A neural network embedding based plasma simulation acceleration method and system

Abstract

The application discloses a kind of plasma simulation acceleration method and system based on neural network embedding, for fixed space topological structure and constant background electromagnetic parameter plasma reaction chamber, based on linear superposition principle, total electromagnetic field is decomposed into plasma current contribution component and external excitation current contribution component, the former is directly predicted by neural network across sub-cycle, the latter is solved offline using frequency domain Maxwell equation set and then linearly reconstructed, the two part field components are linearly superimposed and coupled to plasma state evolution module, to drive the time-domain update of plasma state variable.The application compresses a large number of time-domain finite difference iterations within sub-cycle into single neural network forward propagation, greatly improves the computing efficiency, and training data can be obtained by random generation without relying on actual simulation process, the model can be reused across gas and across working condition, and has strong generalization ability.

Description

Technical Field

[0001] This invention belongs to the field of interdisciplinary technology of plasma numerical simulation and artificial intelligence, specifically relating to a plasma simulation acceleration method and system based on neural network embedding. Background Technology

[0002] Plasma technology is fundamental to cutting-edge industries such as semiconductor micro / nano manufacturing, space electric propulsion, and controlled nuclear fusion. Due to the strong nonlinearity, multi-scale, and multi-physics coupling characteristics of plasma systems, their internal physical processes are difficult to directly observe and accurately characterize through experimental means. To analyze and predict plasma generation, transport, evolution, and their mechanisms of action, numerical modeling methods are typically used to construct corresponding physical and mathematical models. These models are then combined with numerical calculation techniques to obtain the spatiotemporal distribution and variation patterns of key plasma parameters, providing technical support for the design, process optimization, and operational control of related devices.

[0003] However, high-precision plasma numerical simulations face severe efficiency bottlenecks. Due to the inherent multi-scale characteristics and complex internal dynamics of plasma systems, simulations are typically subject to extremely strict time and spatial resolution limitations to ensure the stability and accuracy of numerical solutions. These stringent computational conditions lead to extremely high computational complexity, making a single simulation often consume enormous computational resources and have a long timeframe.

[0004] To overcome the efficiency bottleneck in plasma numerical simulation, existing research has developed various acceleration strategies. Conventional methods mainly include hardware acceleration such as homogeneous parallelism and heterogeneous parallelism, as well as algorithm improvements such as implicit time integration and adaptive mesh refinement. However, hardware acceleration is constrained by communication overhead and Amdahl's law, and algorithm improvements have not escaped the inherent constraints of nonlinear matrix solving or dense mesh discrete iteration, making it difficult to achieve a leapfrog improvement in computational efficiency. In recent years, deep neural networks, with their powerful nonlinear fitting and high-dimensional feature extraction capabilities, have been gradually introduced into the field of plasma numerical simulation. Existing research generally adopts the end-to-end surrogate model construction paradigm, directly establishing the mapping relationship between the initial operating conditions and the physical field distribution through neural networks. However, such end-to-end methods deeply couple the general physical field solution process with specific gas components, chamber configurations, and operating parameters, severely limiting the model's generalization ability; even slight bias in conditions leads to significant accuracy degradation. Furthermore, the acquisition of high-precision plasma simulation data itself relies on large-scale numerical computation, the cost of generating training samples is extremely high, and the cost of remodeling is unbearable, severely restricting the practical application of such methods in complex industrial scenarios.

[0005] A deep analysis of the underlying computational mechanisms of plasma numerical models reveals that in high-fidelity numerical simulations of plasma involving complex electromagnetic evolution, solving for the electromagnetic field is often the core step consuming the most computational resources. Among existing numerical solution methods, the Finite-Difference Time-Domain (FDTD) method has been widely used in all-electromagnetic plasma simulations due to its high accuracy, wide bandwidth applicability, and ability to accurately capture high-frequency transient electromagnetic responses. However, due to the inherent multi-timescale coupling characteristics of plasma physics, the simulation framework based on FDTD faces severe computational efficiency challenges. Specifically, the time-domain advancement step size of FDTD is strictly limited by the Courant-Friedrichs-Lewy (CFL) stability condition, allowing for an extremely small evolution step size; while the characteristic timescales of dynamic processes such as charged particle transport and collisions are much larger, resulting in a mismatch of several orders of magnitude. To alleviate this timescale discrepancy, existing models generally introduce a sub-loop mechanism: within a relatively long macroscopic time step, the plasma current is approximated as a constant as the source term; during this period, the electromagnetic field solution module performs multiple FDTD iterations in small steps to satisfy the CFL condition. Although the sub-loop mechanism alleviates the limitations of the spatiotemporal scale difference on the overall architecture to some extent, as the geometric complexity of the simulation region increases or the mesh resolution becomes finer, the massive FDTD iterations within the sub-loop still account for the majority of the computational overhead, becoming the core bottleneck restricting the overall efficiency of all-electromagnetic plasma numerical simulation. Summary of the Invention

[0006] The purpose of this invention is to address the shortcomings of existing technologies by providing a plasma simulation acceleration method and system based on neural network embedding.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] In a first aspect, the present invention provides a plasma simulation acceleration method based on neural network embedding, comprising the following steps:

[0009] S1, Obtain the geometric topology and background medium parameters of the target plasma reaction chamber, and construct a neural network training dataset: The dataset consists of several sets of input samples and corresponding label samples. Each set of samples contains the plasma electromagnetic field component at the starting point of the sub-cycle and the frozen plasma current density, as well as the plasma electromagnetic field component at the corresponding ending point of the sub-cycle.

[0010] S2, Train a neural network model based on the training dataset, learn the mapping relationship between the plasma electromagnetic field component at the starting point of the sub-cycle and the plasma electromagnetic field component at the freezing current density to the ending point of the sub-cycle, and store the network parameters after training convergence.

[0011] S3, obtain the external excitation current density, which is a known time-varying excitation source generated by an external power source according to a preset rule; perform a Fourier expansion on the external excitation current density, extract each discrete angular frequency and the corresponding frequency domain current density source term, solve the frequency domain Maxwell's equations offline for each angular frequency, obtain the real part and imaginary part of the frequency domain electromagnetic field spatial distribution at each frequency, and store them as an external excitation frequency domain basis function library.

[0012] S4, set the initial state of the target plasma reaction chamber in the entire space at the initial moment, establish the macroscopic evolution time step and its corresponding macroscopic step interval, load the solidified neural network model parameters and external excitation frequency domain basis function library, and complete the initialization of the online simulation system;

[0013] S5, at the current macroscopic update node, input the current plasma electromagnetic field component and the plasma current density frozen within the macroscopic step size interval into the neural network model, and output the plasma electromagnetic field component of the next macroscopic node after crossing the entire sub-cycle interval through a single forward propagation.

[0014] S6. Based on the system physical clock, extract the instantaneous time factor corresponding to each discrete harmonic component, and use the frequency domain basis function library pre-stored in step S3 to perform linear weighting and real part extraction to reconstruct the external excitation electromagnetic field component at the current moment.

[0015] S7. Based on the principle of linear superposition, the plasma electromagnetic field component output in step S5 and the external excitation electromagnetic field component output in step S6 are linearly added point-to-point at the spatial grid level to obtain the total electromagnetic field distribution.

[0016] S8, substitute the total electromagnetic field distribution into the plasma physics evolution module, advance the plasma state variables to the next moment, and calculate the new plasma current density;

[0017] S9, take the global electromagnetic field distribution calculated by the current macroscopic node and the new plasma current density as the initial state input of the next node, return to step S5, until the evolution time reaches the preset maximum physical period or the steady-state convergence criterion is met.

[0018] Preferably, the construction of the training dataset in step S1 includes:

[0019] A random scalar noise field with a uniform distribution is generated in the simulation computation domain. After spatial smoothing filtering, a magnetic vector potential is constructed. The initial magnetic field is calculated analytically using the discrete curl operator, and the initial plasma electric field and current density are generated independently.

[0020] The generated physical field is subjected to boundary truncation and energy scaling is performed using a random scaling factor to form the initial input sample;

[0021] Using this sample as the initial field of the finite-difference time-domain solver, under the condition of freezing the current density source term, the time-domain difference iteration is performed with a microscopic time step that satisfies the Courant-Friedrich-Levi stability condition, and the physical field distribution at the end of the subcycle evolution is extracted as the label sample.

[0022] Preferably, the neural network model in step S2 adopts a network architecture suitable for learning linear mapping operators, the network architecture including Fourier neural operators and their variants.

[0023] Preferably, the neural network model employs a Fourier neural operator with an encoder-decoder structure and skip connections. The neural network model fuses the input physical field with dynamically generated spatial coordinate information and maps it to a high-dimensional feature space. Multi-scale features are extracted through a multi-layer encoder Fourier layer, and then fused step by step through a decoder Fourier layer with corresponding encoded features. Finally, the target physical field output is projected. Each Fourier layer extracts local spatial features and frequency domain modal features through spectral convolution branches, and completes channel transformation through point-by-point residual branches. The results of the two branches are fused and then normalized and nonlinearly activated to form the layer output.

[0024] Preferably, the frequency domain Maxwell's equations in step S3 are:

[0025]

[0026] in For frequency domain electric field, For the frequency domain magnetic field, j is the imaginary unit, ω is the angular frequency, μ is the permeability, and ε is the permittivity.

[0027] Preferably, the total electromagnetic field distribution in step S7 satisfies:

[0028]

[0029] in , The electromagnetic field component excited solely by the plasma current. , The electromagnetic field component excited solely by an external excitation current.

[0030] Preferably, the plasma physics evolution module in step S8 can be any plasma state evolution solution module, such as a particle simulation module, a fluid simulation module, a particle-fluid hybrid simulation module, or a Boltzmann equation model.

[0031] Secondly, the present invention provides a plasma simulation acceleration system based on neural network embedding for implementing the above method, comprising:

[0032] The plasma state evolution module is used to update the plasma state variables and output the plasma current density based on the plasma physics model adopted, taking the total electromagnetic field at the current update node as input.

[0033] The plasma electromagnetic field propulsion module includes a neural network model that has been trained and solidified offline. It is used to take the plasma electromagnetic field component of the current update node and the frozen plasma current density as inputs, and output the plasma electromagnetic field component of the next update node through a single forward propagation.

[0034] The external excitation electromagnetic field reconstruction module is used to load the offline pre-calculated frequency domain basis function library and reconstruct the external excitation electromagnetic field components based on the spectral characteristics of the external excitation current and the current phase.

[0035] The electromagnetic field superposition module is used to linearly superimpose the plasma electromagnetic field components output by the plasma electromagnetic field propulsion module and the external excitation electromagnetic field components output by the external excitation electromagnetic field reconstruction module to obtain the total electromagnetic field distribution and output it to the plasma state evolution module.

[0036] The electromagnetic field frequency domain offline solution module is used to solve the Maxwell's equations in the frequency domain for fixed chamber geometry and background electromagnetic parameters during the offline preparation stage, and obtain the frequency domain electromagnetic field basis response for each frequency component and store it in a fixed manner.

[0037] The neural network offline training module is used to generate a training dataset during the offline preparation phase and to perform supervised training on the neural network model. After the converged network parameters are solidified, they are loaded into the plasma electromagnetic field propulsion module.

[0038] Preferably, the plasma state evolution module, the plasma electromagnetic field propulsion module, the external excitation electromagnetic field reconstruction module, and the electromagnetic field superposition module form an iterative coupling closed loop:

[0039] The plasma state evolution module sends the plasma current density of the current node into the plasma electromagnetic field propulsion module;

[0040] The plasma electromagnetic field propulsion module takes the plasma electromagnetic field component output by the previous node and the frozen plasma current density as input to propel it to obtain the plasma electromagnetic field component of the next node. The result is sent back to the module itself as the initial field for the next sub-cycle, and sent to the electromagnetic field superposition module.

[0041] The external excitation electromagnetic field reconstruction module reconstructs the components of the external excitation electromagnetic field and sends them to the electromagnetic field superposition module.

[0042] The electromagnetic field superposition module linearly superimposes the two field components to obtain the total electromagnetic field, which is then sent back to the plasma state evolution module to drive its state update.

[0043] The plasma state evolution module outputs a new plasma current based on this, and then re-enters the above process to achieve iterative coupling and propulsion.

[0044] Thirdly, the present invention provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described method.

[0045] Fourthly, the present invention provides an electronic device, including a storage component and a processing unit, wherein the storage component stores machine-executable instructions that can be executed by the processing unit, and the processing unit executes the machine-executable instructions to implement the above-described method.

[0046] The beneficial effects of this invention are:

[0047] 1. This invention specifically eliminates the time-step iteration bottleneck in solving plasma electromagnetic fields, significantly improving simulation efficiency. In FDTD-based plasma simulations, due to the order-of-magnitude difference between the electromagnetic field evolution timescale and the plasma response timescale, a large number of repetitive electromagnetic field time-step iterations are required between adjacent plasma current update nodes. During this process, the plasma current remains constant, constituting the main computational overhead in the simulation process. This invention establishes an end-to-end mapping from the current plasma electromagnetic field state and current density distribution to the electromagnetic field state of the next update node through a neural network, and uses Maxwell's equations in the frequency domain to perform offline pre-calculation of the field components contributed by the external excitation current. The two work together to compress the above iterative process into a single neural network forward propagation and a single linear superposition, achieving a speedup of the same order as the number of sub-loop steps.

[0048] 2. The learning object of the neural network in this invention is the electromagnetic field time-progression operator constrained by Maxwell's equations. Since the background electromagnetic parameters of the chamber are constant and the boundary conditions are linear, this electromagnetic evolution system constitutes a strictly linear dynamic system, and the corresponding time-progression operator is a purely linear mapping operator. This operator has a low-dimensional function space and clear physical priors. The linear physical consistency implied in the training data can be applied as a strong implicit regularization constraint to the network parameter space, thereby effectively reducing the complexity of the hypothesis space. Therefore, the network training converges faster, requires fewer training samples, and the numerical stability and generalization consistency of the prediction results are stronger. This overcomes the inherent defects of existing end-to-end surrogate models, which suffer from severely limited generalization ability and significant accuracy degradation due to slight conditional bias, caused by deep coupling of physical field solutions with specific working conditions. Furthermore, since the training data in this invention can be obtained through random generation without relying on high-cost simulations, the sample diversity is sufficient, fundamentally avoiding the bottleneck of difficult training sample acquisition in traditional methods.

[0049] 3. This invention embeds a neural network into the underlying electromagnetic field solution process. Its training data consists of plasma electromagnetic field and current density distributions that satisfy real physical laws. The neural network learns the time-domain evolution operator of the electromagnetic field under the constraints of Maxwell's equations, which is completely independent of specific gas types and process parameters. Therefore, the model trained for the same chamber can be directly applied to different simulation conditions within that chamber without the need for re-collecting data or retraining, demonstrating significant engineering reuse value.

[0050] 4. The acceleration method proposed in this invention operates on the electromagnetic field solution stage, completely decoupling from the control equations and numerical implementation of the plasma state evolution solution module. The coupling interface remains unchanged across various plasma simulation frameworks, including particle simulation, fluid simulation, and hybrid simulation. Therefore, this invention can be embedded as an independent acceleration module into existing simulation frameworks without requiring any modifications to the state evolution solution module. It boasts low engineering implementation costs, flexible deployment, and broad applicability and portability. Attached Figure Description

[0051] Figure 1 This is a block diagram of the system of the present invention.

[0052] Figure 2 This is a flowchart of the method of the present invention.

[0053] Figure 3 The diagram shows the geometry of the reactor in this embodiment.

[0054] Figure 4 The flowchart shows the calculation process of the hybrid method combining particle-Monte Carlo collision (PIC / MCC) and fluid dynamics used for plasma physics modeling.

[0055] Figure 5This is a schematic diagram of the Fourier neural operator (U-FNO) model with an encoder-decoder jump connection structure.

[0056] Figure 6 This is a comparison chart of the neural network prediction results and the true values ​​of FDTD. Detailed Implementation

[0057] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0058] This invention addresses plasma reaction chambers with fixed spatial topology and constant background electromagnetic parameters. In the full electromagnetic numerical simulation of this type of chamber, Maxwell's equations are a strictly linear set of partial differential equations, and their governing equations and constitutive relations are as follows:

[0059] .

[0060] In the formula, For electric field strength, The magnetic field strength, Magnetic flux density It is the electric displacement vector. For free current density, Free charge density, , Let be the permittivity and permeability of the background medium in the chamber, respectively, and t be time. In the above system of equations, the time-domain progression of the electromagnetic field FDTD depends on Faraday's law of electromagnetic induction and Ampère-Maxwell's law, while Gauss's electric field law and Gauss's magnetic field law are only constraints that must be satisfied at the initial moment and do not participate in the time-domain update.

[0061] The total current density injected into the electromagnetic field FDTD solver is the source term in the Ampere-Maxwell equations, which is determined by the plasma current density. With external excitation current density It consists of two parts, namely:

[0062] .

[0063] Among them, plasma current density Calculated and fed back by the plasma state evolution solution module; external excitation current density It is generated by an external power source according to a preset pattern, and is usually a periodic time-varying excitation.

[0064] Since the Maxwell's equations and boundary conditions within this chamber are strictly linear, according to the superposition principle, the total current density... The generated electromagnetic field is strictly equal to and The electromagnetic fields obtained from separately excited states are linearly superimposed. Therefore, the total electromagnetic field within the cavity can be decomposed into two independent field components: one derived from the plasma current density. Individually excited plasma electromagnetic field components and and the external excitation current density Individually excited external excitation electromagnetic field components and The two satisfy the following relationship:

[0065] .

[0066] In this invention, the two sets of electromagnetic field components are solved independently during the simulation process, and they are not coupled at the computational level. The total electromagnetic field is constructed only when transmitting data to the upper-level plasma physics model, based on the superposition principle. and This method enables the separate calculation and unified output of the total electromagnetic field.

[0067] Based on the aforementioned fundamental premises and physical foundations, this invention proposes a plasma simulation acceleration method and system based on neural network embedding, as described above.

[0068] See Figure 1 The all-electromagnetic plasma simulation system for implementing the acceleration method of this invention comprises the following six modules: a plasma state evolution module, a plasma electromagnetic field propulsion module, an external excitation electromagnetic field reconstruction module, an electromagnetic field superposition module, an electromagnetic field frequency domain offline solution module, and a neural network offline training module. The functions of each module and their interconnections are as follows.

[0069] Plasma State Evolution Module. This module carries a specific plasma physics model, updating the total electromagnetic field components at the current node. and As input, the plasma state variables are updated according to the governing equations of the adopted model, and the output plasma current density components are calculated self-consistently. This module updates the state over macroscopic time, with the update interval determined by the characteristic timescale of the plasma dynamics process.

[0070] Plasma electromagnetic field propulsion module. This module replaces the stepwise FDTD iterations within sub-loop intervals with single-step neural network forward propagation, achieving cross-node propagation of the plasma electromagnetic field across the entire sub-loop: using the plasma electromagnetic field components of the currently updated node. and and the plasma current density frozen within the interval As input, directly output the field components of the next update node. and The network parameters are trained offline by the neural network offline training module and then permanently loaded.

[0071] External Excitation Electromagnetic Field Reconstruction Module. This module is used for online reconstruction of the time-domain electromagnetic field generated by an external drive. By utilizing offline pre-calculated frequency-domain response data, this module eliminates the need for online partial differential equation solving and microscopic time-step iteration. Specifically, the module first loads the frequency-domain basis function response matrix calculated offline; then, based on the known external excitation current... Based on the spectral characteristics and the phase corresponding to the current macroscopic time node, the aforementioned frequency domain basis functions are weighted and combined with time domain mapping to directly output the external excitation electromagnetic field components of the current node. and This reconstruction process simplifies the time evolution of the electromagnetic field to a linear weighted summation of basis functions and phase calculation, resulting in extremely low online computational overhead.

[0072] Electromagnetic field superposition module. Due to the linear characteristics of the electromagnetic field evolution system, this module linearly adds the plasma electromagnetic field component output from the plasma electromagnetic field propulsion module to the external excitation electromagnetic field component output from the external excitation electromagnetic field reconstruction module, obtaining the total electromagnetic field distribution at the current update node. , It is then output to the plasma state evolution module.

[0073] An offline frequency domain solution module for electromagnetic fields is used in the offline preparation phase. For a fixed cavity geometry and constant background electromagnetic parameters, this module solves Maxwell's equations in the frequency domain to obtain the frequency domain electromagnetic response of the cavity to a set of basis function excitations. Specifically, the module establishes and solves the corresponding Maxwell's equations for each frequency component of interest, thus obtaining the electromagnetic field basis response at the corresponding frequency. Since the cavity geometry and background medium parameters remain unchanged during the simulation, the above frequency domain response is determined only by the inherent characteristics of the system and is independent of the amplitude, time-domain waveform, and transient state of the plasma from the external excitation. Therefore, this frequency domain solution process only needs to be executed once during the offline preparation phase. The module pre-calculates and stores the obtained electromagnetic field basis responses for each frequency component, which are then called and linearly superimposed and reconstructed by the external excitation electromagnetic field reconstruction module during the online simulation phase.

[0074] The neural network offline pre-training module is used in the offline preparation phase, providing offline data generation and model training functions for the plasma electromagnetic field propulsion engine. For data generation, the module randomly generates field components and current density matrices that strictly adhere to the physical self-consistency constraints of Maxwell's equations as network input. Then, using a linear FDTD solver that satisfies the CFL stability condition, it performs single-step sub-cycle time-domain propulsion under the condition of frozen current source terms, solving for the field distribution at the corresponding time nodes as labels, thus completely constructing training sample pairs. Since the propulsion operators learned by the network depend only on the inherent geometry and topology of the chamber, the background medium, and the time step, the entire data sampling process does not require calling complex plasma transient solutions, achieving zero-dependency decoupling at the computational physics level. For network training, the module uses the aforementioned self-consistent sample set to complete supervised learning, and after model convergence, it performs persistent storage and state freezing of the network parameters, finally loading them into the backbone online simulation framework as an independent operator library.

[0075] The connection relationships and data flow of each module are as follows: Figure 1 As shown. During the online simulation phase, the plasma state evolution module, plasma electromagnetic field propulsion module, external excitation electromagnetic field reconstruction module, and electromagnetic field superposition module form an iteratively coupled closed loop: the plasma state evolution module freezes the plasma current at the current update node. Directly fed into the plasma electromagnetic field propulsion module; the plasma electromagnetic field propulsion module uses the plasma electromagnetic field components output from its previous node. and and frozen As input, the plasma electromagnetic field components of the next node are obtained. and The result is fed back to the module itself as the initial field for the next sub-cycle, and also sent to the electromagnetic field superposition module; the external excitation electromagnetic field reconstruction module is based on the known external excitation current. The external excitation electromagnetic field components are reconstructed by weighting the pre-stored frequency domain basis response with the spectrum of the current node and the phase corresponding to the current node. and The data is fed into the electromagnetic field superposition module; the electromagnetic field superposition module linearly superimposes the two field components to obtain the total electromagnetic field. and The plasma state evolution module is fed back to drive its state update; the plasma state evolution module then outputs a new plasma current. The process is then repeated to achieve iterative coupling and propulsion between electromagnetic field components and plasma state. During the offline preparation phase, the electromagnetic field frequency domain offline solution module solidifies the obtained frequency domain basis responses of each frequency component and loads them into the external excitation electromagnetic field reconstruction module. The neural network offline training module solidifies the converged network parameters and loads them into the plasma electromagnetic field propulsion module.

[0076] See Figure 2 The plasma simulation acceleration method implemented by the above system is divided into an offline preparation stage (steps S1 to S3) and an online simulation stage (steps S4 to S9), as detailed below:

[0077] S1, Construct the neural network training dataset.

[0078] Based on the geometric topology and background medium parameters of the target plasma reaction chamber, this method generates a large-scale and physically consistent training sample set offline by combining random sampling with rigorous physical priors.

[0079] First, a uniformly distributed random scalar noise field is generated within the simulation computational domain, and a multi-level binomial spatial smoothing filter is introduced to remove high-frequency components, thereby obtaining a spatially band-limited smoothed scalar field matching the resolution scale of the discrete grid. To ensure that the initial electromagnetic field strictly falls on the physical manifold and satisfies the divergence-free constraint of Gauss's magnetic law, this method avoids directly sampling the magnetic field randomly. Instead, it prioritizes constructing a spatially smoothed magnetic vector potential based on the aforementioned filtering method, and then analytically calculates the initial magnetic field using a discrete curl operator, thus mathematically guaranteeing the divergence-free characteristics of the generated sample. Similarly, this smoothing method is used to independently generate the initial plasma electric field and current density.

[0080] Next, boundary cutoffs are applied to each physical field to conform to the axisymmetry of the chamber and the boundary conditions of an ideal electrical conductor. The root mean square of each field is then scaled using a random scaling factor to ensure that the gradient contributions of each feature channel are balanced during network training, thus forming the initial input samples.

[0081] Subsequently, using this sample as the initial field for the FDTD solver, and under the condition of freezing the current density source term, a specified number of time-domain difference iterations are continuously performed using a microscopic time step that satisfies CFL stability. The physical field distribution at the end of the sub-cycle evolution is then extracted as the target output label. Since the time-progression operator to be fitted is a low-dimensional linear mapping, its hypothesis space complexity is low. Therefore, repeating the above process can achieve sufficient sampling of the state space at extremely low cost, constructing a large batch of supervised learning training sample sets that strictly satisfy physical self-consistency constraints.

[0082] S2, Neural Network Training.

[0083] Based on the training sample set constructed in step S1, the operator model used to replace the sub-cycle time-domain propagation is trained to learn the mapping relationship from the plasma electromagnetic field components and freezing current density at the sub-cycle starting point to the plasma electromagnetic field components at the sub-cycle ending point. The operator model can employ various neural network architectures and their combinations or variations suitable for learning function space mappings or time-propagation operators. Given that the learned propagation operator has linear properties under constant background electromagnetic parameters and frozen source terms, the model can also adopt a linear or approximately linear network structure that matches this prior property. It should be noted that the neural network model can use nonlinear activation functions without affecting its ability to approximate the target linear mapping; the introduction of nonlinear degrees of freedom is mainly to enhance the model's representational flexibility.

[0084] In a preferred embodiment of the present invention, the neural network model employs a Fourier neural operator (U-FNO) with an encoder-decoder skip connection structure.

[0085] The neural network model dynamically generates spatial coordinate channels with values ​​normalized to a predetermined interval based on the spatial resolution of the input physical field tensor. The input physical field tensor is concatenated with these spatial coordinate channels, and then mapped to a high-dimensional feature space through a lifting layer. These high-dimensional features are sequentially passed through multiple encoder Fourier layers, and the encoded features output by each encoder Fourier layer are saved. Subsequently, they are sequentially passed through multiple decoder Fourier layers, and the corresponding encoded features are selected in reverse order of their saved storage order. The current decoded feature is concatenated with the selected encoded features, mapped to a predetermined number of channels through a fusion layer, and then passed to the next decoder Fourier layer. After decoding, the high-dimensional features are mapped to the target physical field output through a projection layer.

[0086] Each encoder Fourier layer and decoder Fourier layer includes a spectral convolution branch and a pointwise residual branch. In the spectral convolution branch, the input features are first mixed with local spatial features and channel features through spatial convolution, then transformed to the frequency domain through Fourier transform. A linear spectral transform determined by learnable complex weights is applied to the preset Fourier modes, the unselected Fourier modes are set to zero, and the input features are returned to the spatial domain through inverse Fourier transform. In the pointwise residual branch, the input features are transformed into channel linear features through pointwise convolution. The outputs of the spectral convolution branch and the pointwise residual branch are added together and then subjected to grouping normalization and nonlinear activation processing in sequence to obtain the output of the corresponding Fourier layer.

[0087] The training process uses the starting field quantity in the sample pair as the model input and the corresponding sub-loop ending field quantity as the label to construct the prediction error as the basic loss function. Physical constraint terms can be introduced as needed to enhance the model's physical consistency and generalization ability. Based on the obtained loss function, gradient optimization is used to iteratively update the model parameters until the preset convergence accuracy is met. After training, the converged model parameters are stored and saved for loading and use by the plasma electromagnetic field propulsion module in the online simulation phase.

[0088] S3, frequency domain offline solution of external excitation electromagnetic field.

[0089] For external excitation sources with fixed spatial physical locations and known time evolution patterns, this step utilizes the frequency domain mapping characteristics of linear time-invariant systems to completely avoid complex online time-domain solutions. First, the known external excitation current density is mapped along the time axis. Perform a Fourier expansion to decouple it into a linear superposition of spatial complex amplitudes and discrete-time harmonic factors:

[0090] .

[0091] This allows for the precise extraction of the discrete angular frequency set and the corresponding frequency domain current density source terms for each frequency. Subsequently, given that the geometric topology of the target chamber and the electromagnetic parameters of its underlying passive medium maintain strictly linear and time-invariant characteristics throughout their entire lifecycle, for each extracted angular frequency, using... As an independent driving source, solve the following three-dimensional frequency domain Maxwell's equations, which include spatial medium distribution, offline:

[0092] .

[0093] After obtaining the steady-state response using a high-precision frequency domain solver, such as the finite difference frequency domain method (FDFD), the frequency domain electric field at each frequency is extracted. With frequency domain magnetic field The real and imaginary parts of the spatial distribution are persistently stored according to discrete grid coordinates and frequency indices, constructing an offline external excitation frequency domain basis function library. During the online simulation phase, the system only needs to calculate the rotation factor based on physical time, perform low-dimensional algebraic lookup and linear weighted reconstruction on this static basis function library, and directly obtain the time-domain external field components, fundamentally eliminating the stringent constraints of high-frequency complex excitation sources on the micro-step size of online plasma time-domain evolution.

[0094] S4, Online simulation system initialization.

[0095] At the initial moment t0 of the physical evolution, the initial state of the plasma system in all space is set, including the electromagnetic field components inside the plasma. and The plasma dynamic state variables are used to calculate the initial plasma current density distribution using the plasma state evolution module. Subsequently, a macroscopic evolution time step matching the plasma physics response and its corresponding macroscopic step interval were established. The offline-fixed neural network propulsion operator and the pre-computed external excitation frequency domain feature library were loaded into the system's running memory, completing the configuration of the online iterative solution environment.

[0096] S5, plasma electromagnetic field neural network trans-loop propulsion.

[0097] At the current macro update node t n At this point, the current plasma electromagnetic field components and And the plasma current density frozen within this macroscopic step size range. As input, it is injected into the neural network electromagnetic field propulsion module. After a single forward propagation of the neural network, it directly outputs the plasma electromagnetic field of the next macroscopic node after traversing the entire sub-loop interval. and This allows for the complete replacement of traditional micro-time step difference iteration with end-to-end operator reasoning.

[0098] S6, Time-domain dynamic reconstruction of the external excitation electromagnetic field.

[0099] For the next macro update node t n+1 Based on the system's physical clock, the instantaneous time factor corresponding to each discrete harmonic component is extracted. Subsequently, the frequency domain basis function library pre-stored in step S3 is invoked, and linear weighting and real part extraction are performed based on the phase to directly reconstruct the external excitation electromagnetic field at this moment:

[0100]

[0101] This step involves only algebraic mapping operations, which significantly reduces computational complexity compared to the conventional time-domain difference solution of global partial differential equations.

[0102] S7, linear self-consistent superposition of global electromagnetic fields.

[0103] Based on the strict linear superposition principle satisfied by the target chamber system, the plasma internal field components predicted by the neural network (output of step S5) and the external field components reconstructed analytically in the frequency domain (output of step S6) are linearly added point-to-point at the spatial grid level. This linearly superimposes the two independently solved electromagnetic field components to obtain the total electromagnetic field distribution driving the subsequent evolution of the plasma throughout space.

[0104] .

[0105] S8, the time-domain evolution of plasma dynamic states.

[0106] The superimposed total electromagnetic field and Substitute this into the plasma physics evolution module. Based on the system-specific plasma physics control equations, advance the plasma state variables to t. n+1 At what time, the new plasma current density was calculated. , serving as the physical source term for the subsequent evolution of electromagnetic systems.

[0107] S9, System State Feedback and Time-Domain Cyclic Iteration.

[0108] The global electromagnetic field distribution calculated at the current macroscopic node and the new plasma current density are used as the initial state input for the next node to update the system state variables and time. Then, return to step S5 to continuously drive the self-consistent coupling iteration of the electromagnetic field and plasma state until the evolution time reaches the preset maximum physical period or the system macroscopic parameters meet the expected steady-state convergence criterion, at which point the simulation terminates and the final plasma spatiotemporal evolution result is output.

[0109] To further illustrate the method of the present invention, an inductively coupled plasma (ICP) reaction chamber was used as the test object to demonstrate the method. The reactor geometry is as follows: Figure 3 As shown.

[0110] This embodiment uses a miniaturized inductively coupled plasma reaction chamber as the subject of simulation verification. This chamber includes typical ICP reactor structures such as a dielectric window, external induction coil, bottom substrate, and gas discharge region, and can be used to verify the basic mechanism and acceleration effect of the neural network embedded electromagnetic field propulsion method described in this invention.

[0111] In this embodiment, the reaction chamber is modeled as an axisymmetric cylindrical structure. In the two-dimensional cylindrical coordinate system rz plane, r represents the radial coordinate from the central axis of the chamber, and z represents the axial coordinate along the height of the chamber. The origin of the coordinate system is set at the center of the bottom of the chamber. The total radius and total height of the simulation region are 40 mm; the height of the plasma simulation region is 30 mm, the thickness of the quartz dielectric window is 5 mm, and the height of the vacuum region where the coil is located is 5 mm. A substrate is set at the bottom of the chamber, with a radius of 30 mm and a height of 5 mm; in the electromagnetic field solution, the substrate region is equivalent to an alumina dielectric region.

[0112] In this embodiment, the external induction coil is disposed in the vacuum region above the quartz dielectric window, and is equivalent to four ring coils with rectangular cross-sections in the two-dimensional rz plane. The height and width of each coil cross-section are 2.5 mm, and the four coils are arranged radially in sequence. The center-to-center distance between adjacent coil cross-sections is 8.5 mm, and the center of the first coil cross-section is 7.25 mm from the central axis of the chamber. The source RF excitation frequency is 13.56 MHz, and the source RF current amplitude is 30 A.

[0113] In this embodiment, the background gas is argon (Ar) at a pressure of 10 Pa. Among the electromagnetic parameters of the chamber material, the dielectric constant of the quartz dielectric window is 3.9ε0, and the permeability is μ0; the dielectric constant of the equivalent alumina dielectric of the substrate is 9.5ε0, and the permeability is μ0; the background dielectric parameters for the remaining areas are ε0 and μ0.

[0114] In this embodiment, the plasma physics model employs a hybrid approach combining particle-in-cell / Monte Carlo Collisions (PIC / MCC) and fluid dynamics. The specific calculation process is as follows: Figure 4 As shown.

[0115] In this embodiment, the neural network employs a Fourier neural operator (U-FNO) with an encoder-decoder skip connection structure, as shown in the figure below. Figure 5As shown, the neural network dynamically generates normalized spatial coordinate channels based on the spatial resolution of the input physical field tensor, concatenates these channels with the input physical field tensor, maps them to a high-dimensional feature space via a lifting layer P, and then sequentially passes them through multiple encoder Fourier layers and multiple decoder Fourier layers. The encoder extracts and saves encoded features layer by layer; the decoder selects corresponding encoded features in reverse order of feature saving, concatenates the current decoded feature with the selected encoded feature, and maps it to a predetermined number of channels via a skip connection fusion layer. Each intermediate fused feature is passed to the next decoder Fourier layer, and the final fused feature is dimensionality-reduced and mapped to the target physical field output via a projection layer Q, thereby fusing local spatial features and global correlation features at different depths. Each encoder Fourier layer and decoder Fourier layer includes a spectral convolution branch R and a pointwise residual branch W. In the spectral convolution branch R, the input features are first mixed with local spatial features and channel features through spatial convolution, and then transformed to the frequency domain through Fourier transform. A linear spectral transform determined by learnable complex weights is applied to the preset Fourier modes, the unselected Fourier modes are set to zero, and the input features are returned to the spatial domain through inverse Fourier transform. In the pointwise residual branch W, the input features are subjected to channel linear transform through pointwise convolution. The outputs of the two branches are added together and then processed sequentially by a grouping normalization layer and a nonlinear activation function to obtain the output of the corresponding Fourier layer. In this embodiment, the nonlinear activation function is the GELU activation function. In a feasible implementation, the Fourier neural operator includes a lifting layer P, multiple encoder Fourier layers, multiple decoder Fourier layers, multiple skip connection fusion layers, and a projection layer Q. As a preferred configuration, the total number of Fourier layers is set to 6, of which the first 3 layers are encoder Fourier layers and the last 3 layers are decoder Fourier layers; the number of hidden feature channels is set to 64, and the Fourier mode truncation parameters in the two spatial directions are set to 16 respectively. The parameters mentioned above are only one feasible configuration and can be adjusted according to the size of the training data, the accuracy of the target prediction, the spatial resolution of the input physics field, and the computing resources.

[0116] The training dataset was generated by a finite-difference time-domain electromagnetic field solver. Specifically, a uniformly distributed random noise field was generated within the simulation computation domain. After spatial smoothing filtering and energy scaling, initial electric and current densities were independently generated. Simultaneously, a magnetic vector potential was constructed based on the filtered noise field. The initial magnetic field components, which naturally satisfy divergence-free constraints, were analytically calculated using discrete operators and similarly subjected to energy scaling, thus forming a complete initial input sample. In each sample set, the preset initial electromagnetic field components and current density source terms were used as the physical field input to the neural network. Under the condition of freezing the current density source terms, the solver evolved for a preset number of steps, and the target electromagnetic field component at the evolution endpoint was extracted as the label sample. In the formal training, the total number of samples in the dataset was set to 24,000 independent samples, randomly divided into training, validation, and test sets in an 8:1:1 ratio. The training set included 19,200 samples, the validation set included 2,400 samples, and the test set included 2,400 samples. The training set is used to update the neural network parameters, the validation set is used to monitor the training process, adjust the training hyperparameters, and select the best-performing model, and the test set is used to independently evaluate the performance of the neural network after training.

[0117] In one feasible training configuration, the training batch size is set to 32, the Adam optimizer is used to iteratively update the model parameters, and the maximum learning rate is set to 2.0 × 10⁻⁶. -3 The weight decay factor is set to 1.0 × 10. -4 To improve training stability, the gradient norm pruning threshold was set to 1.0. The learning rate was scheduled using a combination of warm-up and cosine decay. The learning rate was gradually increased in the early stages of training and then gradually decreased to approximately 1.0 × 10⁻⁶ after the warm-up period, following a cosine decay rule. -5 The loss function is the relative L2 error, calculated and averaged for each output physical field component.

[0118] After each training epoch, the validation loss of the model is calculated using the validation set, and the optimal model parameters are selected and saved based on the validation loss. Training can be terminated early if the validation loss does not improve within a preset number of consecutive epochs. After training is complete, the optimal model parameters from the validation set are loaded, and the model's prediction accuracy and generalization ability are independently evaluated using the test set.

[0119] The neural network's prediction results on the test set are as follows: Figure 6 As shown.

[0120] The above description of the embodiments is only for the purpose of helping to understand the method and core ideas of the present invention. It should be noted that those skilled in the art can make several improvements and modifications to the present invention without departing from the principles of the present invention, and these improvements and modifications also fall within the protection scope of the claims of the present invention.

Claims

1. A plasma simulation acceleration method based on neural network embedding, characterized in that, Includes the following steps: S1, for the target plasma reaction chamber, generate several sets of plasma electromagnetic field components and frozen plasma current densities at the starting point of sub-cycles, as well as the corresponding plasma electromagnetic field components at the end point of sub-cycles, as training datasets. S2, Train a neural network model based on the training dataset, learn the mapping relationship between the plasma electromagnetic field component at the starting point of the sub-cycle and the plasma electromagnetic field component at the freezing current density to the ending point of the sub-cycle, and store the network parameters after training convergence. S3 performs frequency domain analysis on the known external excitation current density, solves the frequency domain Maxwell's equations offline, and stores the frequency domain electromagnetic field response of each frequency component as the external excitation frequency domain basis function library. S4, set the initial state of the target plasma reaction chamber in the entire space at the initial moment, establish the macroscopic evolution time step and its corresponding macroscopic step interval, and complete the simulation initialization; S5, at the current macroscopic update node, input the current plasma electromagnetic field component and the plasma current density frozen within the macroscopic step size interval into the neural network model, and output the plasma electromagnetic field component of the next macroscopic node after crossing the entire sub-cycle interval through a single forward propagation. S6, Based on the frequency domain basis function library described in step S3, linearly reconstruct the external excitation electromagnetic field components at the current moment; S7. Based on the principle of linear superposition, the plasma electromagnetic field component described in step S5 and the external excitation electromagnetic field component described in step S6 are linearly added point-to-point at the spatial grid level to obtain the total electromagnetic field distribution. S8, substitute the total electromagnetic field distribution into the plasma physics evolution module, advance the plasma state variables to the next moment, and calculate the new plasma current density; S9, take the global electromagnetic field distribution calculated by the current macroscopic node and the new plasma current density as the initial state input of the next node, return to step S5, until the evolution time reaches the preset maximum physical period or the steady-state convergence criterion is met.

2. The plasma simulation acceleration method based on neural network embedding according to claim 1, characterized in that, The construction of the training dataset in step S1 includes: A random scalar noise field with a uniform distribution is generated in the simulation computation domain. After spatial smoothing filtering, a magnetic vector potential is constructed. The initial magnetic field is analytically calculated using the discrete curl operator, and the initial plasma electric field and current density are generated independently. The generated physical field is subjected to boundary truncation and energy scaling is performed using a random scaling factor to form the initial input sample; Using this sample as the initial field of the finite-difference time-domain solver, under the condition of freezing the current density source term, the time-domain difference iteration is performed with a microscopic time step that satisfies the Courant-Friedrich-Levi stability condition, and the physical field distribution at the end of the subcycle evolution is extracted as the label sample.

3. The plasma simulation acceleration method based on neural network embedding according to claim 1, characterized in that, The neural network model described in step S2 employs a network architecture suitable for learning linear mapping operators, which includes Fourier neural operators and their variants.

4. The plasma simulation acceleration method based on neural network embedding according to claim 1, characterized in that, The neural network model is a Fourier neural operator with an encoder-decoder structure and skip connections. The neural network model fuses the input physical field with dynamically generated spatial coordinate information and maps it to a high-dimensional feature space. Multi-scale features are extracted through a multi-layer encoder Fourier layer, and then the corresponding encoded features are fused step by step through a decoder Fourier layer. Finally, the target physical field output is projected. Each Fourier layer extracts local spatial features and frequency domain modal features through spectral convolution branches, and completes channel transformation through point-by-point residual branches. The results of the two branches are fused and then normalized and nonlinearly activated to form the output layer.

5. The plasma simulation acceleration method based on neural network embedding according to claim 1, characterized in that, The frequency domain Maxwell's equations mentioned in step S3 are: in For frequency domain electric field, For the frequency domain magnetic field, j is the imaginary unit, ω is the angular frequency, μ is the permeability, and ε is the permittivity.

6. The plasma simulation acceleration method based on neural network embedding according to claim 1, characterized in that, The total electromagnetic field distribution in step S7 satisfies: in , The electromagnetic field component excited solely by the plasma current. , The electromagnetic field component excited solely by an external excitation current.

7. A plasma simulation acceleration system based on neural network embedding that implements the method as described in any one of claims 1-6, characterized in that, include: The plasma state evolution module is used to update the plasma state variables and output the plasma current density based on the plasma physics model adopted, taking the total electromagnetic field at the current update node as input. The plasma electromagnetic field propulsion module includes a neural network model that has been trained and solidified offline. It is used to take the plasma electromagnetic field component of the current update node and the frozen plasma current density as inputs, and output the plasma electromagnetic field component of the next update node through a single forward propagation. The external excitation electromagnetic field reconstruction module is used to load the offline pre-calculated frequency domain basis function library and reconstruct the external excitation electromagnetic field components based on the spectral characteristics of the external excitation current and the current phase. The electromagnetic field superposition module is used to linearly superimpose the plasma electromagnetic field components output by the plasma electromagnetic field propulsion module and the external excitation electromagnetic field components output by the external excitation electromagnetic field reconstruction module to obtain the total electromagnetic field distribution and output it to the plasma state evolution module. The electromagnetic field frequency domain offline solution module is used to solve the Maxwell's equations in the frequency domain for fixed chamber geometry and background electromagnetic parameters during the offline preparation stage, and obtain the frequency domain electromagnetic field basis response for each frequency component and store it in a fixed manner. The neural network offline training module is used to generate a training dataset and perform supervised training on the neural network model during the offline preparation phase, and then load the converged network parameters into the plasma electromagnetic field propulsion module.

8. A computer-readable storage medium, characterized in that, It stores a computer program that, when executed by a processor, implements the method as described in any one of claims 1-6.

9. An electronic device comprising a storage component and a processing unit, the storage component storing machine-executable instructions executable by the processing unit, the processing unit executing the machine-executable instructions to implement the method as claimed in any one of claims 1-6.

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