A neural network prediction method and device for two-dimensional quantized vortex dynamics

Abstract

The application provides a neural network prediction method and device for two-dimensional quantized vortex dynamics, and the method comprises the following steps: S1. For a flow field to be predicted, a two-dimensional vortex dynamics equation of the flow field is determined, and the two-dimensional vortex dynamics equation is converted into a Schrodinger equation of a quantum system; S2. Wave function evolution data of the quantum system is obtained according to the Schrodinger equation, and a wave function data set is generated; S3. A full connection layer neural network model is constructed, and phase integration is performed on the full connection layer neural network model; S4. A normalization factor is introduced into the full connection neural network model; S5. The full connection layer neural network model is trained by using the wave function data set; S6. The trained full connection layer neural network model is used to predict a wave function according to a given initial value; and S7. The predicted wave function is converted into flow field evolution. According to the method provided by the application, the conservation law of the system can be ensured in the prediction of the Schrodinger system, so that the accuracy of the flow field prediction can be ensured in a long period.

Description

Technical Field

[0001] The present invention belongs to the field of artificial intelligence applications and relates to a neural network prediction method and device for two-dimensional quantized eddy dynamics. Background Art

[0002] Quantum mechanics reveals the fundamental laws of the microscopic world and provides a theoretical foundation for understanding the structure and properties of atoms, molecules, condensed matter, atomic nuclei, and elementary particles. The principles of quantum mechanics can be used not only to describe microscopic particles but also to construct models of classical mechanics. For example, many researchers have mapped the physical quantities of fluid dynamics to wave function fields, thereby converting the fluid dynamics equations into the Schrödinger equation. Solving the Schrödinger equation allows the flow characteristics of a flow field to be predicted, providing a new approach to flow field prediction. Therefore, accurately predicting the wave function evolution of a quantum system is crucial for accurately predicting flow fields.

[0003] In recent years, neural networks have garnered widespread attention for solving partial differential equations (PDEs), owing to their powerful nonlinear modeling and high-dimensional processing capabilities. However, purely data-driven neural networks face multiple limitations in solving PDEs, including their reliance on large amounts of high-quality data, the risk of overfitting, a lack of physical consistency and numerical stability, limited interpretability due to their black-box nature, and high computational resource requirements. These issues restrict their effective application in complex physical systems, often requiring the integration of physical knowledge or other methods to achieve more reliable results.

[0004] The wave function evolution of a quantum system must satisfy two physical constraints: global phase independence and wave function normalization. Classical numerical calculation methods, due to numerical dissipation, require controlling the time step to ensure that the wave function meets these physical constraints during long solution times. This is even more difficult to achieve with purely data-driven neural networks. Therefore, it is necessary to design neural networks that conform to the characteristics of quantum systems to accurately predict their wave function evolution and, in turn, to accurately deduce the flow field. Summary of the Invention

[0005] To solve the above problems, the purpose of the present invention is to provide a neural network prediction method and device for two-dimensional quantized vortex dynamics. By adding phase integration and normalization factors during the neural network training process, it is ensured that the global phase invariance and normalization conditions are met during the wave function prediction process, thereby improving the accuracy in the long-term prediction process.

[0006] The technical solution adopted in the present invention is as follows:

[0007] A neural network prediction method for two-dimensional quantized eddy dynamics comprises the following steps:

[0008] S1. Determine the two-dimensional vortex dynamics equation for the flow field to be predicted, and convert the two-dimensional vortex dynamics equation into the Schrödinger equation of the quantum system;

[0009] S2. Obtain the wave function evolution data of the quantum system based on the Schrödinger equation and generate a wave function dataset;

[0010] S3. Construct a fully connected neural network model and perform phase integration on the fully connected neural network model so that the prediction of the wave function of the quantum system by the fully connected neural network model satisfies the global phase invariance.

[0011] S4. Introduce a normalization factor into the fully connected neural network model so that the prediction of the wave function of the quantum system by the fully connected neural network model satisfies the normalization condition;

[0012] S5. Use the wave function dataset to train the fully connected layer neural network model;

[0013] S6. Use the trained fully connected neural network model to predict the wave function based on the given initial value;

[0014] S7. Convert the predicted wave function into flow field evolution.

[0015] Furthermore, for the flow field to be predicted, the two-dimensional vortex dynamics equation is determined, and the two-dimensional fluid vortex dynamics equation is converted into the Schrödinger equation of the quantum system. The specific steps are:

[0016] For the flow field to be predicted, its two-dimensional vortex dynamics equation is:

[0017]

[0018] Among them, φ j =x j (t)+iy j (t), (x j (t),y j (t)) represents the position of the jth vortex particle in the flow field to be predicted at time t, i represents the imaginary unit; φ i =x i (t)+iy i (t), (x i (t),y i (t)) represents the position of the i-th vortex particle in the flow field to be predicted at time t, Γ i is the vortex intensity corresponding to the i-th vortex particle in the flow field to be predicted; N p is the total number of vortex particles in the flow field to be predicted;

[0019] Perform a nonlinear transformation on the vortex dynamics equations for a two-dimensional fluid:

[0020]

[0021] Among them, ψ i , ψ j , ψ k are the wave functions corresponding to the vortex particles i, j and k in the flow field to be predicted, They represent the wave function ψ i , ψ j and ψ k The complex conjugate of Γ k is the vortex intensity corresponding to the vortex particle k in the flow field to be predicted; i represents the imaginary unit; N p is the total number of vortex particles in the flow field to be predicted; c0 is an arbitrary complex number; α is a normalized real number to ensure that the wave function satisfies

[0022] Convert the vortex dynamics equations into the Schrödinger equation for quantum systems:

[0023]

[0024] Among them, ψ m , ψ n are the wave functions corresponding to the vortex particles m and n in the flow field to be predicted, They represent the wave function ψ m and ψ n The complex conjugate of Γ n is the vortex intensity corresponding to the vortex particle n in the flow field to be predicted.

[0025] Furthermore, the wave function evolution data is obtained through testing or numerical solution methods.

[0026] Furthermore, the specific steps of the numerical solution method are:

[0027] Given the initial wave function for training and time step Δt, N p is the total number of vortex particles, and the result of solving N time steps by applying the fourth-order Runge-Kutta method is in, Represents the wave function of the jth time step, j = 0, 1, ..., N; the wave functions of the two previous and next time steps constitute a set of wave function data set samples: k=1,2,…,N。

[0028] Furthermore, the fully connected layer neural network model is:

[0029] y=G w,b (x)

[0030] Among them, w and b represent the weight and bias parameters to be trained of the fully connected neural network model respectively; x is the input data, that is, the wave function of the k-1th time step y is the output data, that is, the wave function of the kth time step

[0031] Furthermore, the phase integration of the fully connected layer neural network model is specifically performed as follows:

[0032]

[0033] Among them, f k-1 is the increment of the wave function.

[0034] The normalization factor introduced into the fully connected neural network model is specifically:

[0035]

[0036] in, is the wave function of the k-1th time step, λ is the normalization factor, f k-1 is the increment of the wave function, Re represents the real part of a complex number, Im represents the conjugate part of the complex number, and Im represents the real part of the complex number.

[0037] Furthermore, the wave function prediction is performed based on the given initial value using the trained fully connected layer neural network model, specifically:

[0038] Given any initial phase angle θ and time step Δt, the fully connected neural network model calculates the initial wave function and predicts the wave function of the next several time steps.

[0039] Initial wave function ψ during prediction 0 The calculation method is:

[0040]

[0041] Among them, ψ 0 is the initial wave function during prediction, is the initial wave function used in training;

[0042] The wave function at the kth time step is:

[0043]

[0044] Among them, ψ k is the wave function of the kth time step predicted by the fully connected layer neural network model, is the wave function of the k-1th time step, λ is the normalization factor, f k-1 is the increment of the wave function.

[0045] Further, the predicted wave function is converted into a flow field evolution, specifically:

[0046] According to the wave function conversion relationship, the wave function is converted into the evolution result of the vortex particle position,

[0047]

[0048] According to the fluid mechanics related theory, the evolution of the vortex particle position is converted into the flow field evolution.

[0049] A computer device, the computer device comprising:

[0050] One or more processors;

[0051] Memory for storing one or more programs;

[0052] When the one or more programs are executed by the one or more processors, the one or more processors implement the above-mentioned two-dimensional quantized vortex dynamics neural network prediction method.

[0053] A computer readable storage medium storing computer instructions, when the computer instructions are executed by one or more processors, cause the one or more processors to execute the steps in the above-mentioned method.

[0054] The beneficial effects of the present application are:

[0055] The present application can ensure the phase invariance in the evolution process of the quantum system by phase integration of the neural network; and can ensure the wave function normalization in the evolution process of the quantum system by introducing a normalization factor. This ensures the conservation law of the physical system in the learning process of the neural network, so that the neural network has higher physical interpretability, which is important for improving the prediction accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0056] Figure 1 The flowchart of the method in the embodiment of the present application.

[0057] Figure 2 The reference trajectory and network prediction trajectory graph of the wave function evolution in the embodiment of the present application.

[0058] Figure 3 The reference trajectory and network prediction trajectory graph of the vortex particle evolution in the embodiment of the present application.

[0059] Figure 4 The reference result and network prediction result graph of the flow field flow at t=0, 20, 40, 60 in the embodiment of the present application. DETAILED DESCRIPTION

[0060] In order to enable those skilled in the art to better understand the solutions of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts should fall within the scope of protection of the present invention.

[0061] like Figure 1 , a neural network prediction method for two-dimensional quantized vortex dynamics, comprising the following steps:

[0062] S1. Determine the two-dimensional vortex dynamics equation for the flow field to be predicted, and convert the two-dimensional fluid vortex dynamics equation into the Schrödinger equation of the quantum system; the specific steps are:

[0063] For the flow field to be predicted, its two-dimensional vortex dynamics equation is:

[0064]

[0065] Among them, φ j =x j (t)+iy j (t), (x j (t),y j (t)) represents the position of the jth vortex particle in the flow field to be predicted at time t, i represents the imaginary unit; φ i =x i (t)+iy i (t), (x i (t),y i (t)) represents the position of the i-th vortex particle in the flow field to be predicted at time t, Γ i is the vortex intensity corresponding to the i-th vortex particle in the flow field to be predicted; N p is the total number of vortex particles in the flow field to be predicted;

[0066] Perform a nonlinear transformation on the vortex dynamics equations for a two-dimensional fluid:

[0067]

[0068] Among them, ψ i , ψ j , ψ k are the wave functions corresponding to the vortex particles i, j and k in the flow field to be predicted, They represent the wave function ψ i , ψ j and ψ k The complex conjugate of Γ kis the vortex intensity of vortex particle k in the flow field to be predicted; i represents an imaginary unit; N p is the total number of vortex particles in the flow field to be predicted; c0is an arbitrary complex number; a is a normalized real number, which ensures that the wave function satisfies

[0069] The vortex dynamics equation is converted into the Schrodinger equation of a quantum system:

[0070]

[0071] wherein ψ m and ψ n are wave functions corresponding to vortex particles m and n in the flow field to be predicted, respectively, represent the conjugate complex of the wave functions ψ m and ψ n ; Γ n is the vortex intensity of vortex particle n in the flow field to be predicted, and the above equation satisfies the normalization and global phase invariance characteristics of a quantum system.

[0072] S2. Obtain wave function evolution data of a quantum system according to the Schrodinger equation, and generate a wave function data set;

[0073] The wave function evolution data is obtained by a test or a numerical solving method.

[0074] The specific steps of the numerical solving method are as follows:

[0075] Given a training initial wave function and a time step Δt, N p is the total number of vortex particles, and a fourth-order Runge-Kutta method is used to solve the results of N time steps wherein, represents the wave function at the jth time step, j = 0, 1,..., N; and the wave functions at the previous and subsequent two time steps are used to form a set of wave function data set samples: k = 1, 2,..., N.

[0076] S3. Construct an L-layer fully connected layer neural network model, and the number of neurons in the ith layer is m i Integrate the phase of the fully connected layer neural network model, so that the prediction of the fully connected layer neural network model on the wave function of the quantum system satisfies the global phase invariance; the fully connected layer neural network model is:

[0077] y = G w,b (x)

[0078] wherein w and b represent the weight and bias parameters to be trained of the fully connected neural network model, respectively; x is input data, i.e., the wave function at the k-1th time step y is output data, i.e., the wave function at the kth time step The structure is the same complex number as the input data.

[0079] The phase integral is performed on the full connection layer neural network model, specifically:

[0080]

[0081] Where f k-1 is the increment of the wave function.

[0082] S4. A normalization factor is introduced into the full connection neural network model, so that the prediction of the wave function of the quantum system by the full connection layer neural network model satisfies the normalization condition;

[0083] The normalization factor is introduced into the full connection neural network model, specifically:

[0084]

[0085] Where, is the wave function at the k-1th time step, λ is the normalization factor, f k-1 is the increment of the wave function, represents taking the conjugate complex number, Im represents taking the imaginary part of the complex number, and Re represents taking the real part of the complex number.

[0086] S5. The full connection layer neural network model is trained using the wave function data set; further, the loss function of the full connection layer neural network model is The wave function ψ k predicted by the model is k = λψ k-1 + Δtf k-1 and the real wave function is calculated, and the full connection layer neural network model is trained using the Adam algorithm until the loss function is reduced to stable.

[0087] S6. The trained full connection layer neural network model is used to predict the wave function according to the given initial value; specifically:

[0088] Given any initial phase angle θ and time step Δt, the full connection layer neural network model calculates the initial wave function and predicts the wave function ψ k at the next k (k = 1, 2, …, M) time step.

[0089] The calculation method of the initial wave function ψ 0 when predicting is:

[0090]

[0091] Where ψ0 ψk is the initial wave function at the prediction time, ψk is the initial wave function used in the training;

[0092] The wave function at the kth time step is:

[0093]

[0094] where ψk is the wave function at the kth time step predicted by the fully connected neural network model. k

[0095] S7. Convert the predicted wave function into a flow field evolution, specifically:

[0096] According to the wave function conversion relationship, the wave function is converted into the evolution result of the vortex particle position:

[0097]

[0098] According to the fluid mechanics related theory, the evolution of the vortex particle position is converted into the evolution of the flow field.

[0099] Embodiment 1

[0100] To explain the application operation process of the method and the prediction effect in detail, the application of the present application is explained in detail below with a specific actual situation as an example.

[0101] For the flow field to be predicted, the evolution of four vortex particles is considered, the initial position of the vortex particle is x = [0 0 0 0], y = [1 0.3 -1 -0.3], and the vortex intensity is Γ = [1 1 -1 -1]. The following steps are followed for prediction:

[0102] Step 1, take c0 = 2.7, To ensure the normalization of the initial wave function, α = |x + iy + c0| 2 .

[0103] Step 2, take the time step Δt = 0.01, apply the Runge-kutta method to numerically calculate the results of 2000 steps, and pair the time steps before and after to form 2000 groups of samples.

[0104] Step 3, build a fully connected neural network G w,b (x) to predict the real and imaginary parts of complex numbers. It contains 3 layers of hidden layers, each layer has 16 neurons, and the input and output of the network are both 4.

[0105] Step 4, use the trapezoidal method to perform integral operation on the network:

[0106] Step 5, calculate the normalization factor: ​

[0107] Step 6: Predict the next wave function, ψ k =λψ k-1 +Δtf k-1 The network training uses the L1 loss function and the Adam optimization algorithm for 5000 rounds of training, with a batch size of 128 per round and an initial learning rate of 0.1. The learning rate is reduced to the original 0.9 after every 100 rounds of training.

[0108] Step 7: For the trained network G θ (x), given a wave function with an arbitrary initial phase angle θ Take the time step Δt and predict the wave function ψ of the kth step (k=1,2,…,M) according to steps 3-5 in sequence k . Under the initial phase angle θ=0, take Δt=0.02 and predict 5000 steps, the wave function evolution results are as follows Figure 2 As shown in (a) and (b).

[0109] Step 8: Convert the wave function evolution results into vortex particle position evolution, and the prediction results are consistent with Figure 3 (a) and (b) are shown. At the same time, the reverse tracking algorithm is applied to visualize the flow field. The results are shown in Figure 4 As shown in (a) and (b), the results show that the neural network has a high accuracy in flow field prediction over a long period of time.

[0110] It will be understood by those skilled in the art that embodiments of the present invention may be provided as methods, systems, or computer program products. Thus, the present invention may take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware. Furthermore, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to magnetic disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0111] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowcharts and / or block diagrams, as well as combinations of processes and / or blocks in the flowcharts and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowcharts and / or block diagrams. Figure 1 a process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0112] These computer program instructions can also be stored in a computer readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer readable memory produce an article of manufacture including instructions which implement the Figure 1 function specified in the flow or flows and / or blocks Figure 1 of the block or blocks.

[0113] These computer program instructions can also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions that are executed on the computer or other programmable apparatus provide steps for implementing the Figure 1 function specified in the flow or flows and / or blocks Figure 1 of the block or blocks.

[0114] Other embodiments of the application will be apparent to those skilled in the art from consideration of the specification and practice of the application disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the application being indicated by the following claims.

[0115] The specific embodiments have been shown and described for purposes of illustrating the applications, and not for purposes of limitation, the scope of protection sought is defined by the claims.

Claims

1. A neural network prediction method for two-dimensional quantized eddy dynamics, characterized in that: The following steps are involved: S1. Determine the two-dimensional vortex dynamics equation for the flow field to be predicted, and convert the two-dimensional vortex dynamics equation into the Schrödinger equation of the quantum system; S2. Obtain the wave function evolution data of the quantum system based on the Schrödinger equation and generate a wave function dataset; S3. Build a fully connected layer neural network model and perform phase integration on the fully connected layer neural network model; S4. Introducing normalization factors into fully connected neural network models; S5. Use the wave function dataset to train the fully connected layer neural network model; S6. Use the trained fully connected neural network model to predict the wave function based on the given initial value; S7. converting the predicted wave function into flow field evolution; The fully connected layer neural network model is: ， in, and Respectively represent the weights and bias parameters to be trained for the fully connected neural network model; is the input data, that is, the wave function of the k-1th time step ; is the output data, that is, the wave function of the kth time step ; The phase integration of the fully connected layer neural network model is specifically performed as follows: ， in, is the increment of the wave function; The normalization factor introduced into the fully connected neural network model is specifically: ， in, is the wave function of the k-1th time step, is the normalization factor, is the increment of the wave function, Re represents the real part of a complex number, Im represents the conjugate part of the complex number, and Im represents the real part of the complex number.

2. A neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that, The two-dimensional vortex dynamics equation of the flow field to be predicted is determined, and the two-dimensional fluid vortex dynamics equation is converted into the Schrödinger equation of the quantum system. The specific steps are: For the flow field to be predicted, its two-dimensional vortex dynamics equation is: ， in, , Indicates the first Vortex particles in The position of the moment, i represents the imaginary unit; , Indicates the first Vortex particles in The location at the moment, The first The vortex intensity of each vortex particle; is the total number of vortex particles in the flow field to be predicted; Perform a nonlinear transformation on the vortex dynamics equations for a two-dimensional fluid: ， in, 、 、 are the vortex particles in the flow field to be predicted. 、 and The corresponding wave function, 、 、 Represent the wave functions 、 and The complex conjugate of Vortex particles in the flow field to be predicted The corresponding eddy strength; i represents the imaginary unit; is the total number of vortex particles in the flow field to be predicted; Any plural number; is a normalized real number, ensuring that the wave function satisfies ; Convert the vortex dynamics equations into the Schrödinger equation for quantum systems: ， in, 、 are the vortex particles in the flow field to be predicted. and The corresponding wave function, 、 Represent the wave functions and The complex conjugate of Vortex particles in the flow field to be predicted The corresponding vortex strength.

3. A neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The wave function evolution data is obtained through testing or numerical solution methods.

4. A neural network prediction method for two-dimensional quantized eddy dynamics according to claim 3, characterized in that: The specific steps of the numerical solution method are: Given the initial wave function for training and time step , is the total number of vortex particles, and the fourth-order Runge-Kutta method is used to solve The results of the time step ,in, Indicates the The wave function of time steps, ; The wave functions of the two previous and next time steps constitute a set of wave function data set samples: , .

5. A neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The wave function prediction is performed using the trained fully connected layer neural network model according to the given initial value, specifically: Given any initial phase angle and time step , using the fully connected layer neural network model to calculate the initial wave function and predict the wave function of the next several time steps; Initial wave function during prediction The calculation method is: ， No. The wave function for a time step is: ， in, The first The wave function of time steps, is the wave function of the k-1th time step, is the normalization factor, is the increment of the wave function.

6. A neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The predicted wave function is converted into flow field evolution, specifically: According to the wave function conversion relationship, the wave function is converted into the evolution result of the vortex particle position, and then according to the relevant fluid mechanics theory, the evolution of the vortex particle position is converted into the flow field evolution.

7. A computer device, characterized in that: The computer device comprises: one or more processors; a memory for storing one or more programs; When the one or more programs are executed by the one or more processors, the one or more processors implement a neural network prediction method for two-dimensional quantized vortex dynamics as described in any one of claims 1 to 6.

8. A computer-readable storage medium storing computer instructions, characterized in that: When the computer instructions are executed by one or more processors, the one or more processors are caused to perform the steps in the method according to any one of claims 1 to 6.

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