Flow field solving method based on time sequence physical information neural network

Abstract

The invention provides a flow field solving method based on a time sequence physical information neural network, and belongs to a partial differential equation calculation method. The method comprises the following steps: firstly, determining a flow field motion law and a flow field computational domain based on a fluid motion control equation, configuring data points for training and performing data fusion operation; secondly, constructing a physical constraint residual equation, taking the training data set as input data of a neural network, calculating a data residual term, weighting a calculation result of the physical constraint residual equation and a calculation result of the data residual term, and constructing a data-physical mixed loss function; and finally, constructing a time sequence-based physical information neural network, performing training optimization on parameters of the time sequence-based physical information neural network, completing training and obtaining simulated flow field features. By introducing the time sequence module, the accuracy of flow field dynamic simulation based on the time sequence physical information neural network is improved, the requirement for the model solving speed can be met, and the feasibility and accuracy of the model are guaranteed.

Description

Technical Field

[0001] This invention pertains to partial differential equation calculation methods, specifically a flow field solution method based on a time-series physical information neural network. Background Technology

[0002] The Navier-Stokes equations (NS), as the core set of partial differential equations describing fluid motion, present significant computational challenges for direct numerical solutions due to their strong nonlinearity and multi-scale characteristics. In complex flow scenarios involving turbulent separation and transient eddy shedding, traditional computational fluid dynamics (CFD) methods are computationally expensive and struggle to meet the real-time optimization needs of engineering projects.

[0003] In recent years, Physics-informed Neural Networks (PINNs) have provided a new data-driven paradigm for solving partial differential equations. By embedding the residual terms of the governing equations as physical constraints into the loss function, PINNs can integrate sparse observation data with physical laws, significantly reducing the dependence on high-fidelity data and demonstrating great potential in the reconstruction of steady-state and simple transient flow fields [Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378: 686-707]. However, the classical PINN method still faces significant challenges in handling high Reynolds numbers and complex flow fields. Specifically, the problems are as follows: First, the dynamic feature capture capability is insufficient. Existing methods, relying solely on simple fully connected neural networks, struggle to effectively learn the rapidly evolving multi-scale dynamic features in the flow field, leading to decreased solution accuracy. Second, numerical instability occurs at high Reynolds numbers, and the "low-frequency preference" of neural networks fails to recover high-frequency details of the flow field. Therefore, a novel neural network architecture is urgently needed to enhance the model's ability to capture the spatiotemporal dynamic features of the flow field while strictly ensuring the physical consistency of the solution, thereby achieving high-precision and high-efficiency transient flow field solutions. Summary of the Invention

[0004] To address the shortcomings of current PINN models in simulating flow fields under turbulent conditions, particularly in solving turbulent problems, due to insufficient dynamic information and feature shifts, this invention provides a flow field solution method based on a time-series physical information neural network, aiming to improve the accuracy of PINN model simulations to a certain extent.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for solving the flow field around a network based on time-series physical information includes the following steps:

[0007] Step 1: Determine the computational domain of the flow field, the governing equations, and the corresponding initial and boundary conditions. This provides physical constraints and training objectives for constructing a physical information neural network based on time series data. Specifically:

[0008] The governing equations are a set of partial differential equations describing the flow field motion, preferably the incompressible Navier-Stokes equations, which take the following form:

[0009] (1)

[0010] Where, vector Represents the velocity variable in the flow field. express Flow velocity in the direction, express Flow velocity in the direction; Represents a spatial domain, by Vie Euclidean space represents, For example, spatial dimension Represents a two-dimensional space; Let represent the computational domain of the flow field, and let represent a spatially bounded open set with smooth boundaries. ; It is a pressure field. Indicates the pressure gradient; Represents a time variable. For discrete spatial coordinates; Represents the Reynolds number. It is determined by fluid density Characteristic velocity Flow characteristic length and dynamic viscosity Together they describe the relative importance of inertial forces and viscous forces in fluid motion. and These represent the gradient, divergence, and Laplacian operator, respectively. Represents the velocity variable in the flow field; This represents the initial conditions of the flow field control equations at the initial moment; This represents the boundary conditions that the flow field control equations must satisfy.

[0011] The flow field computation domain The NS equations are defined within the spatial region involved in the flow field motion, and are applied in this flow field computational domain. The initial conditions are established and serve as the governing equations describing the physical laws of the flow field; With boundary conditions The specific physical parameters involved (such as inflow velocity and fluid density) need to be determined based on the specific experimental scenario to be simulated, measured flow field data, or engineering application requirements. These elements together constitute the physical constraints and training objectives of a time-series physical information neural network.

[0012] Step Two: Based on the flow field control equations obtained in Step One, obtain the flow field motion laws and the flow field computational domain, and configure data points for training and data fusion operations within the flow field computational domain. Specifically:

[0013] Step 2.1: Random sampling is performed within and at the boundary of the flow field computational domain to obtain feature data, providing sparse data points of the true physical field values ​​for network architecture training. The feature data includes time... Spatial coordinates Constitutes spacetime coordinates Flow field characteristic data includes the flow field velocity field. and flow field and pressure field Abbreviated as flow field velocity field and flow field and pressure field , representing the spatiotemporal coordinates One-to-one correspondence position With time Feature data. Randomly selected within the flow field computational domain. These spatiotemporal coordinate points serve as spatial and temporal references, and are only used as subscripts in step 2.1 for ease of description. , Distinguish between different spatial coordinates and points in time, i.e. Represents spatial coordinates and time points Above, feature data of the location and time point are sampled; for each spatial point Above, the time window size is extracted from the feasible time range. Subsequently, the data was extracted from... arrive The continuous historical characteristic data sequence, and The feature data at each moment are then used as the same spatial point At the target time The actual flow field value.

[0014] Step 2.2: Using the feature data obtained from the random sampling in Step 2.1, the feature data is fused to obtain the feature data required for training the time-series physical information neural network. Specifically:

[0015] Considering the time window size is The sequence, therefore the flow field during random sampling The velocity sequence before time step is represented as The pressure sequence is represented as .in, Indicates in Continuity before time The abbreviation for the flow field velocity sequence at each time step, with superscript. The continuous time step is determined during the experiment, and the subscript represents the time step. The characteristic of the last moment of the flow field velocity sequence, i.e., in spatial coordinates. Above, respectively with time points The corresponding flow velocity field; Shown Continuity before the moment The time-step flow field pressure sequence is abbreviated as superscript. The continuous time step is determined during the experiment, and the subscript represents the time step. The characteristic of the last moment of the flow field pressure sequence, i.e., in spatial coordinates. Above, respectively with time points The corresponding flow field and pressure field. That is... express Flow field velocity field transpose at time t , express Flow field velocity field transpose at time t , express Flow field velocity field transpose at time t Similarly, express The flow field pressure field at any given moment express The flow field pressure field at any given moment express The flow field and pressure field at any given moment.

[0016] The sampled feature data is fused with the corresponding spatiotemporal coordinate data to obtain the training dataset. ,in , Representing the spacetime coordinates respectively The speed and pressure feature data are used as the true values ​​of the training target feature data. Spatial coordinates contained within the flow field computation domain and time and in spatial coordinates superior One-to-one correspondence before time Flow field velocity sequence of historical flow field characteristics for each time window and flow field pressure sequence The goal of training a time-series physical information-based neural network is to fuse the training dataset. The spatiotemporal coordinates and flow field feature sequences enable a neural network based on time-series physical information to learn the temporal evolution of the flow field and solve for the flow field. Flow field characteristic data at any given time.

[0017] Step 2.3: Perform random sampling within the flow field computation domain to obtain the configuration point dataset. Each sample point is a spatiotemporal coordinate. The configuration point dataset The physical constraint residuals (the equations for which are given in step three) are used to calculate the physical constraint residuals of the time-series physical information neural network, which force the learning results of the time-series physical information neural network to obey the basic physical laws.

[0018] Step 3: Based on the governing equations of fluid motion defined in Step 1, i.e., Equation (1), are used as the physical constraints of the flow field motion, and the physical constraint residual equations are constructed, as specifically described in Step 3.1; based on the training dataset obtained in Step 2... As input data for a time-series physical information neural network, the flow field at the target time obtained by the time-series physical information neural network is compared with the true values ​​in the dataset to calculate the data residual term; the configuration point dataset obtained in step two is then used. Calculate the physical constraint residual equations, and then weight the results of the physical constraint residual equation calculations with the data residual term calculations to obtain the total loss function of the time-series physical information neural network. Specifically:

[0019] Step 3.1: Construct the residual expression for solving the Navier-Stokes equations as follows:

[0020] (2)

[0021] This represents the residual due to mass conservation. Represents the residuals of the momentum equation; Represents a spatial domain, by Vie Euclidean space represents, For spatial dimension; when hour, Represents a one-dimensional spatial domain; Represents the velocity field of the flow field; It is a pressure field. Indicates the pressure gradient; and These represent the gradient, divergence, and Laplacian operator, respectively. Specifically:

[0022] Calculate the constructed formula (2), and select the mean squared error (MSE) as the evaluation index of the loss function. Then the physical constraint residual equation is as follows: As shown:

[0023] (3)

[0024] in, Represents the physical constraint residual equation; The cardinality of the set given in this article; set This represents the configuration point dataset; This represents the mass-conserving residual after training with a neural network based on time-series physical information; This represents the residual of the momentum equation after training with a neural network based on time-series physical information; express consecutive moments before A sequence of flow field velocity characteristics for historical flow field features within a time window; express consecutive moments before A sequence of historical flow field characteristics within a given time window; symbol express -norm; This represents the parameters of a neural network based on time-series physical information.

[0025] Step 3.2: Construct the data residual term, with the following expression:

[0026] (4)

[0027] in, For data residuals; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and velocity field at any given moment; This indicates the solution obtained after training a neural network based on time-series physical information. Flow field and pressure field at any time; symbol express -norm; express consecutive moments before A sequence of flow field velocity characteristics for historical flow field features within a time window; express consecutive moments before Flow field pressure sequence of historical flow field characteristics over a time window; This represents the parameters of a neural network based on time-series physical information.

[0028] Step 3.3: Construct the data-physical hybrid loss function, specifically:

[0029] (5)

[0030] in, For data loss, The physical constraint residual equation. The data-physical hybrid loss function. Optimization for training neural networks based on time-series physical information.

[0031] Step 4: Construct a neural network based on time-series physical information, including a time-series module and a fully connected neural network module. Specifically:

[0032] Step 4.1: Construct a temporal feature extraction module, where the input data is spatial coordinates. Shanghe Continuous feature data before time step Historical velocity sequence of a time window and historical pressure sequence , mapped to ,in Represents the time-series module operator. Represents the network parameters of the timing module. and These represent the dynamic information of the velocity and pressure characteristics of the flow field, respectively.

[0033] Step 4.2: Establish a nonlinear mapping relationship from spatiotemporal coordinates and dynamic information to transient physical fields, that is, spatiotemporal coordinates and the dynamic features obtained from the time-series module training in step 4.1. and With input, the mapping relationship is as follows: ,in Represents a nonlinear mapping operator. The network parameters represent the nonlinear modules of a fully connected neural network. and They are respectively and Abbreviated form, i.e. This indicates the solution obtained after training a neural network based on time-series physical information. Flow field velocity field at any time This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and pressure field at any given moment; The parameters of the neural network based on time-series physical information are: ,in Represents the network parameters of the timing module. This represents the parameters of the nonlinear modules in a fully connected neural network.

[0034] Combining steps 4.1 and 4.2, the overall input-output mapping relationship of the time-series physical information-based neural network can be simplified as follows: ,in Represents a neural network operator based on time-series physical information. These represent the parameters of a neural network based on time-series physical information. Represents the network parameters of the timing module. Parameters representing the nonlinear modules of a fully connected neural network; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and velocity field at any given moment; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and pressure field at any given moment.

[0035] Step 5: Based on the data-physical hybrid loss function in Step 3 And step four involves constructing a neural network architecture based on time-series physical information, training and optimizing the parameters of the neural network based on time-series physical information, specifically:

[0036] Assumption and The network parameters are in the first place. In each iteration, the parameters of the network architecture are optimized during training based on the following mathematical formula:

[0037] (6)

[0038] in, The parameters of the timing module are in the first... Iteration; The parameter representing the nonlinear module of a fully connected neural network is in the th order. Iteration; Indicates the network parameters of the timing module; These represent the parameters of the nonlinear modules in a fully connected neural network. Indicates data loss. This represents the physical constraint residual equation.

[0039] Step Six: Determine the computational domain, governing equations, and corresponding initial and boundary conditions based on Step One; Step Two yields the training dataset. and configuration point dataset Step 3 yields the data-physical hybrid loss function; Step 4 completes the construction of a neural network based on time-series physical information; Step 5 yields the parameter optimization formula for the neural network based on time-series physical information. Therefore, in this step, the data-physical hybrid loss function described in Step 3... The value is lower than the preset threshold When the model converges, the training is complete and the simulated flow field characteristics are obtained. Those skilled in the art can reasonably determine this threshold based on the required level of accuracy (e.g., approximate solutions in engineering applications versus exact solutions in high-fidelity simulations). Preferred to .

[0040] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0041] (1) By introducing a time-series module, as described in the data fusion in step two and the network architecture construction in step four, this invention can explicitly learn the dynamic patterns in the historical flow field compared with PINN which only uses transient spatiotemporal coordinates. Therefore, it compensates for the feature offset and error accumulation problems caused by the lack of dynamic prior information in traditional PINN, and improves the accuracy of flow field dynamic simulation based on time-series physical information neural network.

[0042] (2) The time-series physical information neural network proposed in this invention, compared with the pure data-driven model, constructs physical constraints of the control equation in step three, introduces the residual into the training process, and ensures that the solution results of the network structure have physical consistency.

[0043] (3) The adaptive parameter update strategy proposed in this invention, such as the parameter optimization in step five, ensures that the model parameters are adaptively adjusted during network training and ultimately obtains the optimal model parameters. This satisfies the model solution speed requirements while ensuring the feasibility and accuracy of the model. Attached Figure Description

[0044] Figure 1 The flowchart shows a flow field solution method based on a neural network of time-series physical information.

[0045] Figure 2 This is a schematic diagram of the network structure of the timing module.

[0046] Figure 3 This is a schematic diagram of a network architecture based on time-series physical information networks;

[0047] Figure 4 This refers to the point-by-point error between the flow field data obtained using a time-series physical information network and the actual values. Detailed Implementation

[0049] To clarify the implementation details of the technical solution of this invention, in accordance with the appendix... Figure 1-3 Preferred embodiments will be described. It should be specifically noted that the embodiments described in this specification are merely exemplary supports for the scope of the claims and are not intended to limit the invention.

[0050] refer to Figure 1 This invention provides a flow field solution method based on a time-series physical information neural network, comprising the following steps:

[0051] Step 1: Determine the computational domain of the flow field, the governing equations, and the corresponding initial and boundary conditions. This provides physical constraints and training objectives for constructing a physical information neural network based on time series data. Specifically:

[0052] This implementation example uses the Reynolds number. The following is a specific application scenario of two-dimensional turbulent flow around a cylindrical Karman vortex street, aiming to solve the flow field using a time-series-based physical information neural network. The cylinder diameter is the characteristic length. Its center is located at the origin of the coordinate system. The spatial computational domain is defined as a rectangular domain. The time computation domain is a finite interval. and with a fixed step size Discretization is performed for data sampling and network architecture evaluation.

[0053] The governing equations are a set of partial differential equations describing the fluid motion, preferably the incompressible Navier-Stokes equations, which take the following form:

[0054] (1)

[0055] Where, vector Represents the velocity variable in the flow field. express Flow velocity in the direction, express Flow velocity in the direction; Represents a spatial domain, by Vie Euclidean space represents, For example, spatial dimension Represents a two-dimensional space; Let represent the computational domain of the flow field, and let represent a spatially bounded open set with smooth boundaries. ; It is a pressure field. Indicates the pressure gradient; Represents a time variable; For discrete spatial coordinates; Represents the Reynolds number, which describes the relative importance of inertial and viscous forces in fluid motion. and These represent the gradient, divergence, and Laplacian operator, respectively. Represents the velocity variable in the flow field; This represents the initial conditions of the flow field control equations at the initial moment; This represents the boundary conditions that the flow field control equations must satisfy.

[0056] The flow field computation domain The spatial region involved in the flow field motion The NS equations in this flow field computation domain The initial conditions are established and serve as the governing equations describing the physical laws of the flow field; With boundary conditions The specific physical parameters involved (such as inflow velocity, fluid density, etc.) and the flow field reference data required in this embodiment are all derived from the high-precision direct numerical simulation calculation results.

[0057] Step Two: Based on the flow field control equations obtained in Step One, obtain the flow field motion laws and the flow field computational domain, and configure data points for training and data fusion operations within the flow field computational domain. Specifically:

[0058] Step 2.1: Random sampling is performed within and at the boundary of the flow field computational domain to obtain feature data, providing sparse data points of the true physical field values ​​for network architecture training. The feature data includes time... Spatial coordinates Constitutes spacetime coordinates Flow field characteristic data includes the flow field velocity field. and flow field and pressure field Abbreviated as flow field velocity field and flow field and pressure field , representing the spatiotemporal coordinates One-to-one correspondence position With time Feature data. Randomly selected within the flow field computational domain. These spatiotemporal coordinate points serve as spatial and temporal references, and are only used as subscripts in step 2.1 for ease of description. , Distinguish between different spatial coordinates and points in time, i.e. Represents spatial coordinates and time points Above, feature data of the location and time point are sampled; for each spatial point Above, the time window size is extracted from the feasible time range. Subsequently, the data was extracted from... arrive The continuous historical characteristic data sequence, and The feature data at each moment are then used as the same spatial point At the target time The actual flow field value.

[0059] Step 2.2: Using the feature data obtained from the random sampling in Step 2.1, the feature data is fused to obtain the feature data required for training the time-series physical information neural network. Specifically:

[0060] Considering the time window size is The sequence, therefore the flow field during random sampling The velocity sequence before time step is represented as The pressure sequence is represented as .in, Indicates in The flow field velocity sequence of the six consecutive time steps prior to time point is abbreviated as follows: subscript The characteristic of the last moment of the flow field velocity sequence, i.e., in spatial coordinates. Above, respectively with time points The corresponding flow velocity field; Shown Continuity before the moment The time-step flow field pressure sequence is abbreviated as follows: the superscript 6 indicates that the continuous time step is determined during the experiment; the subscript... The characteristic of the last moment of the flow field pressure sequence, i.e., in spatial coordinates. Above, respectively with time points The corresponding flow field and pressure field. That is... express Flow field velocity field transpose at time t , express Flow field velocity field transpose at time t , express Flow field velocity field transpose at time t Similarly, express The flow field pressure field at any given moment express The flow field pressure field at any given moment express The flow field and pressure field at any given moment.

[0061] The sampled feature data is fused with the corresponding spatiotemporal coordinate data to obtain the training dataset. ,in , Representing the spacetime coordinates respectively The speed and pressure feature data are used as the true values ​​of the training target feature data. Spatial coordinates contained within the flow field computation domain and time and in spatial coordinates superior Flow field velocity sequence of historical flow field characteristics corresponding to six consecutive time windows prior to time point 1. and flow field pressure sequence The goal of training a time-series physical information-based neural network is to fuse the training dataset. The spatiotemporal coordinates and flow field feature sequences enable a neural network based on time-series physical information to learn the temporal evolution of the flow field and solve for the flow field. Flow field characteristic data at any given time.

[0062] Step 2.3: Perform random sampling within the flow field computation domain to obtain the configuration point dataset. Each sample point is a spatiotemporal coordinate. The configuration point dataset The physical constraint residuals (the equations for which are given in step three) are used to calculate the physical constraint residuals of the time-series physical information neural network, which force the learning results of the time-series physical information neural network to obey the basic physical laws.

[0063] Step 3: Based on the governing equations of fluid motion defined in Step 1, i.e., Equation (1), are used as the physical constraints of the flow field motion, and the physical constraint residual equations are constructed, as specifically described in Step 3.1; based on the training dataset obtained in Step 2... As input data for a time-series physical information neural network, the flow field at the target time obtained by the time-series physical information neural network is compared with the true values ​​in the dataset to calculate the data residual term; the configuration point dataset obtained in step two is then used. Calculate the physical constraint residual equation, and weight the result of the physical constraint residual equation calculation with the result of the data residual term calculation to obtain the specific total loss function of the neural network based on time series physical information:

[0064] This implementation example uses the autograd() module in the PyTorch framework to calculate the first-order partial derivatives of the network output data with respect to spatiotemporal coordinates and time. Then, the second-order partial derivatives were calculated using the first-order partial derivatives. Substitute the calculation results into formulas (3) and (4) to calculate the residual terms based on the time series physical information neural network.

[0065] Step 3.1: Construct the residual expression for solving the Navier-Stokes equations as follows:

[0066] (2)

[0067] This represents the residual due to mass conservation. Represents the residuals of the momentum equation; Represents a spatial domain, by Vie Euclidean space represents, For spatial dimension; when hour, Represents a one-dimensional spatial domain; Represents the velocity field of the flow field; It is a pressure field. Indicates the pressure gradient; and These represent the gradient, divergence, and Laplacian operator, respectively. Specifically:

[0068] Calculate the constructed formula (2), and select the mean squared error (MSE) as the evaluation index of the loss function. Then the physical constraint residual equation is as follows: As shown:

[0069] (3)

[0070] in, Represents the physical constraint residual equation; The cardinality of the set given in this article; set This represents the configuration point dataset; This represents the mass-conserving residual after training with a neural network based on time-series physical information; This represents the residual of the momentum equation after training with a neural network based on time-series physical information; express The flow field velocity sequence of historical flow field characteristics over six consecutive time windows prior to the specified time. express The flow field pressure sequence of historical flow field characteristics over six consecutive time windows prior to time point; symbol express -norm; This represents the parameters of a neural network based on time-series physical information.

[0071] Step 3.2: Construct the data residual term, with the following expression:

[0072] (4)

[0073] in, For data residuals; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and velocity field at any given moment; This indicates the solution obtained after training a neural network based on time-series physical information. Flow field and pressure field at any time; symbol express -norm; express The flow field velocity sequence of historical flow field characteristics over six consecutive time windows prior to the specified time. express The flow field pressure sequence of historical flow field characteristics over six consecutive time windows prior to the specified moment; This represents the parameters of a neural network based on time-series physical information.

[0074] Step 3.3: Construct the data-physical hybrid loss function, specifically:

[0075] (5)

[0076] in, For data loss, The physical constraint residual equation. The data-physical hybrid loss function. Optimization for training neural networks based on time-series physical information.

[0077] Step 4: Construct a neural network based on time-series physical information, including a time-series module and a fully connected neural network module. Specifically:

[0078] Step 4.1: Construct a temporal feature extraction module. This scheme uses one-dimensional convolutional neural networks (1D-CNNs) to extract dynamic features of the time-series flow field. The kernel size is fixed at 3. The first layer generates a 16-channel feature map, the second layer outputs a 32-channel feature map, and the final layer uses a max-pooling layer to compress the temporal dimension and extract flow evolution features. The activation function chosen is ReLU. The input data is in spatial coordinates... Shanghe Continuous feature data prior to time point, consisting of historical velocity sequences spanning six time windows. and historical pressure sequence , mapped to ,in Represents the time-series module operator. Represents the network parameters of the timing module. and These represent the dynamic information of the velocity and pressure characteristics of the flow field, respectively.

[0079] Step 4.2: Establish a nonlinear mapping relationship from spatiotemporal coordinates and dynamic information to the transient physical field. In this implementation example, a fully connected neural network is selected as the nonlinear mapping module. Preliminary flow field characteristic simulation results are obtained through the mapping of neurons. Among them, the fully connected hidden layer... Each hidden layer contains 30 neurons, and the activation function chosen is tanh. Spatiotemporal coordinates... and the dynamic features obtained from the time-series module training in step 4.1. and With input, the mapping relationship is as follows: ,in Represents a nonlinear mapping operator. The network parameters represent the nonlinear modules of a fully connected neural network. and They are respectively and Abbreviated form, i.e. This indicates the solution obtained after training a neural network based on time-series physical information. Flow field velocity field at any time This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and pressure field at any given moment; The parameters of the neural network based on time-series physical information are: ,in Represents the network parameters of the timing module. These represent the parameters of the nonlinear modules in a fully connected neural network. The timing module of this invention is as follows: Figure 2 As shown.

[0080] Combining steps 4.1 and 4.2, the overall input-output mapping relationship of the time-series physical information-based neural network can be simplified as follows: ,in Represents a neural network operator based on time-series physical information. These represent the parameters of a neural network based on time-series physical information. Represents the network parameters of the timing module. Parameters representing the nonlinear modules of a fully connected neural network; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and velocity field at any given moment; This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and pressure field at any given time. The network architecture of this invention, based on a time-series physical information network, is as follows: Figure 3 As shown.

[0081] Step 5: Based on the data-physical hybrid loss function in Step 3 And step four involves constructing a neural network architecture based on time-series physical information, training and optimizing the parameters of the neural network based on time-series physical information, specifically:

[0082] This implementation example uses an adaptive moment estimator (Adam) to iteratively update parameters, with a learning rate set to 0.001. Assume... and The network parameters are in the first place. In each iteration, the parameters of the network architecture are optimized during training based on the following mathematical formula:

[0083] (6)

[0084] in, The parameters of the timing module are in the first... Iteration; The parameter representing the nonlinear module of a fully connected neural network is in the th order. Iteration; Indicates the network parameters of the timing module; These represent the parameters of the nonlinear modules in a fully connected neural network. Indicates data loss. This represents the physical constraint residual equation.

[0085] Step Six: Determine the computational domain, governing equations, and corresponding initial and boundary conditions based on Step One; Step Two yields the training dataset. and configuration point dataset Step 3 yields the data-physical hybrid loss function; Step 4 completes the construction of a neural network based on time-series physical information; Step 5 yields the parameter optimization formula for the neural network based on time-series physical information. Therefore, in this step, the data-physical hybrid loss function described in Step 3... The value is lower than the preset threshold When the model has converged, training is complete and the simulated flow field characteristics are obtained, the model is considered to have converged. Those skilled in the art can reasonably determine this threshold based on the required level of accuracy (e.g., approximate solutions in engineering applications versus exact solutions in high-fidelity simulations).

[0086] After the neural network based on time-series physical information is trained, the spatiotemporal coordinates in the benchmark dataset will be used. By inputting historical data within the corresponding time window into the model, the target values ​​of the flow field velocity and pressure fields under complete spatiotemporal coordinates are obtained. To quantify the model accuracy, simulated values ​​are calculated. Compared with baseline flow field characteristic data relative - Norm error:

[0087] (7)

[0088] The two-dimensional flow field characteristics around a cylinder under simulated turbulent conditions are obtained by the time-series physical information neural network model proposed in this invention. relative - Norm errors are respectively The relative results obtained by comparing with the baseline PINN model -norm error The point-by-point error between the flow field data solved using the time-series physical information network of this invention and the true value is as follows: Figure 4 As shown, it provides accurate dynamic information about the flow field.

[0089] In summary, the method described in this invention learns the transient characteristics of the flow around the flow using historical flow field data within a time window, and compensates for the missing high-frequency and dynamic information in the discrete data while maintaining the constraints of the Navier-Stokes equations, thereby reducing the iterative computation requirements for solving the flow field.

[0090] The above embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. A method for solving a flow field based on a time series physical information neural network, characterized in that, The flow field solving method comprises the following steps: Step 1: determining a flow field calculation domain, control equations, and corresponding initial conditions and boundary conditions, obtaining control equations of fluid motion, and providing physical constraints and training targets for constructing a physical information neural network based on time series; Step 2: obtaining a flow field motion law and a flow field calculation domain based on the flow field control equations obtained in step 1, and configuring data points and data fusion operations for training in the flow field calculation domain; specifically: Step 2.1: randomly sampling feature data in the flow field calculation domain and on the boundary to provide sparse data points of physical field true values for network architecture training; Step 2.2: The feature data obtained by random sampling in step 2.1 is fused to obtain the feature data required for training the time series physical information neural network; the sampled feature data is fused with the corresponding space-time coordinate data to obtain a training data set ; Step 2.3: Random sampling in the flow field calculation domain to obtain the configuration point dataset where each sample point is a space-time coordinate ; Step three: take the control equation of step one as the physical constraint of the flow field motion, and construct a physical constraint residual equation; based on the training data set obtained in step two As the input data of the time series physical information neural network, the flow field at the target time solved by the time series physical information neural network is compared with the true value in the data set, so as to calculate the data residual term; using the configuration point data set obtained in step two Calculate the physical constraint residual equation, and weight the calculation results of the physical constraint residual equation and the data residual term to construct a data-physical hybrid loss function ; Step 4: constructing a time series-based physical information neural network, including a time series module and a fully connected neural network module; Step five: data-physical hybrid loss function based on step three and step four, based on the construction of the time series physical information neural network architecture, the parameters of the time series physical information neural network are trained and optimized. Step six: when the value of the data-physical hybrid loss function in step three is lower than a preset threshold , it is considered that the model has converged, the training is completed, and the simulated flow field characteristics are obtained.

2. The time-series physical information neural network based flow field solving method according to claim 1, wherein, The step 1 specifically: The control equations of fluid motion are partial differential equation sets for describing the flow field motion law, specifically the incompressible NS equation, which has the following form: （1）； Where, vector Represents the velocity variable in the flow field. express Flow velocity in the direction, express Flow velocity in the direction; Represents a spatial domain, by Vie Euclidean space represents, For example, spatial dimension Represents a two-dimensional space; Let represent the computational domain of the flow field, and let represent a spatially bounded open set with smooth boundaries. ; It is a pressure field. Indicates the pressure gradient; Represents a time variable. For discrete spatial coordinates; Represents the Reynolds number. It is determined by fluid density Characteristic velocity Flow characteristic length and dynamic viscosity Together they describe the relative importance of inertial forces and viscous forces in fluid motion; and These represent the gradient, divergence, and Laplacian operator, respectively. Represents the velocity variable in the flow field; This represents the initial conditions of the flow field control equations at the initial moment; This represents the boundary conditions that the flow field control equations must satisfy.

3. The time-series physical information neural network based flow field solving method according to claim 2, characterized in that, the flow field calculation domain a spatial region involved in the flow field motion; The NS equation is established in the flow field calculation domain and serves as a control equation describing the physical law of the flow field; the initial condition and the boundary condition involved in the specific physical parameters need to be determined according to the specific experimental scene, the measured flow field data or the engineering application requirements required for simulation; the above elements together constitute the physical constraints and training targets of the time series physical information neural network.

4. The time-series physical information neural network based flow field solving method according to claim 3, characterized in that, In the step 2.1: The feature data includes time with spatial coordinates , constituting a space-time coordinate ; The flow field characteristic data includes a flow field velocity field and a flow field pressure field , which are abbreviated as a flow field velocity field and a flow field pressure field , respectively, and represent characteristic data at a position corresponding to a space-time coordinate and a time ; a space-time coordinate point is randomly selected in the flow field calculation domain as a space and time reference, and is distinguished by a subscript , , wherein represents that the characteristic data at the position and the time point is sampled; for each space point , a time window size of is intercepted from a feasible time range, and then a continuous historical characteristic data sequence from to is intercepted, and the characteristic data at is taken as the real flow field value at the same space point at a target time .

5. The time-series physical information neural network based flow field solving method according to claim 4, characterized in that, In the step 2.2: Consider the time window size is , so the velocity sequence of the flow field before the time is represented as , and the pressure sequence is represented as ; wherein represents the flow field velocity sequence of the continuous time steps before the time , and the superscript represents the continuous time steps determined in the experimental process, and the subscript is the feature of the last time of the flow field velocity sequence, that is, on the spatial coordinates , respectively, the corresponding flow field velocity field at the time point ; represents the flow field pressure sequence of the continuous time steps before the time , and the superscript represents the continuous time steps determined in the experimental process, and the subscript is the feature of the last time of the flow field pressure sequence, that is, on the spatial coordinates , respectively, the corresponding flow field pressure field at the time point ; that is represents the flow field velocity field at the time , represents the flow field velocity field at the time , represents the flow field velocity field at the time ; similarly represents the flow field pressure field at the time , represents the flow field pressure field at the time , The sampled feature data is fused with the corresponding spatiotemporal coordinate data to obtain the training dataset. ,in , Representing the spacetime coordinates respectively The speed and pressure feature data are used as the true values ​​of the training target feature data; among them, Spatial coordinates contained within the flow field computation domain and time and in spatial coordinates superior One-to-one correspondence before time Flow field velocity sequence of historical flow field characteristics for each time window and flow field pressure sequence The goal of training a time-series physical information-based neural network is to integrate the training dataset. The spatiotemporal coordinates and flow field feature sequences enable a neural network based on time-series physical information to learn the temporal evolution of the flow field and solve for the flow field. Flow field characteristic data at any given time.

6. The time-series physical information neural network based flow field solving method according to claim 5, wherein, In step 2.3, the configuration point dataset a physical constraint residual for computing the time-series based physical information neural network.

7. The time-series physical information neural network based flow field solving method according to claim 6, characterized in that, The step 3 specifically: Step 3.1: constructing a residual expression for solving the NS equation as follows: （2）； denotes the mass conservation residual; denotes the momentum equation residual; denotes the spatial domain, by denotes the Euclidean space of dimension denotes the spatial dimension; when denotes the one-dimensional spatial domain; denotes the pressure gradient; denotes the flow field velocity field; is the pressure field, denotes the pressure gradient; and denote the gradient, divergence and Laplacian operators, respectively; in particular: The constructed formula (2) is calculated, and mean squared error (MSE) is selected as an evaluation index of the loss function, and the physical constraint residual equation is as shown in the formula ​ （3）； wherein, represents a physical constraint residual equation; denotes cardinality of a given set; set represents a configuration point dataset; represents a mass conservation residual trained by a time-series based physical information neural network; represents a momentum equation residual trained by a time-series based physical information neural network; represents a flow field velocity sequence of historical flow field features of consecutive time windows before the time instant t; represents a flow field pressure sequence of historical flow field features of consecutive time windows before the time instant t; notation represents a L2-norm; denotes parameters of a time-series based physical information neural network; Step 3.2: constructing a data residual term, which has the following expression: （4）； wherein, is a data residual; denotes a flow field velocity field solved by a time series based physical information neural network after training at time t; denotes a flow field pressure field solved by a time series based physical information neural network after training at time t; denotes a norm; denotes a flow field velocity sequence of historical flow field features of consecutive time windows before time t; denotes a flow field pressure sequence of historical flow field features of consecutive time windows before time t; denotes a parameter of a time series based physical information neural network; Step 3.3: constructing a data-physical hybrid loss function, specifically: （5）； wherein, is a data loss, is a physical constraint residual equation; a data-physical hybrid loss function optimization for neural network training based on time series physical information.

8. The time-series physical information neural network based flow field solving method according to claim 7, characterized in that, The step 4 specifically: Step 4.1: Construct a temporal feature extraction module, where the input data is spatial coordinates. Shanghe Continuous feature data before time step Historical velocity sequence of a time window and historical pressure sequence , mapped to ,in Represents the time-series module operator. Represents the network parameters of the timing module. and These represent the dynamic information of the velocity and pressure characteristics of the flow field, respectively. Step 4.2: Establish a nonlinear mapping relationship from spatiotemporal coordinates and dynamic information to transient physical fields, that is, spatiotemporal coordinates and the dynamic features obtained from the time-series module training in step 4.

1. and With input, the mapping relationship is as follows: ,in Represents a nonlinear mapping operator. Network parameters representing the nonlinear modules of a fully connected neural network. and They are respectively and Abbreviated form, i.e. This indicates the solution obtained after training a neural network based on time-series physical information. Flow field velocity field at any time This indicates the solution obtained after training a neural network based on time-series physical information. The flow field and pressure field at any given moment; The parameters of the neural network based on time-series physical information are: ,in Represents the network parameters of the timing module. These represent the parameters of the nonlinear modules in a fully connected neural network. Combining steps 4.1 and 4.2, the overall time-series based physical information neural network input and output mapping relationship is simplified as wherein represents the time-series based physical information neural network operator, represents the parameters of the time-series based physical information neural network; represents the network parameters of the time-series module, represents the parameters of the fully connected neural network module nonlinear module; represents the flow field velocity field at time t solved by the time-series based physical information neural network after training; represents the flow field pressure field at time t solved by the time-series based physical information neural network after training.

9. The time-series physical information neural network based flow field solving method according to claim 8, characterized in that, The step 5 specifically: Assume and are network parameters at the iteration of the first , the parameters of the network architecture are optimized during the training process based on the following mathematical formula: （6）； wherein, parameters of the recurrent module are iterated at step parameters of the nonlinear module of the fully connected neural network module are iterated at step network parameters of the recurrent module; parameters of the nonlinear module of the fully connected neural network module; data loss, physical constraint residual equation.​​ 10. The time-series physical information neural network based flow field solving method according to claim 9, characterized in that, The preset threshold Preferably To .

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