One-dimensional plasma simulation method based on deep learning optimization and related equipment

Abstract

The invention discloses a one-dimensional plasma simulation method based on deep learning optimization and related equipment, and the method comprises the steps: firstly constructing a one-dimensional basic model containing specific parameters, and then constructing and training a neural network of a multi-layer sensor (MLP) structure for predicting a potential function; then, a particle simulation method (DL-PIC) based on deep learning optimization is executed. Under the condition that 1000 grid points are included, compared with a traditional particle simulation method (PIC), the simulation time of the DL-PIC neural network is shortened by about 33.37%. According to the invention, complex physical phenomena such as reflection oscillation caused by heat effect and space charges can be effectively processed, and the method has good expansibility. The method is not only suitable for one-dimensional simulation, but also can be popularized to higher-dimension and electromagnetic particle simulation. Therefore, the invention provides an efficient and accurate solution for plasma simulation research, and can be widely applied to the technical field of plasma simulation.

Description

Technical Field

[0001] The present invention relates to the field of plasma simulation technology, and in particular to a one-dimensional plasma simulation method based on deep learning optimization and related equipment. Background Art

[0002] In plasma physics research, the transport characteristics of charged particles between two electrodes is a fundamental and critical issue. It has been widely used in many fields such as virtual cathode oscillators, high-power microwaves, thermionic energy converters, magnetron discharges, and electrostatic plasma probes. The particle-in-cell (PIC) method, as a basic algorithm in the field of plasma simulation, determines the self-consistent motion trajectories of a large number of electrons and protons by solving Maxwell's equations. However, the computational cost of this simulation method is relatively high, especially when exploring charged particle transport scenarios with finite initial velocity. The computational complexity and resource consumption increase significantly, resulting in its efficiency being limited in practical applications. Therefore, new methods need to be developed to improve simulation efficiency while maintaining simulation accuracy.

[0003] With the development of artificial intelligence (AI) technology, machine learning has gradually been applied in the natural sciences, but its application in plasma simulation is still in the exploratory stage. Traditional numerical solutions for plasma simulations have certain limitations in terms of efficiency and computational complexity. Therefore, how to combine emerging machine learning technologies, such as deep learning, with traditional PIC algorithms to improve the efficiency and accuracy of plasma simulations has become an urgent problem to be solved in this field. Deep learning has powerful data processing capabilities and can automatically extract features from large amounts of simulation data and optimize parameter adjustments during the simulation process, thereby reducing the amount of computation and improving the accuracy of the results. Therefore, research on plasma simulation methods based on deep learning optimization can effectively address the limitations of existing methods and provide more efficient and accurate tools for plasma physics research. Summary of the Invention

[0004] In order to solve at least one of the technical problems existing in the prior art to a certain extent, the purpose of the present invention is to provide a one-dimensional plasma simulation method and related equipment based on deep learning optimization.

[0005] The first technical solution adopted by the present invention is:

[0006] A one-dimensional plasma simulation method based on deep learning optimization comprises the following steps:

[0007] Build a basic model and parameter settings, and determine the solution framework of Maxwell's equations simplified to Poisson's equation in the electrostatic limit for charged particle transport between parallel electrodes in a one-dimensional model;

[0008] The ratio of the electron's initial kinetic energy to its average potential energy is introduced and combined with the space charge limited (SCL) current expression. The thermal effect is considered, the heat diffusion rate is added to the initial velocity equation, and its probability density function is clarified to describe the particle motion state.

[0009] Build and train a neural network. A multilayer perceptron (MLP) neural network is constructed to predict the electric potential function. The network consists of an input layer, three hidden layers, and an output layer. The number of neurons in the output layer matches the dimensions of the multivariate regression function of the grid point potential.

[0010] Obtain training data, randomly shuffle it, and divide it into training, validation, and test sets. During the training process, perform forward propagation, calculate loss, perform backpropagation, and update weights, optimizing network weights through multiple iterations.

[0011] Execute the DL-PIC simulation process, update the electron velocity and coordinates in sequence in the DL-PIC simulation, insert the electrons into the grid points to calculate the charge density, use the trained neural network to solve the Poisson equation to obtain the potential distribution, and update the electric field at the electron position; all PIC variables are dimensionless, the electron specific charge q / m and the number of grid points are fixed, the time step is determined, and the number of particles injected in each time step is determined based on the β value.

[0012] Furthermore, the ratio of the electron initial kinetic energy to the average potential energy is:

[0013]

[0014] Where m is the mass of the particle, v0 is the initial velocity of the electron, q is the electron charge, and φ is the potential at the anode;

[0015] The space charge limited (SCL) current expression is:

[0016]

[0017] Where β is the ratio of the electron's initial kinetic energy to its average potential energy, J CL is the limiting current value when the electron has no initial velocity, J dc is the limiting current value when the electron has an initial velocity;

[0018] The probability density function of the heat diffusion rate is:

[0019]

[0020] Where A is the normalization constant, k B is the Boltzmann constant, T is the temperature, v T is the initial velocity under thermal effect.

[0021] Furthermore, the forward propagation of the neural network calculates the neuron output according to the following formula:

[0022]

[0023] Where, is the output of the neuron, σ is the activation function, M is the number of neurons in the upper layer, is the bias parameter 1, is the input of this neuron, is the bias parameter 2;

[0024] The mean squared error (MSE) loss function is used to evaluate the network performance; the weights are updated through backpropagation and the Levenberg-Marquardt algorithm.

[0025] Furthermore, the loss function is expressed as:

[0026] L(θ)=L oDE (θ)+L BC (θ)

[0027] Where, L ODE (θ) is the mean square error between the actual value and the predicted value, L BC (θ) is the Poisson equation;

[0028] The weight update formula is:

[0029]

[0030] Where η n is the learning rate, J(w) is the error function component, e(w) is the Jacobian matrix of the error function relative weight, μI is a constant scaling factor used to control the amplitude of weight adjustment, and T is the transpose.

[0031] Furthermore, the electron velocity and coordinates are updated according to the following formula:

[0032]

[0033] Where, is the position of particle i at time n, is the particle position at time n+1, is the velocity of particle i at time n+1 / 2, is the particle velocity at time n-1 / 2, is the electric field value on particle i at time n;

[0034] Inserting electrons into the grid points calculates the charge density according to the following formula:

[0035]

[0036] in

[0037] Where x n is the particle position, x i is the grid point position, n is the total number of particles, Δx is the distance from the grid point, and |x| is the unit grid length;

[0038] The trained neural network is used to solve the Poisson equation as φ = f(n; θ) to obtain the potential distribution; according to the formula Update the electric field at the electron position; perform dimensionless processing, fix the electron ratio (q / m) and the number of grid points, and determine the time step and the number of injected particles per time step.

[0039] Furthermore, the obtaining of training data includes:

[0040] The training samples are obtained from the traditional PIC method modeling, including the electron concentration n0 and the corresponding potential φ i , the dataset covers a variety of β and φ values.

[0041] The second technical solution adopted by the present invention is:

[0042] A one-dimensional plasma simulation system based on deep learning optimization, comprising:

[0043] The model building module is used to build the basic model and set parameters. In a one-dimensional model, for charged particle transport between parallel electrodes, it determines the solution framework of Maxwell's equations simplified to Poisson's equation in the electrostatic limit.

[0044] The ratio of the electron's initial kinetic energy to its average potential energy is introduced and combined with the space charge limited (SCL) current expression. The thermal effect is considered, the heat diffusion rate is added to the initial velocity equation, and its probability density function is clarified to describe the particle motion state.

[0045] A neural network training module is used to build and train a neural network. A neural network with a multilayer perceptron (MLP) structure is constructed to predict the electric potential function. The network consists of an input layer, three hidden layers, and an output layer. The number of neurons in the output layer matches the dimensions of the multivariate regression function of the grid point potential.

[0046] Obtain training data, randomly shuffle it, and divide it into training, validation, and test sets. During the training process, perform forward propagation, calculate loss, perform backpropagation, and update weights, optimizing network weights through multiple iterations.

[0047] The simulation execution module is used to execute the DL-PIC simulation process. In the DL-PIC simulation, the electron velocity and coordinates are updated sequentially, the electrons are inserted into the grid points to calculate the charge density, the Poisson equation is solved using the trained neural network to obtain the potential distribution, and the electric field at the electron position is updated. All PIC variables are dimensionless, the electron charge ratio q / m and the number of grid points are fixed, the time step is determined, and the number of particles injected in each time step is determined based on the β value.

[0048] The third technical solution adopted by the present invention is:

[0049] An electronic device comprises a processor and a memory, wherein the memory stores at least one instruction, at least one program, a code set, or an instruction set, and the at least one instruction, at least one program, the code set, or the instruction set is loaded and executed by the processor to implement a one-dimensional plasma simulation method based on deep learning optimization as described above.

[0050] The fourth technical solution adopted by the present invention is:

[0051] A computer-readable storage medium stores at least one instruction, at least one program, a code set, or an instruction set, wherein the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by a processor to implement a one-dimensional plasma simulation method based on deep learning optimization as described above.

[0052] The fifth technical solution adopted by the present invention is:

[0053] A computer program product or computer program includes computer instructions stored in a computer-readable storage medium. A processor of a computer device can read the computer instructions from the computer-readable storage medium and execute the computer instructions, causing the computer device to perform the aforementioned one-dimensional plasma simulation method based on deep learning optimization.

[0054] The beneficial effects of the present invention are as follows: the present invention first constructs a one-dimensional basic model containing specific parameters, and then builds and trains a neural network with a multi-layer perceptron (MLP) structure to predict the electric potential function. Subsequently, a particle simulation method based on deep learning optimization (DL-PIC) is executed. The present invention can effectively handle complex physical phenomena such as reflection oscillations caused by thermal effects and space charges, and has good scalability. The present invention is not only applicable to one-dimensional simulations, but can also be extended to higher dimensions and electromagnetic particle simulations (PIC). Therefore, the present invention provides an efficient and accurate solution for plasma simulation research. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following introduction is made to the drawings of the embodiments of the present invention or the related technical solutions in the prior art. It should be understood that the drawings introduced below are only for the convenience of clearly describing some embodiments of the technical solutions of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative work.

[0056] Figure 1 It is a schematic diagram of the system of charged particles flowing between parallel electrodes of the present invention;

[0057] Figure 2 This is a flow chart of the PIC method based on deep learning of the present invention;

[0058] Figure 3 This is a comparison chart of the average time of 100 iterations of the traditional PIC algorithm and the neural network-based PIC algorithm;

[0059] Figure 4 Comparison of spatial potential and electron density between traditional PIC and DL-PIC simulations at different β values;

[0060] Figure 5 is a graph showing changes in normalized threshold current density with β and T according to the present invention;

[0061] Figure 6 is a graph showing the changing trend of the current passing through the virtual cathode of the present invention;

[0062] Figure 7 This is a graph showing the change of spatial average electron density over time at different temperatures in the present invention. DETAILED DESCRIPTION

[0063] The embodiments of the present application are described in detail below, and examples of the embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are only used to explain the present application and are not to be construed as limiting the present application. For the step numbers in the following embodiments, they are provided only for the convenience of explanation and are not intended to limit the order of the steps. The order of execution of the steps in the embodiments can be adaptively adjusted according to the understanding of those skilled in the art.

[0064] The terms used in the embodiments of the present application are only for the purpose of describing specific embodiments and are not intended to limit the embodiments of the present application. The singular forms of "a", "said", and "the" used in the embodiments of the present application and the appended claims are also intended to include the plural forms, unless the context clearly indicates otherwise. In addition, unless otherwise clearly defined, words such as setting, installing, and connecting should be understood in a broad sense, and those skilled in the art can reasonably determine the specific meanings of the above words in the present invention in combination with the specific content of the technical solution.

[0065] In the description of this application, it should be understood that descriptions involving orientations, such as up, down, front, back, left, right, etc., indicating orientations or positional relationships, are based on the orientations or positional relationships shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation. Therefore, they cannot be understood as limitations on this application.

[0066] In the description of this application, "several" means one or more, "many" means more than two, "greater than," "less than," and "exceed" are understood to exclude the number itself, while "above," "below," and "within" are understood to include the number itself. The terms "first" and "second" are used solely to distinguish technical features and are not to be construed as indicating or implying relative importance, or as implicitly specifying the number or order of the technical features indicated.

[0067] In the description of this application, "and / or" describes the relationship between associated objects, indicating that three possible relationships exist. For example, "A and / or B" can represent: A exists alone, A and B exist simultaneously, and B exists alone. The character " / " generally indicates that the associated objects are in an "or" relationship.

[0068] Explanation of terms:

[0069] DL-PIC: PIC algorithm based on deep learning neural network.

[0070] Example 1

[0071] For existing technical issues, see Figure 2 This embodiment provides a one-dimensional plasma simulation method based on deep learning optimization, which uses deep learning technology to optimize PIC simulation to reduce simulation time, improve simulation accuracy, and solve the problems of high computational cost and low efficiency of traditional PIC simulation. By combining the advantages of deep learning, the method of this embodiment can significantly improve the efficiency of plasma simulation and effectively overcome the computational bottleneck faced by existing methods when dealing with complex physical phenomena. The method specifically includes the following steps:

[0072] S101. Build basic model and set parameters.

[0073] Specifically, see Figure 1 In this embodiment, when constructing the basic model and setting parameters, a solution framework under the electrostatic limit is established for the transport characteristics of charged particles between parallel electrodes, which simplifies Maxwell's equations into Poisson's equations to describe the potential distribution. Key parameters such as the ratio of the initial kinetic energy of electrons to the average potential energy are introduced. Used to characterize the initial energy state of electrons and combined with the space charge limited (SCL) current expression ( in It is used to clarify the limiting conditions of the maximum extractable current to accurately describe the current transport process. When considering the thermal effect, the initial velocity equation is improved by introducing the thermal diffusion velocity v0 = v d +v T Supplement the initial velocity distribution of the particles and use As the probability density function of the thermal diffusion rate, where A is the normalization constant, m is the particle mass, k B is the Boltzmann constant and T is the temperature, which can more accurately describe the thermal motion characteristics of particles and thus improve the physical accuracy of the simulation.

[0074] S102. Build and train a neural network.

[0075] In some embodiments, a neural network for predicting electric potential function is constructed using a multilayer perceptron (MLP) structure, including an input layer, three hidden layers, and an output layer. The output layer has 512 neurons corresponding to the potentials of 512 grid points to meet the computational requirements of multiple regression; the three hidden layers contain 64, 128, and 256 neurons, respectively, to enhance the network's expressive power and enable it to accurately fit complex functional relationships. The training data comes from the traditional PIC method, including the electron concentration n0 and the corresponding potential φ i The dataset covers a variety of β and φ values, with a total of 36 different experimental conditions and a total data volume of 18,000 samples, which are divided into 17,000 training samples, 1,000 validation samples and 500 test samples after random shuffling (β = 0.6, φ = 0.8).

[0076] During the training process, the forward propagation is based on Calculate the neuron output, where w ij is the connection weight, b i is the bias term, M is the total number of neurons in the previous layer, and the activation function Used to map input values ​​to between -1 and 1 to accelerate training convergence; using the mean square error (MSE) loss function Evaluate the prediction error of the network and optimize the weights through backpropagation and Levenberg-Marquardt algorithm. The weight update formula is Δw(n) = -η n [J T (w)J(w)+μI] -1 ×J(w)e(w), the network parameters are optimized through multiple iterative training to improve the prediction accuracy.

[0077] S103: Execute the DL-PIC simulation process.

[0078] As an implementation method, in the DL-PIC simulation, key steps are performed sequentially. When updating the electron velocity and coordinates, the equation is:

[0079]

[0080] Insert electrons into the grid points to calculate the charge density using the formula:

[0081]

[0082] in

[0083] With the help of the trained neural network, the Poisson equation φ=f(n;θ) is solved to obtain the potential distribution; finally, according to the formula Update the electric field at the electron position. During the simulation, all PIC quantities are dimensionless, the electron q / m ratio is fixed to 1, and the number of grid points N is x =512, and according to Determine the time step to meet the CFL condition, and inject the number of particles N per time step m When the system is stable, the value of β is between 500-1000.

[0084] This embodiment also provides a method for using the above method, which includes the following steps:

[0085] Step S201: Demand confirmation and environment preparation.

[0086] Step S201.1: Clarify simulation requirements.

[0087] Before using this system to conduct simulations, users must clearly define their research objectives. For example, they may want to study the particle transport patterns in a plasma under a specific electric field, or explore the interaction between plasma and external electromagnetic fields. Once this goal is clear, they can further determine the ranges of key simulation parameters, such as electrode voltage, electrode spacing, initial particle velocity, and temperature. Proper settings for these parameters directly impact the accuracy and effectiveness of the simulation results.

[0088] Step S201.1: Build the operating environment.

[0089] Choose the appropriate computing equipment based on the scale and complexity of your simulation. For small-scale simulations, a standard multi-core CPU server can meet basic computing needs. For larger simulations or those requiring high computational speed, a GPU cluster is recommended to fully utilize its powerful parallel computing capabilities. Also, ensure that the necessary software and libraries are installed on the computing equipment, such as the Python programming environment, deep learning frameworks (such as TensorFlow or PyTorch), and the NumPy library for numerical computation, to provide a stable operating foundation for the simulation.

[0090] Step S202: Data preparation and processing.

[0091] Step S202.1: Generate initial data

[0092] Using the traditional particle-in-cell (PIC) method, a series of simulation data is generated according to different combinations of the ratio of the electron's initial kinetic energy to the average potential energy and the potential value. This data should cover the electron concentration and corresponding potential under various physical conditions, providing a rich sample for subsequent neural network training.

[0093] Step S202.2:

[0094] The large datasets generated are randomly shuffled to ensure data randomness and representativeness. Next, the datasets are divided into training, validation, and test sets according to a specific ratio. The training set is used for neural network learning, allowing the model to extract patterns and regularities from the data; the validation set is used to evaluate the model's performance in real time during training to detect problems such as overfitting; and the test set is used to ultimately evaluate the trained model's ability to generalize to unseen data. When dividing the data, ensure that the physical meaning of each dataset is clear, that the data in different sets are representative and independent, and that data duplication or bias is avoided.

[0095] Step S203: neural network training.

[0096] Step S203.1: Construct a neural network model.

[0097] The system includes a built-in multilayer perceptron (MLP) neural network for predicting electric potential functions. Users do not need to manually build a complex network structure. Based on simulation requirements, users simply set key parameters on the system interface, such as the number of neurons in the output layer (to match the dimensions of the multivariate regression function of the grid point potential) and the number of neurons in each hidden layer. The system automatically constructs an appropriate neural network model based on the user's settings.

[0098] Step S203.2: Set training parameters.

[0099] Select an appropriate loss function within the system to measure the discrepancy between the neural network's predictions and the actual data. Also, select an optimizer and set its parameters. The optimizer adjusts the neural network's weights based on the loss function, ensuring that the model's predictions consistently approach the actual data. Furthermore, determine the number of training rounds—the number of times the neural network learns from the training data. A reasonable number of training rounds balances the model's training effectiveness and time.

[0100] Step S203.3: Execute the training process.

[0101] The preprocessed training data is input into the constructed neural network, and the training process is initiated. During training, the system automatically completes forward propagation (calculating predictions based on the input data), loss calculation (using a loss function to measure the difference between the predictions and the actual data), backpropagation (calculating the direction and size of weight adjustments based on the loss results), and weight updates. Simultaneously, the system regularly evaluates model performance using validation data. If the model performance on the validation set stops improving or even declines, indicating overfitting, the system will automatically stop training and save the currently performing model.

[0102] Step S204: DL-PIC simulation runs.

[0103] Step S204.1: Initialize simulation parameters.

[0104] Before starting a deep learning-optimized particle-in-cell (DL-PIC) simulation, the system initializes the simulation parameters. This includes setting the electron charge-to-mass ratio to a fixed value and determining the number of grid points in the PIC simulation. The system automatically calculates the appropriate time step based on specific rules and physical conditions to ensure simulation stability and accuracy. Furthermore, the ratio of the electron's initial kinetic energy to its average potential energy determines the number of particles injected into the simulation at each time step, simulating realistic particle transport.

[0105] Step S204.2: Execute the simulation steps in a loop.

[0106] The system automatically executes the following steps within the DL-PIC simulation, executing each time step in sequence. First, the electron's velocity and coordinates are updated to reflect its motion under the influence of the electric and magnetic fields. Next, the electron is inserted into the grid points and the charge density at those points is calculated. The trained neural network is then used to solve the Poisson equation to obtain the potential distribution. Finally, the electric field at the electron's location is updated based on the potential distribution. This cycle repeats, gradually advancing the simulation process.

[0107] Step S204.3: Record simulation results.

[0108] During the simulation, the system regularly records data on key physical quantities, such as electron position, velocity, potential distribution, and electron density. This data is stored in system-specified files, such as HDF5 format, to facilitate subsequent data analysis and processing.

[0109] Step S205: Result analysis and optimization adjustment.

[0110] Step S205.1: Result analysis.

[0111] See also Figure 4 and Figure 7 After the simulation is complete, the recorded data is processed and analyzed using the analysis tools provided by the system. By comparing the potential distribution and electron density distribution under different simulation conditions (such as different initial velocities and different temperatures), their changing patterns can be observed. The system automatically calculates indicators such as the average relative error of particle density between the simulation results based on deep learning optimization and the simulation results based on traditional PIC methods to evaluate the accuracy of the simulation. At the same time, users can intuitively view the changing trends of various physical quantities and gain a deeper understanding of the transport characteristics and behavioral patterns of the plasma.

[0112] Step S205.2: Optimization and adjustment.

[0113] Improve the model's ability to fit the root form; or adjust training parameters, such as the optimizer's learning rate, to improve the model's training results. Based on feedback from analysis results, if simulation accuracy is insufficient, users can adjust relevant neural network parameters within the system. This can include increasing the number of hidden layer neurons, enhancing the model's filter, and increasing the amount of training data to improve the model's generalization. If simulation efficiency is low, users can optimize the code implementation through the system, for example, by enabling parallel computing to accelerate the simulation process. Once adjustments are made, retrain the neural network and repeat the DL-PIC simulation until satisfactory results are achieved.

[0114] As a further preferred embodiment, this embodiment provides a one-dimensional plasma simulation method based on deep learning optimization, which specifically includes the following steps:

[0115] Step S301: Preparation stage.

[0116] Step S301.1: Define simulation objectives and parameters.

[0117] As a preferred approach described in the present invention, before commencing plasma simulation, the primary task is to clearly define the research objective. This could be to explore the transport characteristics of charged particles between electrodes in a specific plasma device, or to study the energy distribution of plasma under different conditions. Based on the actual physical system conditions, the ranges of values ​​for a series of key parameters should be carefully determined.

[0118] As a preferred embodiment of the present invention, the electrode voltage has a significant impact on the acceleration and trajectory of the particles. Different plasma application scenarios have significantly different requirements for electrode voltage. The electrode spacing determines the size of the particle movement space and the distribution characteristics of the electric field. The initial velocity of the particle injection is directly related to the initial energy state of the particle, while the temperature will change the thermal motion state of the particle, thereby affecting its overall transport behavior. Figure 6 For example, in a virtual cathode oscillator, according to its working principle and performance requirements, the electrode voltage is set in the range of several hundred volts to several thousand volts, and the electrode spacing is set in the range of several millimeters to tens of millimeters.

[0119] Step S301.2: Prepare computing resources.

[0120] As a preferred solution described in the present invention, the computing equipment is reasonably selected based on the scale and complexity of the simulation. For small-scale one-dimensional simulations, ordinary multi-core CPU servers can meet basic computing needs with their multi-core parallel processing capabilities. However, when the simulation involves large-scale calculations or has high requirements on computing time, GPU clusters are a more suitable choice. GPUs have powerful parallel computing capabilities and can greatly accelerate the training of deep learning models and the process of simulation calculations.

[0121] As a preferred embodiment of the present invention, it is necessary to ensure that the computing device has the necessary software and libraries installed. The Python programming environment provides a wealth of scientific computing and deep learning development tools for simulation work. Deep learning frameworks such as TensorFlow or PyTorch provide convenient interfaces and efficient computing engines for building and training neural networks. The NumPy library plays an important role in numerical computing, enabling efficient processing of array and matrix operations.

[0122] Step S301.3: Data preprocessing.

[0123] As a preferred embodiment of the present invention, the method utilizes the traditional particle-in-cell (PIC) method to generate simulation data based on different combinations of the ratio of the electron's initial kinetic energy to the average potential energy and the potential value. This data covers electron concentrations and corresponding potentials under various physical conditions and serves as an important sample for subsequent neural network training.

[0124] As a preferred solution described in the present invention, in order to ensure the randomness and representativeness of the data, it is necessary to perform a random shuffling operation on the large number of generated data sets. Then, these data sets are divided into training sets, validation sets, and test sets according to a certain ratio. The training set is used to allow the neural network to learn the patterns and regularities in the data, the validation set is used to evaluate the performance of the model during the training process and to promptly detect problems such as overfitting, and the test set is used to finally evaluate the generalization ability of the trained model. When dividing the data, it is necessary to ensure that the physical meaning of each data set is clear, that the data in different sets are representative and independent, and to avoid data duplication or deviation, thereby improving the accuracy and generalization ability of neural network training.

[0125] Step S302: Neural network training process.

[0126] Step S302.1: Neural network construction.

[0127] As a preferred embodiment of the present invention, the neural network for predicting the potential function employs a multilayer perceptron (MLP) architecture. This architecture comprises an input layer, three hidden layers, and an output layer. The number of neurons in the output layer is carefully configured to match the dimensions of the multivariate regression function of the grid point potential, thereby enabling accurate prediction of the potential at each grid point.

[0128] As a preferred embodiment of the present invention, the first, second, and third hidden layers are configured with different numbers of neurons, determined through multiple experiments and optimizations. This configuration enables the neural network to effectively learn and fit complex functional relationships in the data, while ensuring the model's ability to fit the data while avoiding overfitting, a phenomenon where the model performs well on the training data but poorly on new data.

[0129] Step S302.2: Setting training parameters.

[0130] As a preferred solution of the present invention, before training a neural network, a series of key training parameters must be determined. The first step is to select an appropriate loss function, which measures the degree of discrepancy between the neural network's predictions and the actual data. The choice of loss function directly impacts the model's training performance and convergence speed, and thus the final prediction accuracy.

[0131] As a preferred solution of the present invention, a suitable optimizer must also be selected. The optimizer's function is to adjust the neural network's weights based on the value of the loss function, continuously reducing the loss function and thereby making the model's predictions increasingly close to the actual data. The type of optimizer and the settings of its related parameters have a significant impact on the model's training efficiency and ultimate performance.

[0132] As a preferred embodiment of the present invention, the number of training rounds must also be determined, that is, the number of times the neural network learns the training data. Setting the number of training rounds requires a balance between model training effectiveness and training time. Too few rounds may lead to insufficient model learning, while too many rounds may lead to overfitting, thereby affecting the model's generalization ability.

[0133] Step S302.3: Training execution.

[0134] As a preferred embodiment of the present invention, training begins by inputting the prepared training data set into the constructed neural network. During each round of training, the neural network performs forward propagation, calculating predictions based on the input data. A loss function is then used to calculate the difference between the predictions and the actual data, measuring the model's predictive performance.

[0135] As a preferred embodiment of the present invention, back propagation is then performed to calculate the direction and size of each weight adjustment based on the result of the loss function, and then the weights of the neural network are updated by the optimizer. During the training process, the validation set data is regularly used to evaluate the performance of the model to ensure the generalization ability of the model on unseen data. If it is found that the performance of the model on the validation set no longer improves or even begins to decline, this may mean that overfitting has occurred. At this time, it is necessary to stop training and save the current best-performing model to ensure that the final model has the best predictive ability.

[0136] Step S303: DL-PIC simulation implementation.

[0137] Step S303.1: Initialize simulation parameters.

[0138] As a preferred embodiment of the present invention, before performing the deep learning optimized particle-in-cell (DL-PIC) simulation, a series of key simulation parameters need to be initialized. First, the charge-to-mass ratio of the electron is set to a fixed value, and the number of grid points in the PIC simulation is determined to ensure the discretization accuracy of the simulation space.

[0139] As a preferred embodiment of the present invention, the appropriate time step is calculated based on specific rules and physical conditions. The selection of the time step is crucial, as it directly affects the stability and accuracy of the simulation. If the time step is too large, it may lead to numerical instability and affect the physical rationality of the simulation; if the time step is too small, it will significantly increase the computational cost and reduce the simulation efficiency. Therefore, properly setting the time step is crucial to obtaining efficient and accurate simulation results.

[0140] At the same time, the number of particles injected in each time step is determined based on the ratio of the electron's initial kinetic energy to the average potential energy to ensure the rationality of the particle number, so that the simulation can accurately characterize the transport behavior of the plasma and truly reflect the dynamic characteristics of the plasma in the actual physical environment.

[0141] Step S303.2: Simulate the cycle advancement.

[0142] As a preferred embodiment of the present invention, the DL-PIC simulation steps are followed, with a series of operations performed sequentially within each time step. First, the electron's velocity and coordinates are updated. This step reflects the electron's motion under the influence of electric and magnetic fields. Combined with the potential field prediction mechanism optimized by deep learning, it improves the accuracy and efficiency of particle motion. Next, the electron is inserted into the grid points, and the charge density at the grid points is calculated. The charge density calculation is the basis for the subsequent electric field solution. It can accurately reflect the charge distribution in the plasma and provide important information for the electric field calculation. Then, the trained neural network is used to solve the Poisson equation to obtain the potential distribution. During the training process, the neural network has learned the complex relationship between electron concentration and potential and can accurately predict the potential in a short time. Finally, based on the calculated potential distribution, the electric field at the electron's position is updated. The updated electric field affects the subsequent motion of the electron, thereby driving the continuous progress of the simulation process and improving the accuracy of particle motion and electric field distribution through continuous iteration. The optimization and efficient execution of the entire process enable DL-PIC simulation to better cope with complex physical scenarios.

[0143] Step S303.1: Record simulation results.

[0144] During the simulation process, it is necessary to regularly record key physical quantities, such as electron position, velocity, potential distribution, and electron density. This data can reflect the changes and evolution of the plasma state during the simulation. To efficiently store and manage large amounts of simulation data, this recorded data can be stored in a suitable file format, such as HDF5. HDF5 is well-suited for large-scale scientific data storage. It efficiently organizes data and supports fast read and write operations, facilitating subsequent data analysis and processing.

[0145] For example, after every certain number of time steps, the current electron position, potential distribution, and other relevant physical quantity data can be saved to an HDF5 file. This stored data can include simulation results at different time points, providing a rich foundation for subsequent in-depth research. Using the HDF5 format ensures efficient data storage and supports a variety of tools for further data analysis, such as visualization and statistical analysis, helping researchers better understand plasma behavior.

[0146] Step S304: Result analysis and optimization

[0147] Step S304.1: Analyze simulation results.

[0148] As a preferred embodiment of the present invention, the use of professional simulation result analysis tools is used to comprehensively and deeply process and analyze the recorded data. By comparing the simulation results under conditions such as different initial velocities and different temperatures, and observing the changing patterns of potential distribution and electron density distribution, we can gain a deeper understanding of the effects of various physical conditions on plasma behavior. For example, analyzing the changes in potential distribution under different initial velocities and temperatures can reveal the effects of thermal effects and initial kinetic energy on particle transport characteristics, helping researchers explore the dynamic behavior of plasma from different angles.

[0149] As a preferred embodiment of the present invention, the accuracy of the simulation is evaluated by calculating the average relative error of the simulation results based on deep learning optimization and the traditional PIC method simulation results to particle density. This process helps to verify the performance improvement of the deep learning optimization method relative to the traditional method. By quantifying the simulation error, the effectiveness of the optimization algorithm can be intuitively evaluated to ensure the accuracy and reliability of the simulation results. At the same time, the changing trend of each physical quantity (such as electron density, electric field, velocity distribution, etc.) in the simulation process is analyzed to help gain a deeper understanding of the transport characteristics and behavior of plasma, and provide theoretical basis and practical reference for further research and practical application.

[0150] Step S304.2: Model and parameter optimization.

[0151] As a preferred embodiment of the present invention, the simulation parameters or neural network are optimized and adjusted based on the feedback information from the analysis results. If the simulation accuracy is found to be insufficient, one can try increasing the number of neurons in the hidden layer of the neural network to enhance the model's fitting ability; or adjusting training parameters, such as the optimizer's learning rate, to improve the model's training results. These optimization steps help improve the accuracy of the neural network, enabling it to better handle different simulation scenarios and physical conditions.

[0152] As a preferred solution of the present invention, it is also possible to consider increasing the amount of training data, allowing the neural network to learn more data features and patterns, thereby improving the model's generalization ability. If the simulation efficiency is low, the code implementation can be optimized, and methods such as parallel computing technology can be used to accelerate the simulation process. Parallel computing technology can significantly increase the execution speed of simulation tasks, allowing large-scale plasma simulations to be completed in a shorter time. These optimization measures help improve the performance of the entire simulation process and ensure the real-time and accuracy of the simulation results.

[0153] After these adjustments, the neural network training and DL-PIC simulation were repeated until satisfactory simulation results were obtained. Through repeated adjustments and optimizations, efficient and accurate simulation results were ultimately achieved.

[0154] In summary, compared with the prior art, the method of the present invention has at least the following advantages and beneficial effects:

[0155] (1)Simulation time reduction: see Figure 3 In a simulation scenario with 1,000 grid points, the DL-PIC neural network, through its parallel computing capabilities and efficient electric potential field prediction mechanism, achieved a 33.37% reduction in simulation time compared to the traditional PIC algorithm. This is primarily due to the neural network's ability to quickly process large amounts of data and accurately predict the electric potential field, thereby reducing the complex iterative calculation time required by traditional algorithms. Through this optimization, DL-PIC improves the real-time performance and computational efficiency of simulations, enabling large-scale simulations to be completed in a shorter time.

[0156] (2) Improved simulation accuracy: See Figure 4 Under three different initial velocity conditions, the average relative errors of particle density between DL-PIC and traditional PIC methods were 4.346%, 0.792%, and 0.773%, respectively. The powerful function fitting capabilities of neural networks enable them to more accurately capture the complex nonlinear relationship between particle density and electric potential field, effectively reducing simulation errors. Compared to traditional methods, DL-PIC significantly improves simulation accuracy, providing more reliable simulation data and enabling researchers to delve deeper into plasma physics phenomena.

[0157] (3) Enhanced ability to process complex physical phenomena: Figure 5 The proposed method excels in handling complex physical phenomena such as reflection oscillations caused by thermal effects and space charge. By accounting for the influence of thermal-related parameters on particle initial velocity and the neural network's ability to model complex physical processes, it is possible to more accurately simulate inter-electrode particle transport. For example, thermal effects increase electron thermal motion, altering velocity distribution and space charge confinement. This method effectively accounts for these changes, making simulation results more consistent with actual physical conditions.

[0158] (4) Scalability advantage: The method of the present invention demonstrates superior performance when dealing with complex physical phenomena such as reflection oscillations caused by thermal effects and space charges. By introducing parameters related to thermal effects, the distribution characteristics of the initial velocity of particles are accurately characterized, and by utilizing the powerful modeling capabilities of neural networks, the transport process of particles between electrodes can be simulated more accurately. For example, thermal effects can enhance the thermal motion of electrons, affect the velocity distribution, and change the space charge confinement effect. This method can effectively capture these changes, making the simulation results closer to the actual physical situation, thereby improving the accuracy and reliability of the simulation.

[0159] Example 2

[0160] This embodiment provides a one-dimensional plasma simulation system based on deep learning optimization, including:

[0161] The model building module is used to build a complete one-dimensional plasma simulation basic model and accurately determine various parameters, including β, SCL current expressions, and thermal effect related parameters, providing a solid theoretical foundation for simulation;

[0162] The neural network training module is used to undertake the construction and training tasks of the neural network, including data processing, network structure construction, and performs training operations according to the standard process of forward propagation, loss calculation, backpropagation and weight update, and iterative training to ensure that the neural network can accurately predict the potential function;

[0163] The simulation execution module is used to carry out one-dimensional plasma simulation based on the trained neural network and strictly follow the DL-PIC simulation steps, including operations such as electron velocity and coordinate update, electron insertion into grid points, Poisson equation solution, and electric field update. It also accurately performs dimensionless processing and time step control to ensure the accuracy and stability of the simulation process;

[0164] The result analysis module is used to conduct in-depth analysis of the simulation results, compare the potential distribution, electron density distribution and other results under different simulation conditions (such as different initial velocities and different temperatures), and calculate indicators such as the average relative error of particle density between DL-PIC and traditional PIC methods to evaluate the accuracy and efficiency of the simulation and provide a basis for optimizing the simulation results.

[0165] Since the device is a one-dimensional plasma simulation system based on deep learning optimization in an embodiment of the present invention, and the principle of solving the problem by the system is similar to that of the method, the implementation of the device can refer to the implementation process of the above-mentioned method embodiment, and the repeated parts will not be repeated.

[0166] Example 3

[0167] An embodiment of the present invention also provides an electronic device, comprising a processor and a memory, wherein the memory stores at least one instruction, at least one program, a code set, or an instruction set, and the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by the processor to implement a one-dimensional plasma simulation method based on deep learning optimization as described in Example 1.

[0168] It is understood that the memory may include random access memory (RAM) or read-only memory (ROM). Optionally, the memory includes a non-transitory computer-readable storage medium. The memory may be used to store instructions, programs, codes, code sets, or instruction sets. The memory may include a program storage area and a data storage area, wherein the program storage area may store instructions for implementing an operating system, instructions for at least one function, instructions for implementing the various method embodiments described above, etc.; the data storage area may store data created based on the use of the server, etc.

[0169] The processor may include one or more processing cores. The processor utilizes various interfaces and circuits to connect various components within the server. It executes various server functions and processes data by running or executing instructions, programs, code sets, or instruction sets stored in memory, as well as accessing data stored in memory. Optionally, the processor may be implemented using at least one of the following hardware forms: digital signal processing (DSP), field-programmable gate array (FPGA), and programmable logic array (PLA). The processor may integrate one or a combination of a central processing unit (CPU) and a modem. The CPU primarily processes the operating system and application programs, while the modem handles wireless communications. It is understood that the modem may not be integrated into the processor and may be implemented separately via a single chip.

[0170] Since the electronic device is an electronic device corresponding to the one-dimensional plasma simulation method based on deep learning optimization in an embodiment of the present invention, and the principle of solving the problem by the electronic device is similar to that of the method, the implementation of the electronic device can refer to the implementation process of the above-mentioned method embodiment, and the repeated parts will not be repeated.

[0171] Example 4

[0172] An embodiment of the present invention also provides a computer-readable storage medium, which stores at least one instruction, at least one program, a code set, or an instruction set. The at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by a processor to implement a one-dimensional plasma simulation method based on deep learning optimization as described in Example 1.

[0173] Those skilled in the art will appreciate that all or part of the steps in the various methods of the above embodiments can be completed by instructing related hardware through a program. The program can be stored in a computer-readable storage medium, and the storage medium includes a read-only memory (ROM), a random access memory (RAM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), a one-time programmable read-only memory (OTPROM), an electronically erasable programmable read-only memory (EEPROM), a compact disc read-only memory (CD-ROM) or other optical disc storage, magnetic disk storage, magnetic tape storage, or any other computer-readable medium capable of carrying or storing data.

[0174] Since the storage medium is a storage medium corresponding to a one-dimensional plasma simulation method based on deep learning optimization in an embodiment of the present invention, and the principle of solving the problem by the storage medium is similar to that of the method, the implementation of the storage medium can refer to the implementation process of the above-mentioned method embodiment, and the repeated parts will not be repeated.

[0175] Example 5

[0176] In some possible implementations, various aspects of the methods of the embodiments of the present invention may also be implemented in the form of a program product, which includes program code. When the program product is executed on a computer device, the program code is used to cause the computer device to perform the steps of the one-dimensional plasma simulation method based on deep learning optimization according to various exemplary embodiments of the present application as described above in this specification. The executable computer program code or "code" used to perform the various embodiments may be written in a high-level programming language such as C, C++, Python, Smalltalk, Java, JavaScript, Visual Basic, Structured Query Language (e.g., Transact-SQL), Perl, or various other programming languages.

[0177] It should be understood that various parts of the present invention can be implemented using hardware, software, firmware, or a combination thereof. In the above-described embodiments, multiple steps or methods can be implemented using software or firmware stored in a memory and executed by a suitable instruction execution system. For example, if implemented using hardware, as in another embodiment, any one of the following technologies known in the art or a combination thereof can be used: a discrete logic circuit having a logic gate circuit for implementing a logic function on a data signal, an application-specific integrated circuit having a suitable combination of logic gate circuits, a programmable gate array (PGA), a field programmable gate array (FPGA), etc.

[0178] In the description of this specification, the reference terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" mean that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine different embodiments or examples described in this specification and features of different embodiments or examples without contradiction.

[0179] The above embodiments are intended only to illustrate the technical concepts and features of the present invention. Their purpose is to enable those skilled in the art to understand the contents of the present invention and implement them accordingly. They are not intended to limit the scope of protection of the present invention. Any equivalent changes or modifications made based on the essence of the present invention are intended to be covered by the scope of protection of the present invention.

Claims

1. A one-dimensional plasma simulation method based on deep learning optimization, characterized in that: The following steps are involved: Build a basic model and parameter settings, and determine the solution framework of Maxwell's equations simplified to Poisson's equation in the electrostatic limit for charged particle transport between parallel electrodes in a one-dimensional model; The ratio of the electron's initial kinetic energy to its average potential energy is introduced, and combined with the space charge-limited current expression; thermal effects are considered, the heat diffusion rate is added to the initial velocity equation, and its probability density function is clarified to describe the particle motion state; Build and train a neural network. A multilayer perceptron neural network is constructed to predict the electric potential function. The network consists of an input layer, three hidden layers, and an output layer. The number of neurons in the output layer matches the dimensions of the multivariate regression function of the grid point potential. Get the training data, randomly shuffle it and divide it into training set, validation set and test set; During the training process, forward propagation, loss calculation, backpropagation and weight update are performed to optimize the network weights through multiple iterations; Execute the DL-PIC simulation process, in which the electron velocity and coordinates are updated in sequence, the electron is inserted into the grid point to calculate the charge density, the trained neural network is used to solve the Poisson equation to obtain the potential distribution, and the electric field at the electron position is updated; All PIC variables are dimensionless, the electron charge ratio q / m and the number of grid points are fixed, the time step is determined, and the number of particles injected in each time step is determined according to the β value.

2. The one-dimensional plasma simulation method based on deep learning optimization according to claim 1, characterized in that: The ratio of the electron's initial kinetic energy to its average potential energy is: Where m is the mass of the particle, v0 is the initial velocity of the electron, q is the electron charge, and φ is the potential at the anode; The space charge limited current expression is: Where β is the ratio of the electron's initial kinetic energy to its average potential energy, J CL is the limiting current value when the electron has no initial velocity, J dc is the limiting current value when the electron has an initial velocity; The probability density function of the heat diffusion rate is: Where A is the normalization constant, k B is the Boltzmann constant, T is the temperature, v T is the initial velocity under thermal effect.

3. The one-dimensional plasma simulation method based on deep learning optimization according to claim 1, characterized in that: The forward propagation of the neural network calculates the neuron output according to the following formula: Where, is the output of the neuron, σ is the activation function, M is the number of neurons in the upper layer, is the bias parameter 1, is the input of this neuron, is the bias parameter 2; The mean squared error loss function is used to evaluate the network performance; the weights are updated through backpropagation and the Levenberg-Marquardt algorithm.

4. The one-dimensional plasma simulation method based on deep learning optimization according to claim 3, characterized in that: The expression of the loss function is: L(θ)=L oDE (θ)+L BC (i) Where, L ODE (θ) is the mean square error between the actual value and the predicted value, L BC (θ) is the Poisson equation; The weight update formula is: Δw(n)=-η n [J T (w)J(w)+μI] -1 ×J(w)e(w) Where η n is the learning rate, J(w) is the error function component, e(w) is the Jacobian matrix of the error function relative weight, μI is a constant scaling factor used to control the amplitude of weight adjustment, and T is the transpose.

5. The one-dimensional plasma simulation method based on deep learning optimization according to claim 1, characterized in that: Update the electron velocity and coordinates according to the following formula: Where, is the position of particle i at time n, is the particle position at time n+1, is the velocity of particle i at time n+1 / 2, is the particle velocity at time n-1 / 2, is the electric field value on particle i at time n; Inserting electrons into the grid points calculates the charge density according to the following formula: in Where x n is the particle position, x i is the grid point position, n is the total number of particles, Δx is the distance from the grid point, and |x| is the unit grid length; The trained neural network is used to solve the Poisson equation as φ = f(n; θ) to obtain the potential distribution; according to the formula Update the electric field at the electron position; Perform dimensionless processing, fix the electron ratio and the number of grid points, and determine the time step and the number of injected particles per time step.

6. The one-dimensional plasma simulation method based on deep learning optimization according to claim 1, characterized in that: The obtaining of training data includes: The training samples are obtained from the traditional PIC method modeling, including the electron concentration n0 and the corresponding potential φ i , the dataset covers a variety of β and φ values.

7. A one-dimensional plasma simulation system based on deep learning optimization, characterized in that: include: The model building module is used to build the basic model and set parameters. In a one-dimensional model, for charged particle transport between parallel electrodes, it determines the solution framework of Maxwell's equations simplified to Poisson's equation in the electrostatic limit. The ratio of the electron's initial kinetic energy to its average potential energy is introduced, and combined with the space charge-limited current expression; thermal effects are considered, the heat diffusion rate is added to the initial velocity equation, and its probability density function is clarified to describe the particle motion state; A neural network training module is used to build and train a neural network. A neural network with a multilayer perceptron structure is constructed to predict the electric potential function. The network consists of an input layer, three hidden layers, and an output layer. The number of neurons in the output layer matches the dimension of the multivariate regression function of the grid point potential. Get the training data, randomly shuffle it and divide it into training set, validation set and test set; During the training process, forward propagation, loss calculation, backpropagation and weight update are performed to optimize the network weights through multiple iterations; The simulation execution module is used to execute the DL-PIC simulation process. In the DL-PIC simulation, the electron velocity and coordinates are updated in sequence, the electrons are inserted into the grid points to calculate the charge density, the Poisson equation is solved using the trained neural network to obtain the potential distribution, and the electric field at the electron position is updated. All PIC variables are dimensionless, the electron charge ratio q / m and the number of grid points are fixed, the time step is determined, and the number of particles injected in each time step is determined according to the β value.

8. An electronic device, characterized in that: The electronic device includes a processor and a memory, wherein the memory stores at least one instruction, at least one program, a code set, or an instruction set, and the at least one instruction, the at least one program, the code set, or the instruction set is loaded and executed by the processor to implement the method according to any one of claims 1 to 6.

9. A computer-readable storage medium, characterized in that The storage medium stores at least one instruction, at least one program, a code set or an instruction set, and the at least one instruction, the at least one program, the code set or the instruction set is loaded and executed by the processor to implement the method according to any one of claims 1 to 6.

10. A computer program product, characterized in that The computer program product comprises computer instructions, which are used to perform the method according to any one of claims 1 to 6 when executed by a processor.

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