Neural network prediction method and device for two-dimensional quantization vortex dynamics

Abstract

The invention provides a neural network prediction method and device for two-dimensional quantization vortex dynamics, and the method comprises the following steps: S1, determining a two-dimensional vortex dynamics equation for a to-be-predicted flow field, and converting the two-dimensional vortex dynamics equation into a Schrodinger equation of a quantum system; s2, obtaining wave function evolution data of the quantum system according to a Schrodinger equation, and generating a wave function data set; s3, constructing a full-connection layer neural network model, and performing phase integration on the full-connection layer neural network model; s4, introducing a normalization factor into the full-connection neural network model; s5, training the full connection layer neural network model by using the wave function data set; s6, performing wave function prediction according to a given initial value by using the trained full-connection layer neural network model; and S7, converting the predicted wave function into flow field evolution. According to the method provided by the invention, the conservation law of the system can be ensured in the prediction of the Schrodinger system, so that the flow field prediction precision can be ensured in a long time period.

Description

Technical Field

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

[0002] Quantum mechanics reveals the basic laws of the microscopic world and provides a theoretical basis 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 scholars map the physical quantities of fluid mechanics to the wave function field, thereby converting the fluid mechanics equation into the Schrödinger equation. By solving the Schrödinger equation, the flow characteristics of the flow field can be predicted, which provides a new method for predicting the flow field. Therefore, accurately predicting the evolution of the wave function of a quantum system plays an important role in accurately predicting the flow field.

[0003] In recent years, neural networks have attracted extensive attention in solving partial differential equations (PDEs) due to their powerful nonlinear modeling capabilities and high-dimensional processing capabilities. Purely data-driven neural networks face multiple limitations in solving PDEs, including dependence on large amounts of high-quality data, risk of overfitting, lack of physical consistency and numerical stability, lack of interpretability due to black-box characteristics, and high computational resource requirements. These issues limit their effective application in complex physical systems, and usually require the combination of physical knowledge or other methods to obtain more reliable results.

[0004] For the evolution of the wave function of a quantum system, it must satisfy two physical constraints, namely global phase independence and normalization of the wave function. Due to the existence of numerical dissipation, the classical numerical calculation method needs to control the time step to ensure that the wave function satisfies the physical constraints during the long-term solution process, which is even more difficult to ensure with a purely data-driven neural network. For this reason, it is necessary to design a neural network that conforms to the characteristics of the quantum system so as to accurately predict the evolution of its wave function, and then deduce the flow of the flow field with high precision. 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 integral and normalization factors in the neural network training process to ensure that the global phase invariance and normalization conditions are met in the wave function prediction process, thereby improving the accuracy in the long-term prediction process.

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

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

[0008] S1. for the flow field to be predicted, determine its two-dimensional vortex dynamics equation, 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 according to the Schrödinger equation and generate a wave function data set;

[0010] S3. Construct a fully connected layer neural network model, perform phase integration on the fully connected layer neural network model, so that the prediction of the wave function of the quantum system by the fully connected layer 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 layer neural network model meets 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 layer neural network model to predict the wave function according to 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, and the specific steps are:

[0016] For the flow field to be predicted, the 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 an 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 , j , 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 , j and ψ k The conjugate complex number of Γ k is the vortex intensity corresponding to the 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; c0 is an arbitrary complex number; α is a normalized real number to ensure that the wave function satisfies

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

[0023]

[0024] Among them, ψ m , 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 conjugate complex number 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 fourth-order Runge-Kutta method is used to solve the result of N time steps 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, i.e. the wave function at 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 conjugate complex number, Im represents the imaginary part of the complex number, and Re represents the real part of the complex number.

[0037] Furthermore, the wave function prediction is performed using the trained fully connected layer neural network model according to the given initial value, 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 for the next several time steps.

[0039] The initial wave function ψ is 0 The calculation method is:

[0040]

[0041] Among them, ψ 0 is the initial wave function for 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 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] Furthermore, the predicted wave function is converted into 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] Then, according to the relevant theories of fluid mechanics, the evolution of the vortex particle positions is converted into the flow field evolution.

[0049] A computer device, comprising:

[0050] one or more processors;

[0051] A 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 neural network prediction method for two-dimensional quantized vortex dynamics.

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

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

[0055] The present invention can ensure the phase invariance of the quantum system evolution process by performing phase integration on the neural network, and can ensure the normalization of the wave function in the quantum system evolution process by introducing a normalization factor. This ensures the conservation law of the physical system in the neural network learning process, thus making the neural network more physically interpretable, which is of great help in improving the prediction accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 Flow chart of the method in the embodiment of the present invention.

[0057] Figure 2 It is a reference trajectory and network prediction trajectory diagram of the wave function evolution in the embodiment of the present invention.

[0058] Figure 3 It is a reference trajectory and network prediction trajectory diagram of the vortex particle evolution in the embodiment of the present invention.

[0059] Figure 4 Graphs showing reference results and network prediction results of flow field at times t=0, 20, 40, and 60 in an embodiment of the present invention. DETAILED DESCRIPTION

[0060] In order to enable those skilled in the art to better understand the scheme of the present invention, the technical scheme 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 described embodiments 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 creative work 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. For the flow field to be predicted, determine its two-dimensional vortex dynamics equation, 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, the 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 an 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 , j , 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 , j and ψ k The conjugate complex number of Γ kis the vortex intensity corresponding to the 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; c0 is an arbitrary complex number; α is a normalized real number to ensure that the wave function satisfies

[0069] Transform the vortex dynamics equations into the Schrödinger equation for quantum systems:

[0070]

[0071] Among them, ψ m , 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 conjugate complex number of Γ n To predict the vortex intensity corresponding to the vortex particle n in the flow field, the above equations satisfy the normalization and global phase invariance characteristics of the quantum system.

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

[0073] The wave function evolution data is obtained through testing or numerical solution methods.

[0074] The specific steps of the numerical solution method are:

[0075] Given the initial wave function for training and time step Δt, N p is the total number of vortex particles, and the fourth-order Runge-Kutta method is used to solve the result of N time steps 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.

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

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

[0078] 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, i.e. the wave function at the kth time step Its structure is the same complex number as the input data.

[0079] The phase integration of the fully connected layer neural network model is specifically as follows:

[0080]

[0081] Among them, f k-1 is the increment of the wave function.

[0082] 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 layer neural network model meets the normalization condition;

[0083] The normalization factor introduced into the fully connected neural network model is specifically:

[0084]

[0085] 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 conjugate complex number, Im represents the imaginary part of the complex number, and Re represents the real part of the complex number.

[0086] S5. Use the wave function data set to train the fully connected layer neural network model; further, the loss function of the fully connected layer neural network model is The wave function ψ predicted by the model k , ψ k =λψ k-1 +Δtf k-1 and the real wave function It is calculated that the Adam algorithm is used to train the fully connected layer neural network model until the loss function drops to a stable level.

[0087] S6. Use the trained fully connected layer neural network model to predict the wave function according to the given initial value; specifically:

[0088] 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 ψ of the next k-th (k=1, 2, ..., M) time step k ;

[0089] The initial wave function ψ is 0 The calculation method is:

[0090]

[0091] Among them, ψ0 is the initial wave function for prediction, is the initial wave function used in training;

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

[0093]

[0094] Among them, ψ k The wave function of the kth time step predicted by the fully connected neural network model.

[0095] S7. Convert the predicted wave function into 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] Then, according to the relevant theories of fluid mechanics, the evolution of the vortex particle positions is converted into the flow field evolution.

[0099] Example 1

[0100] In order to explain the application operation process and prediction effect of the method in detail, the application of the present invention is explained in detail below by taking a specific actual situation as an example.

[0101] For the flow field to be predicted, consider the evolution of four vortex particles, the initial position of the vortex particles is x = [0 0 0 0], y = [1 0.3 -1 -0.3], and the vortex intensity is Γ = [1 1 -1 -1]. The prediction is performed step by step as follows:

[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 them into 2000 groups of samples according to the previous and next time steps.

[0104] Step 3: Build a fully connected neural network G w,b (x), predict the real and imaginary parts of complex numbers. There are 3 hidden layers in total, each with 16 neurons, and the network input and output are both 4.

[0105] Step 4: Use the trapezoidal method to integrate 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. It is trained for 5000 rounds, 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 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 to 5 in sequence. k . Under the initial phase angle θ = 0, Δt = 0.02 is selected to predict 5000 steps, and the obtained 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 the vortex particle position evolution, and the prediction results are consistent with Figure 3 (a) and (b). 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 higher accuracy in flow field prediction over a long period of time.

[0110] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, 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 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 flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing device to produce a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 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 may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture including an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0113] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0114] Those skilled in the art will readily appreciate other embodiments of the present application after considering the description and practicing the contents disclosed herein. The present application is intended to cover any modification, use or adaptation of the present application, which follows the general principles of the present application and includes common knowledge or customary techniques in the art that are not disclosed in the present application. The description and examples are intended to be exemplary only, and the true scope and spirit of the present application are indicated by the claims.

[0115] The above specific implementation modes are used to explain the present invention rather than to limit the present invention. Any modification and change made to the present invention within the spirit of the present invention and the protection scope of the claims shall fall within the protection scope of the present invention.

Claims

1. A neural network prediction method for two-dimensional quantized eddy dynamics, characterized in that: The following steps are involved: S1. for the flow field to be predicted, determine its two-dimensional vortex dynamics equation, 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 according to the Schrödinger equation and generate a wave function data set; S3. construct 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 layer neural network model to predict the wave function according to the given initial value; S7. Convert the predicted wave function into flow field evolution.

2. The neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The specific steps of determining the two-dimensional vortex dynamics equation of the flow field to be predicted and converting the two-dimensional fluid vortex dynamics equation into the Schrödinger equation of the quantum system are as follows: For the flow field to be predicted, the two-dimensional vortex dynamics equation is: 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 an 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 of 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; Perform a nonlinear transformation on the vortex dynamics equations for a two-dimensional fluid: Among them, ψ i , j , 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 , j and ψ k The conjugate complex number of Γ k is the vortex intensity corresponding to the 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; c0 is an arbitrary complex number; α is a normalized real number to ensure that the wave function satisfies Transform the vortex dynamics equations into the Schrödinger equation for quantum systems: Among them, ψ m , 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 conjugate complex number of Γ n is the vortex intensity corresponding to the vortex particle n in the flow field to be predicted.

3. The 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. The 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 Δt, N p is the total number of vortex particles, and the fourth-order Runge-Kutta method is used to solve the result of N time steps 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.

5. The neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The fully connected layer neural network model is: y=G w,b (x) 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, i.e. the wave function at the kth time step 6. The neural network prediction method for two-dimensional quantized eddy dynamics according to claim 1, characterized in that: The phase integration of the fully connected layer neural network model is specifically as follows: Among them, f k-1 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, λ is the normalization factor, f k-1 is the increment of the wave function, Re represents the conjugate complex number, Im represents the imaginary part of the complex number, and Re represents the real part of the complex number.

7. The 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 θ and time step Δt, the fully connected neural network model is used to calculate the initial wave function and predict the wave function of the next several time steps; The initial wave function ψ is 0 The calculation method is: The wave function at the kth time step is: Among them, ψ k is the wave function of the kth time step predicted by the fully connected 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.

8. The 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.

9. 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 1-8.

10. 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 execute the steps in the method according to any one of claims 1 to 8.

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