshold, the computational domain of each corresponding problem is identified as a shock wave region. When the gradient value is less than or equal to the preset threshold, the computational domain of each corresponding problem is identified as a smooth flow field region; The total loss construction module is configured to introduce DG data into the shock wave region as the data loss term in the loss function; combine the partial differential equation loss term, the initial condition loss term, and the boundary condition loss term, and introduce a gradient weight factor to automatically balance the gradient contribution of each loss term to construct the total loss function. The output module is configured to minimize the total loss function and ultimately output the density, velocity, and pressure of the entire flow field space and each time step.

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