Low-pressure plasma particle simulation method based on variational auto-encoder

Abstract

The invention discloses a low-pressure plasma particle simulation method based on a variational auto-encoder, and the method comprises the following steps: firstly, building a plasma particle model which is used for generating a data set needed by neural network training; constructing a neural network framework based on a variational auto-encoder to reconstruct a source field of the model and derive a corresponding solution field; designing a loss function according to a closure relationship of the neural network, and optimizing hyper-parameters of the neural network; and training the network until the loss function converges to a predetermined threshold. According to the method, the source field of the plasma particle model can be accurately reconstructed, and the solution field can be accurately solved. Compared with a traditional particle simulation method based on a neural network, the problems that a complex source field is difficult to express and the equation solving precision is insufficient are effectively solved, and the accurate mapping relation can be efficiently trained through limited sample data.

Description

Technical Field

[0001] This invention relates to the interdisciplinary field of artificial intelligence and plasma, specifically to a method for simulating low-pressure plasma particles based on a variational autoencoder. Background Technology

[0002] Low-pressure plasma technology is a core foundation for many high-end industrial applications, playing an irreplaceable role, especially in chip etching in semiconductor manufacturing and energy conversion devices in the new energy field. The generation and maintenance of low-pressure plasma is a complex physical field problem involving the coupling of electromagnetic fields, chemical kinetics, fluid transport, and thermodynamics, inherently characterized by high nonlinearity and instability. Furthermore, to meet specific process requirements, reaction chambers are often designed with irregular geometries and require the introduction of complex source terms. While pursuing performance optimization, these structural and source term designs significantly increase the complexity of the physical field within the reaction chamber, making it difficult to achieve uniform, stable, and controllable plasma density and active particle distribution.

[0003] Particle simulation is a commonly used numerical method for studying low-pressure plasmas. This method reveals the physical mechanisms of low-pressure plasmas by sampling the plasma into a finite number of macroparticles, placing them within a numerical grid, and then driving the macroparticles' motion based on first-principles calculations. However, this method often suffers from significant computational time when solving specific physical models due to the difficulty in converging the field equations when dealing with charge source terms formed by particle deposition and externally applied electromagnetic source terms.

[0004] In recent years, deep learning-based neural network methods have offered a potential avenue for accelerating particle simulations. However, existing techniques typically input the complete physical field or linearly dimensionality-reduced data directly into the network. The former consumes a large amount of computer memory, hindering efficient network training; the latter often fails to effectively represent complex physical fields, leading to decreased accuracy in simulation results. Therefore, in practical applications, there is an urgent need for a new method that can efficiently train and accurately establish mapping relationships under limited sample conditions to address the difficulties in representing complex source terms and insufficient equation solving accuracy faced by neural networks in accelerating particle simulations. Summary of the Invention

[0005] The purpose of this invention is to propose a low-pressure plasma particle simulation method based on a variational autoencoder, which can accurately reconstruct the source field of a plasma particle model and accurately solve its solution field. Compared with traditional neural network-based particle simulation methods, this invention effectively overcomes the difficulties in representing complex source fields and the insufficient accuracy of equation solving, and can efficiently train accurate mapping relationships using limited sample data.

[0006] To achieve the above-mentioned technical objectives, the technical solution adopted by the present invention is as follows:

[0007] A method for simulating low-pressure plasma particles based on a variational autoencoder, the method comprising:

[0008] Step 1: Establish a low-pressure plasma particle model, take random seeds in Gaussian random fields of different ranges as training and testing sets respectively, and generate the dataset required for neural network training.

[0009] Step 2: Based on the low-pressure plasma particle model in Step 1, construct a corresponding neural network based on variational autoencoder to reconstruct the source field of the plasma particle model and derive the corresponding solution field. Specifically, the source field of the plasma particle model is reconstructed using the closure model constructed by variational autoencoder, and the latent vector representation of the source field is output. Then, the latent vector representation is used as the input of the deep operator network to calculate the solution field of the plasma particle model.

[0010] Step 3: Design the loss function based on the closure relationship of the neural network in Step 2, and optimize the hyperparameters of the neural network;

[0011] Step 4: Train the neural network set up in Step 3 until the loss function converges to a predetermined value; reconstruct the source field of the plasma particle model using the trained neural network and solve its solution field, outputting the simulation results of the low-pressure plasma particle model.

[0012] Step 1 further includes:

[0013] Establish a plasma equation model, and then rewrite the plasma equation model into a plasma particle model:

[0014]

[0015] The boundary conditions are:

[0016]

[0017] The initial conditions are:

[0018]

[0019] In the formula, ζ represents the source field of the plasma particle model, t represents the time quantity, u represents the solution to the equation, the meaning of which depends on the type of the corresponding multiphysics equation, and λ represents the variable plasma parameters in the equation. These are the nonlinear operators required for different physical fields parameterized by λ. These are the corresponding boundary values, and β is the corresponding initial value; Ω represents the spatial region, u b U represents the solution at the boundary. i This represents the solution at the initial moment.

[0020] Furthermore, in step 2, the source field of the plasma particle model is reconstructed using a closure model constructed by a variational autoencoder. The process of simultaneously outputting the latent vector representation Z of the source field includes:

[0021] Two variational autoencoders are used as the forward and feedback paths of the closure model, respectively. The encoders of the variational autoencoders are both ordinary convolutional neural networks, and the decoders are both transposed convolutional neural networks. The variational autoencoder used as the forward path employs a plasma particle model source field. As input, the outputs are the latent principal vector Z' and the source field reconstructed based on the latent principal vector. The variational autoencoder, used as a feedback path, employs a plasma particle model source field. Source field reconstructed based on latent principal vector The difference is taken as input, and the output is the latent error vector Z” and the source field reconstructed based on the latent error vector. The sum of the obtained potential principal vector and potential error vector is used as the potential vector representation of the source field of the plasma particle model: Z = Z' + Z.

[0022] Furthermore, in step 2, the deep operator network includes a branch network and a backbone network, both of which are composed of feedforward neural networks.

[0023] Further, in step 3, the loss function is:

[0024]

[0025] Where MSE is the mean squared error function.

[0026] Furthermore, the physical field corresponding to the low-pressure plasma particle model is the physical field in the semiconductor chip etching reaction chamber or the physical field in the reaction chamber of the energy conversion device.

[0027] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0028] The present invention provides a low-pressure plasma particle simulation method based on variational autoencoders. This method utilizes variational autoencoders to accurately reconstruct the source field of a plasma particle model and precisely solve for its solution field. Compared to traditional neural network-based particle simulation methods, this invention effectively overcomes the difficulties in representing complex source fields and the insufficient accuracy of equation solving, and can efficiently train accurate mapping relationships using limited sample data. Attached Figure Description

[0029] Figure 1 This is a schematic diagram of the low-pressure plasma particle simulation method based on variational autoencoder of the present invention.

[0030] Figure 2 This is a diagram of the variational autoencoder framework for the particle random field electrostatic model in this invention.

[0031] Figure 3 This is a comparison of source field reconstructions from a conventional variational autoencoder.

[0032] Figure 4 This is a comparison diagram of source field reconstruction using the method of this invention;

[0033] Figure 5 This is a comparison diagram of the two-dimensional distribution results of the solution field obtained by the method of this invention and the simulation of ordinary particle fields. Detailed Implementation

[0034] The embodiments of the present invention will be described in further detail below with reference to the accompanying drawings.

[0035] In this embodiment, the particle random field is taken as the research object and the electrostatic particle model is taken as the research sample. The simulation results of the electrostatic particle model are obtained by batch training.

[0036] Please see Figure 1 It illustrates a flowchart of a particle simulation method based on a variational autoencoder, which includes the following steps:

[0037] Step 1: Establish a plasma particle model to generate the dataset required for neural network training. Take a Gaussian random field with random seeds in the range [1, 500] as the training set and a Gaussian random field in the range (500, 600] as the test set.

[0038] Step 1.1: Establish the particle model field equations based on Gaussian random fields and the Poisson equation:

[0039]

[0040] The boundary conditions are as follows:

[0041]

[0042] In the formula, ρ is the electric potential, ε0 is the vacuum permittivity, x and y are two-dimensional spatial coordinates, GP represents a Gaussian random process, (x1,y1) and (x2,y2) represent the coordinates of two points in space, and σ and l are the standard deviation and length parameter of the Gaussian random process, respectively.

[0043] Step 2: Then, based on the plasma particle model from Step 1, construct the corresponding neural network framework based on variational autoencoders, such as... Figure 2 As shown, details are as follows:

[0044] Step 2.1: Reconstruct the source field of the plasma particle model using the closure model constructed by the variational autoencoder. Simultaneously outputs the latent vector representation Z of the source field;

[0045] Step 2.1.1: Use two variational autoencoders as the forward and feedback paths of the closure model, respectively. The encoders of the variational autoencoders are both ordinary convolutional neural networks, and the decoders are both transposed convolutional neural networks.

[0046] Step 2.1.1.1: Set the number of convolutional layers in the variational autoencoder to 4, with the number of channels being [1,128,32,64,128], the kernel size being [12,9,6,3], all convolutional strides being set to 2, and the latent vector length being 256.

[0047] Step 2.1.1.2: Set the number of convolutional layers in the decoder of the variational autoencoder to 4, with the number of channels being [128, 64, 32, 128, 1], the kernel size being [3, 6, 9, 12], and all convolutional strides being set to 2;

[0048] Step 2.1.2: The variational autoencoder, serving as the forward path, uses a plasma particle model source field. As input, the outputs are the latent principal vector Z' and the source field reconstructed based on the latent principal vector.

[0049] Step 2.1.3: The variational autoencoder, serving as the feedback path, uses a plasma particle model source field. Source field reconstructed based on latent principal vector The difference is taken as input, and the output is the latent error vector Z” and the source field reconstructed based on the latent error vector.

[0050] Step 2.1.4: The sum of the obtained potential principal vector and potential error vector is the potential vector representation of the source field of the plasma particle model, Z = Z' + Z”.

[0051] Step 2.2: Using the latent vector Z as input to the deep operator network, calculate the solution field of the plasma particle model.

[0052] Step 2.2.1: The deep operator neural network specifically includes a branch network and a backbone network, both of which are composed of feedforward neural networks.

[0053] Step 3: Design the loss function based on the closure relationship of the neural network in Step 2, and optimize the hyperparameters of the neural network. The loss function is expressed as follows:

[0054]

[0055] Where MSE is the mean squared error function.

[0056] Step 4: Train the network set up in Step 3. Set the number of training iterations to 500,000 and the learning rate to 10. -4 This method allows the loss function to converge to a predetermined value. It can accurately reconstruct the source field of a plasma particle model and precisely solve for its solution field; the final output of the neural network is the result of the low-pressure plasma particle simulation.

[0057] At this point, the source function reconstruction results of the variational autoencoder with closure structure and the ordinary variational autoencoder in this method are, for example... Figure 3 and Figure 4 As shown, ordinary autoencoders cannot accurately reconstruct the source field of a particle model, resulting in a large and uniformly distributed error. In contrast, the variational autoencoder with a closure structure outputs the main components of the source field through VAE1 and the error components through VAE2, thus achieving accurate reconstruction of the source field and reducing the absolute error by an order of magnitude.

[0058] After training, random seeds of 530, 560, and 590 were selected. The results of the two-dimensional potential distribution in the low-pressure plasma particle simulation method based on variational autoencoders were compared with those of ordinary particle simulations. Figure 5 As shown, when calculating the solution field of the particle model under different random fields, the neural network framework based on variational autoencoder can obtain accurate solution fields with uniform error distribution, and has the ability to handle complex source fields.

[0059] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0060] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

Claims

1. A method for simulating low-pressure plasma particles based on a variational autoencoder, characterized in that, The method includes: Step 1: Establish a low-pressure plasma particle model, take random seeds in Gaussian random fields of different ranges as training and testing sets respectively, and generate the dataset required for neural network training. Step 2: Based on the low-pressure plasma particle model in Step 1, construct a corresponding neural network based on variational autoencoder to reconstruct the source field of the plasma particle model and derive the corresponding solution field. Specifically, the source field of the plasma particle model is reconstructed using the closure model constructed by variational autoencoder, and the latent vector representation of the source field is output. Then, the latent vector representation is used as the input of the deep operator network to calculate the solution field of the plasma particle model. Step 3: Design the loss function based on the closure relationship of the neural network in Step 2, and optimize the hyperparameters of the neural network; Step 4: Train the neural network set up in Step 3 until the loss function converges to a predetermined value; reconstruct the source field of the plasma particle model using the trained neural network and solve its solution field, outputting the simulation results of the low-pressure plasma particle model.

2. The low-pressure plasma particle simulation method based on variational autoencoder according to claim 1, characterized in that, Step 1 further includes: Establish a plasma equation model, and then rewrite the plasma equation model into a plasma particle model: The boundary conditions are: The initial conditions are: In the formula, ζ represents the source field of the plasma particle model, t represents the time quantity, u represents the solution to the equation, the meaning of which depends on the type of the corresponding multiphysics equation, and λ represents the variable plasma parameters in the equation. These are the nonlinear operators required for different physical fields parameterized by λ. These are the corresponding boundary values, and β is the corresponding initial value; Ω represents the spatial region, u b U represents the solution at the boundary. i This represents the solution at the initial moment.

3. The low-pressure plasma particle simulation method based on variational autoencoder according to claim 1, characterized in that, Step 2, which involves reconstructing the source field of the plasma particle model using a closure model constructed with a variational autoencoder and simultaneously outputting the latent vector representation of the source field, includes: Two variational autoencoders are used as the forward and feedback paths of the closure model, respectively. The encoders of the variational autoencoders are both ordinary convolutional neural networks, and the decoders are both transposed convolutional neural networks. The variational autoencoder used as the forward path employs a plasma particle model source field. As input, the outputs are the latent principal vector Z' and the source field reconstructed based on the latent principal vector. The variational autoencoder, used as a feedback path, employs a plasma particle model source field. Source field reconstructed based on latent principal vector The difference is taken as input, and the output is the latent error vector Z” and the source field reconstructed based on the latent error vector. The sum of the obtained potential principal vector and potential error vector is used as the potential vector representation of the source field of the plasma particle model: Z = Z' + Z.

4. The low-pressure plasma particle simulation method based on variational autoencoder according to claim 1, characterized in that, In step 2, the deep operator network includes a branch network and a backbone network, both of which are composed of feedforward neural networks.

5. The low-pressure plasma particle simulation method based on variational autoencoder according to claim 3, characterized in that, In step 3, the loss function is: Where MSE is the mean squared error function.

6. The low-pressure plasma particle simulation method based on variational autoencoder according to claim 1, characterized in that, The physical field corresponding to the low-pressure plasma particle model is the physical field in the semiconductor chip etching reaction chamber or the physical field in the reaction chamber of the energy conversion device.

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