Physics-informed neural networks for solving forward and inverse Vlasov–Poisson equation via fully kinetic simulation

The Vlasov–Poisson equation is one of the most fundamental models in plasma physics. It has been widely used in areas such as confined plasmas in thermonuclear research and space plasmas in planetary magnetospheres. In this study, we explore the feasibility of the physics-informed neural networks for solving forward and inverse Vlasov–Poisson equation (PINN-Vlasov). The PINN-Vlasov method employs a multilayer perceptron (MLP) to represent the solution of the Vlasov–Poisson equation. The training dataset comprises the randomly sampled time, space, and velocity coordinates and the corresponding distribution function. We generate training data using the fully kinetic PIC simulation rather than the analytical solution to the Vlasov–Poisson equation to eliminate the correlation between data and equations. The Vlasov equation and Poisson equation are concurrently integrated into the PINN-Vlasov framework using automatic differentiation and the trapezoidal rule, respectively. By minimizing the residuals between the reconstructed distribution function and labeled data, and the physically constrained residuals of the Vlasov–Poisson equation, the PINN-Vlasov method is capable of dealing with both forward and inverse problems. For forward problems, the PINN-Vlasov method can solve the Vlasov–Poisson equation with given initial and boundary conditions. For inverse problems, the completely unknown electric field and equation coefficients can be predicted with the PINN-Vlasov method using little particle distribution data.

In addition to the traditional numerical methods, physics-informed deep learning algorithms [21][22][23][24] have emerged as an essential branch for solving partial differential equations (PDEs) like the Vlasov-Poisson equation.Raissi et al [21] proposed the physics-informed neural networks (PINNs) to solve forward and inverse problems involving nonlinear PDEs.In PINNs, the solution to a PDE is approximated by the MLP with a small amount of data constrained by the PDE.Based on physics-informed deep learning, PINNs can not only reconstruct the accurate solution but also provide a better physical interpretation for the results than purely data-driven neural networks.More recently, PINNs have been applied to construct data-driven solutions to the Boltzmann equation [25][26][27].Lou et al [25] employed PINNs to solve the Boltzmann equation with the Bhatnagar-Gross-Krook [28] collision model (PINN-BGK).In PINN-BGK, two MLPs were used to approximate the equilibrium distribution function and the non-equilibrium distribution function, respectively.The results show that PINN-BGK works well to solve forward and inverse Boltzmann equations.Consequently, applying PINNs to deal with the Vlasov-Poisson equation is promising, which is quite similar to the Boltzmann equation.
To the best of our knowledge, few studies related PINNs can be utilized to solve the Vlasov-Poisson equation with an additional electric field term compared to the Boltzmann equation.In this work, we explore the feasibility of the PINNs for solving the Vlasov-Poisson equation (PINN-Vlasov).In addition to the physics-informed network structure itself, the way the data is generated is very important for this work.However, whether in [21] or in [25], the data used to train PINNs were virtually generated by the analytical/numerical solutions of the associated PDEs.To establish a more convincing validation, it is advantageous to utilize data obtained from sources such as experimental measurements or numerical simulations that are not intrinsically tied to the underlying PDEs.Consequently, in this study, the fully kinetic PIC method is employed to generate phase space data for the training of the PINNs applied to the Vlasov equation, rather than relying on the direct solutions of the PDEs.The fully kinetic PIC method can provide first-principles descriptions of the plasma dynamics.There is no need for the PIC method to assume any PDEs a priori, and PIC has been proved to be a particle method for solving the Vlasov-Poisson equation theoretically [29].
The PINN-Vlasov method can be applied to two aspects: forward and inverse problems.For forward problems, the PINN-Vlasov method can solve the Vlasov-Poisson equation with given initial and boundary conditions.The inverse problem is that some unknown physical quantities, such as the electric field and equation coefficients, are derived using the phase space data.In addition to PINNs, Alves and Fiuza [30] used a sparse-regression method named PDE-FIND to predict the equation coefficients of the Vlasov equation from the fully kinetic PIC simulations.However, both phase space and electric field data were used to conduct the prediction, and the relationship between the two (Poisson equation) was not considered.Theoretically, considering both the Vlasov equation and Poisson equation, only the phase space data is sufficient to predict the equation coefficients.In our work, the Vlasov equation and Poisson equation are concurrently integrated into the PINN-Vlasov framework using automatic differentiation [31] and the trapezoidal rule [32], respectively.For inverse problems, the completely unknown electric field and equation coefficients can be accurately predicted with the PINN-Vlasov method using only little particle distribution data.
The remainder of the paper is organized as follows.Section 2 introduces the Vlasov-Poisson equation, the two-stream instability case used as an example, the PINN-Vlasov framework, and data processing.Section 3 evaluates the performance of the PINN-Vlasov method for handling forward and inverse problems.Section 4 presents the concluding remarks of the PINN-Vlasov method.

Vlasov-Poisson equation and fully kinetic particle-in-cell method
In the plasma kinetic theory, the Vlasov-Poisson equation describes the evolution of the distribution function f(t, x, v) (where t, x, and v are time, space, and velocity coordinates) and the electric field of a collisionless plasma in phase space.The Vlasov-Poisson equation is expressed by where q is the particle charge, m is the particle mass, E is the electric field, B is the magnetic field, ρ is the charge density, and ε is the dielectric constant of the medium, respectively.The charge density can be computed by where e is the elementary charge, n e is the electron number density, n i is the ion number density, and Z i is the ion charge number.In this study, the one-dimensional electrostatic two-stream instability (electrons and single-charge ions) is considered, and it is governed by where m e is the electron mass.
The fully kinetic PIC method is a first-principles algorithm that resolves the distribution function statistically.In this method, ions and electrons are treated as macro particles, and particle charges are mapped to discrete grid points to compute the self-consistent electromagnetic field using Poisson's or Maxwell's equation (8).The fully kinetic PIC simulation is equivalent to solving the Vlasov-Poisson equation directly as long as the spatial grid is fine enough and the number of macro particles in each grid is large enough [29].

Description of the two-stream instability
The two-stream instability will take place in the scenario that two counter-streaming electron streams travel in a homogenous background of immobile ions, and the imaginary solution of the dispersion relation satisfies where k is the wavenumber, ω p is the plasma frequency, and v 0 is the stream velocity of electrons.The two-stream instability case is simulated using the fully kinetic PIC method in 1D1V.In the simulation, the parameters are normalized by where x, v, t, m, q, and E denote the normalized value of the particle position, particle velocity, simulation time, particle mass, particle charge, and electric field, respectively.x * , v * , t * , m * , q * , and E * are the original value of the aforementioned parameters.λ D , v th , ω p , k, and T e refer to the Debye length, electron thermal velocity, plasma frequency, Boltzmann constant, and electron temperature, respectively.Based on equations (3) and ( 5), the normalized Vlasov-Poisson equation for electrons can be expressed by [1] The electron number density n e can be derived through the zeroth-order moment of the distribution function The computational domain and boundary conditions used in the PIC simulation are shown in figure 1.Two different domain lengths of 10λ D and 20λ D are selected to examine the adaptability of PINN-Vlasov to different levels of nonlinearity.The time step is 0.125ω −1 p , and the total simulation time is 62.5ω −1 p .Ions and electrons are randomly distributed in the computational domain, and the number of ions and electrons per cell is 1 × 10 5 .In the simulation, electrons are divided into two counter-streaming beams with steam velocities to be v th and −v th , respectively.The thermal velocities of the two beams are both 0.02v th .The ions' stream and thermal velocity are zero, and the ion mass and charge are 1836m e and e, respectively.This two-stream instability case is chosen mainly for its relative simplicity and also as a classic case in plasma.It should be noted that this example is based on an idealized model.While this serves as a useful benchmark problem, it may not fully capture the complexity of real-world plasma systems.Real plasmas often involve a multitude of additional physical effects, such as magnetic fields, collisions, and non-uniformities, which are not considered in our idealized simulation.Ignoring these effects limits the realism of our model.This example is used here to validate the feasibility of PINNs in the plasma benchmark case, but specialized studies are needed for specific plasma problems.

PINNs for solving the Vlasov-Poisson equation
Based on equation ( 3), a PINNs-based framework for solving forward and inverse Vlasov-Poisson equation (PINN-Vlasov) is proposed, as shown in figure 2. The PINN-Vlasov method employs a MLP to represent the solution of the Vlasov-Poisson equation.The training dataset comprises the randomly sampled time, space, and velocity coordinates and the corresponding distribution function (and the electric field for forward problems).The loss function of the PINN-Vlasov method includes not only the residual between output and labeled data but also the physical constrained equation loss, which is a crucial feature of such PINNs.
For forward problems, the PINN-Vlasov method is used to solve the Vlasov-Poisson equation given initial and boundary conditions.In such cases, the output of MLP is only the particle distribution.Through training, the Vlasov-Poisson equation is solved by minimizing the loss function where where L eq , L IC , and L BC represent the loss function for the residual of the Vlasov-Poisson equation, the initial condition, and the boundary condition.N eq , N IC , and N BC denote the number of the training data for the aforementioned loss functions.f r IC , f r BC and f t IC , f t BC are the reconstructed and exact values of the distribution function for the initial condition and the boundary condition, respectively.The integral and partial differential terms in equation ( 9) are calculated by automatic differentiation and the trapezoidal rule, respectively.
For inverse problems, the PINN-Vlasov method is applied to predict the electric field and equation coefficients, when we only have access to a limited number of particle distribution data.In such cases, in addition to the particle distribution, the electric field can be obtained by MLP in the meantime.The parametrized Vlasov-Poisson equation is considered in inverse problems.Based on the simulation parameters mentioned in section 2.2, the exact value of λ 1 and λ 2 in equation ( 10) is The loss function for inverse problems is where where L ′ eq and L f represent the loss function for the residual of the parameterized Vlasov-Poisson equation and the sampled particle distribution.N ′ eq and N f are the number of the training data for the Vlasov-Poisson equation and the sampled particle distribution, respectively.f p and f t are the reconstructed and exact value of the distribution function for the sampled particle distribution, respectively.
The neural network architecture of the PINN-Vlasov method for both forward and inverse problems consists of 8 layers with 100 neurons in each layer.The bias and weights of each neuron are initialized to zero.The activation function used in the PINN-Vlasov method is the hyperbolic tangent function tanh (y) = e y − e −y e y + e −y .(14) where y denotes the output of the neural unit before the activation function.The loss function is minimized by AdamW [33] optimization method, whose learning rate and weight decay are set to be 0.001 and 0.005, respectively.

Data processing
In this subsection, we first present the extraction process of the distribution function from raw particle-in-cell data, and then the data sampling methods are discussed.

Derivation of the distribution function from raw particle-in-cell data
The distribution function in the Vlasov-Poisson equation cannot be determined directly from the fully kinetic PIC simulation since the raw data collected is the particle positions and velocities at each time step.The distribution function f(t, x, v) is an implicit function of the time, space, and velocity coordinates (t, x, v), and can be obtained by bilinear interpolation [8]. Figure 3 shows the schematic of bilinear interpolation.In the space-velocity coordinate system, the grids in the velocity coordinate are divided into 256 cells, while those in the space coordinate are divided into 256 and 512 cells for two different domain lengths, respectively.The ranges of the space coordinates are (0, 10λ D ) and (0, 20λ D ), and the velocity coordinate range is (−5v th , 5v th ).The normalized distribution function can be derived by where ∆x and ∆vare the cell lengths of space coordinate and velocity coordinate, respectively.N total is the total number of particles in the simulation.N(t, x, v) can be computed by where x 0 and v 0 denote the position and velocity of a single particle at the moment t.

Data sampling
Based on the number of cells and the total time steps, the total number of data points can be computed by Note that the total number of data points in equation ( 17) is computed based on the domain length of 10λ D .The number of data points needs to be doubled for the length of 20λ D .
For forward problems, the sampled data is composed of the distribution function at the initial moment (t = 0), that on the boundary of the space and velocity coordinate, and that inside the computational domain.The sampled points at the initial moment and on the boundary are 700 and 1100, respectively.The inner data points are determined by global random sampling, as shown in figure 4(a).The sampled number of the inner data is 70 000, representing 0.21% of the total number.We have performed a parametric analysis of the number of data points, as detailed in appendix A.
For inverse problems, only the distribution function inside the computational domain is sampled because the initial and boundary conditions are uncertain.Both global (figure 4(a)) and local (figure 4(b)) random sampling are considered for inverse problems.The scope for local random sampling is which is the core region of the two-stream instability.The sampled data points for global and local random sampling are 70 000 and 30 000 (0.09% of the total number), respectively.The epochs of training for global and local random sampling are 500 000 and 200 000, respectively.Actually, a substantial dataset of 30 000-70 000 training data points in the (t, x, v) phase space is hard to obtain.Although this is not the main scope of this paper, we offer a preliminary idea here.The amount of data required in this study cannot be obtained through experiment alone.Therefore, it is necessary to expand the available experimental data with the help of numerical simulations.In this way, the PINN-Vlasov analysis serves as a valuable complement to traditional simulations.Firstly, by embedding the physical equations, PINN-Vlasov helps us ensure that the simulation results align with the underlying physical principles.Secondly, PINN-Vlasov enables the integration of simulation data with existing experimental observations, making it possible to yield computational results with higher accuracy than simulation data alone.Lastly, PINN-Vlasov offers a valuable capability to reconstruct global data from a relatively small number of particles, which is beneficial to large data storage.
For example, by analyzing the current-voltage characteristics of Langmuir probes, you can obtain information about the electron velocity distribution function, which provides small amounts of (t, x, v) data.Then, you can create numerical simulations of the plasma system using computational methods such as PIC simulations or fluid simulations.These simulations should mirror the experimental conditions as closely as possible.Ensure that the simulations provide information in the same phase space (t, x, v) as your experimental data.Finally, you can augment the combined dataset by applying various data augmentation techniques.This could involve perturbing the existing data points within physically plausible ranges, introducing random noise, or simulating different plasma conditions to increase the dataset's diversity.

Results and discussions
In this section, the PINN-Vlasov method is applied to solve the forward and inverse problems of the Vlasov-Poisson equation.For forward problems, the distribution function of the whole computational domain can be solved with few initial, boundary, and inner data points.For inverse problems, the electric field and equation coefficients (λ 1 and λ 2 ), can be obtained with limited inner data points.

Forward problems
In the PINN-Vlasov framework, a mapping relationship is constructed between the distribution function and the coordinates of time, space, and velocity for forward problems.After training networks fed with little original PIC data, the PINN-Vlasov method can reconstruct the distribution function at any moment, space, and velocity coordinate.According to our simulation settings mentioned in section 2.2, the ions are immobile, so the electron distribution function (EDF) is examined.Figure 5 depicts the EDF (domain length: 10λ D ) computed by PIC and PINN-Vlasov at different moments.The reconstructed contour patterns (figures 5(b), (e) and (h)) of the EDF show a good match with the PIC results (figures 5(a), (d) and (g)).In addition, the residual contours (figures 5(c), (f) and (i)) of the reconstructed EDF are below 0.03 in most regions.The reconstructed and exact electron velocity distribution profiles (x = 4λ D ) at three representative times, i.e. t = 12.5ω −1 p , 25ω −1 p , and 37.5ω −1 p , are illustrated in figure 6.It is shown that the reconstructed profiles correspond well with the exact value.We further compute the relative errors (err f ) for the reconstructed EDF, where err f is defined as where f p (t, x, v) and f t (t, x, v) are the reconstructed (PINN-Vlasov) and exact value (PIC) of the EDF, respectively.In the evolution of the two-stream instability, the relative error is computed to be about 3.05% based on equation (18), demonstrating the PINN-Vlasov method's good accuracy for handling forward problems.Figure 7 gives the EDF (domain length: 20λ D ) computed by PIC and PINN-Vlasov, respectively.First, it can be seen that there are two electron phase-space holes in the nonlinear evolution.As time progresses, the reconstructed results are consistent with the PIC results, despite the increasing nonlinearization of the electron velocity distribution function.However, it does need to be recognized that the deviation of the reconstructed results from the PIC results becomes larger after the computational domain is increased from 10λ D to 20λ D .According to equation ( 18), the relative error is computed to be about 5.56%.Concerning the efficiency of the PINN-Vlasov method, the computation times of PIC and PINN-Vlasov are compared.For PIC, a one-dimensional fully kinetic PIC code [9] is used for evaluation in a workstation whose CPU is AMD EPYC 7T83 (2.45 GHz, 64-Core).The code is computed ten times for the case mentioned in section 2.3.1 in parallel on the CPU using 64 cores, and the average running time is 221.54 s.For PINN-Vlasov, we use PaddlePaddle [34] to evaluate the efficiency in a single Nvidia Quadro RTX-A6000 GPU.The training time for global and local random sampling is about 3.1 h and 0.8 h, respectively.After training, the trained PINN-Vlasov model can be used for solving forward and inverse problems with the same initial and boundary conditions.Considering only the model inference time, the calculation is executed 10 times with an average time of 0.97 s, and the acceleration ratio is 228.39.Note that the model calculation time here represents the time to input the (t, x, v) coordinates into the trained model to get the reconstructed distribution function and it depends on the number of input coordinate points.In our case, the trained model is tested in the full range [t ∈ (0, 62.5ω −1 p ), x ∈ (0, 10λ D ), v ∈ (−v th , v th )] with the number of input coordinate points to be 32 768 000 (500 × 256 × 256).In this case, the EDF of different moments can be obtained simultaneously.Considering both training time and model inference time, the total computation time of PINN-Vlasov is larger than that of PIC.However, in practice, training only needs to be performed once, while inferencing often needs to be performed multiple times.PINN-Vlasov allows the distribution function to be reconstructed from a small subset of particles efficiently, which may be helpful to mitigate the need for huge amounts of data storage.

Inverse problems
In this subsection, we test the performance of the PINN-Vlasov method for inverse problems.Regarding the two-stream instability, the main concern is using the limited distribution function data to derive the completely unknown electric field and equation coefficients of the Vlasov-Poisson equation.Both global (figure 4(a)) and local (figure 4(b)) random sampling methods are evaluated.Figure 8 shows the electric field computed by PIC and PINN-Vlasov with different sampling methods.The electric fields predicted with different sampling methods are almost identical to the one calculated by PIC, and the maximum residual is no more than 0.02.No electric field data is used in the training process of the networks, and the predicted electric field is derived entirely based on physical constraints in equation (13).The electric field profiles (domain length: 10λ D ) obtained by inversion and calculated by PIC at t = 25ω −1 p , 37.5ω −1 p , and 50ω −1 p , are illustrated in figure 9.The results show that the predicted electric field profiles are also in good agreement  with the exact values.The reason may be that the data structure of the electric field is two-dimensional (x, v), while that of the EDF is three-dimensional (t, x, v).Similar to equation ( 18), the relative errors (err E ) for the predicted electric field are computed to be about 0.73% from global random sampling and 3.3% from local random sampling.For the prediction of the electric field in the case of 20λ D , the local random sampling is used.Figure 10 gives the electric field computed by PIC and PINN-Vlasov, respectively.The predictions are also in agreement with the PIC results, and the contour lines are drawn in the figure to show the difference between the two results.In this case, the relative errors (err E ) for the predicted electric field are computed to be about 3.96%.We have also performed a simulation in a frame drifting relative to the coordinates to avoid any potential bias introduced by the fact that there are particular locations where the phase-space density is atypically high (or low), as detailed in appendix B.
As for the inversion of the equation coefficients, table 1 shows the derived results for global random sampling and local random sampling, respectively.For domain length of 10λ D , the derived equation coefficients for local random sampling (λ 1 : 0.42%, λ 2 : 0.30%) are more precise than those for global random sampling (λ 1 : 2.05%, λ 2 : 1.25%), which may be because the local sampling method is to sample at  the location where there are more particles ([x ∈ (4λ D , 6λ D ), v ∈ (−v th , v th )]).For global random sampling, the number of particles in most areas is minimal (figure 5).It is suggested that global random sampling is more suitable for predicting vector fields while local random sampling is more precise in predicting the equation coefficients.For domain length of 20λ D , the derived equation coefficients for local random sampling (λ 1 : 1.94%, λ 2 : 1.89%) are less accurate than those of 10λ D case.In [30], the relative errors of predicted equation coefficients are 3.1% and 1.3%, respectively.Therefore, it is demonstrated that the PINN-Vlasov method can achieve good accuracy for inverse problems.Finally, in order to explore the predicting performance of PINN-Vlasov for unknown times, a preliminary study of forward and backward prediction has also been conducted, as detailed in appendix C. How to improve the performance of the model for unknown times is also a direction that needs to be focused in the subsequent research.

Conclusions
Different from purely data-driven neural networks, PINNs incorporate prior knowledge of the governing physical principles into the neural network architecture.In our case, this means encoding the fundamental laws and equations governing plasma behavior, such as the Vlasov-Poisson equation.By doing so, our model is inherently guided by the known physical laws, ensuring that predictions adhere to these principles.Furthermore, we can not only predict plasma behaviors but also assess the confidence or uncertainty associated with collecting data through PINNs.This adds a layer of transparency to collecting data, helping us identify regions where our model is confident and where it may require further investigation.That is, it is this ability of PINNs to embed physical governing equations that allow us to test the plausibility of the data and dig deeper into its characteristics.Finally, by training our neural networks to not only match data but also satisfy physical laws, we ensure that the learned representations align with our understanding of plasma physics.This allows us to extract meaningful physical insights from the model, providing a better physical explanation of the observed behaviors than purely data-driven neural networks.
In our study, a PINNs-based framework for solving forward and inverse Vlasov-Poisson equation (PINN-Vlasov) is proposed.By exploiting the benefits of the PINNs, the governing equations, i.e., the Vlasov-Poisson equation, are embedded into the networks, making the PINN-Vlasov method have a powerful ability to predict unknown physical information.
For forward problems, the Vlasov-Poisson equation can be solved with few initial, boundary, and inner data points.With the same initial and boundary conditions, the EDF reconstructed by the PINN-Vlasov method matches the PIC results well and the relative errors are about 3.05% (10λ D ) and 5.56% (20λ D ).Although it cannot be reconstructed out of thin air, the PINN-Vlasov can efficiently reconstruct the EDF given a small amount of data, which helps alleviate the challenges associated with large data storage while preserving essential global insights.
For inverse problems, the electric field and equation coefficients, can be obtained with only a limited sampling of the distribution function.No data of the electric field and equation coefficients are used in the training process of the networks, and they are all the output of the PINN-Vlasov networks.The sampling methods of global random sampling and local random sampling are tested.Our results demonstrate that the relative errors of the predicted electric field are about 0.73% and 3.3% (3.96% for 20λ D ) for global and local random sampling, respectively.In addition, the PINN-Vlasov method can precisely predict the equation coefficients of the Vlasov-Poisson equation, with relative errors λ 1 : 2.05%, λ 2 : 1.85% and λ 1 : 0.42%, λ 2 : 0.30% (λ 1 : 1.94%, λ 2 : 1.89% for 20λ D ) for global and local random sampling, respectively.
In a word, the PINN-Vlasov method can not only solve the Vlasov-Poisson equation but also deal with inverse problems such as the inversion of the completely unknown electric field and equation coefficients.Significantly, the inverse problems mentioned in this study are very hard to solve utilizing traditional methods.These scenarios may happen in experiments where we only have a few interior sensors for particle positions and velocities, and we have no information on positions and velocities at the wall or other physical parameters.

Appendix B
To avoid any potential bias introduced by the fact that there are particular locations where the phase-space density is atypically high (or low), the same simulation was performed in a frame drifting relative to the coordinates.In this case, the two electron beams have initial velocities of −0.9v th and +1.1v th , respectively.We used local sampling of the same region [x ∈ (4λ D , 6λ D ), v ∈ (−v th , v th )] for training while keeping other settings constant.Figure B1 gives the electron distribution function evolution results.As time progresses, the electron hole characterizing the late nonlinear stage of evolution moves gradually from left to right.However, it can be seen that even in this case, PINN-Vlasov can still capture the characteristics of the evolution of the electron velocity distribution function well.According to equation (18), the relative error is computed to be about 4.11%.The electric field evolution results in figure B2 show a rightward drift due to the initial velocities setting of the electron beams.PINN-Vlasov results with local sampling match very well with PIC results, with a relative error to be 3.25%.between PINN-Vlasov and PIC.The results demonstrate that PINN-Vlasov has a limited ability to predict the evolution of past moments using only data from the nonlinear regime.How to achieve a better prediction of the previous electron velocity distribution using only data from the nonlinear region is also what we will focus on in the following study.

Figure 1 .
Figure 1.Computational domain and boundary conditions of the two-stream instability simulation.

Figure 2 .
Figure 2. Schematic of the physics-informed neural networks-based framework for solving forward and inverse Vlasov-Poisson equation (PINN-Vlasov).

Figure 4 .
Figure 4. Schematic of the (a) global and (b) local random sampling.

Figure B2 .
Figure B2.The electric field with initial beam velocities to be −0.9vth and +1.1v th .(a) PIC results, (b) PINN-Vlasov results with local sampling.

Figure C2 . 1 p
Figure C2.Results of the initial condition and the electron distribution at t = 3.125ω −1 p and t = 5.625ω −1 p .(Top: PIC results, Bottom: PINN-Vlasov results).

Table A1 .
Relative errors of the reconstructed distribution function using different numbers of inner points, initial condition (IC) points, and boundary condition (BC) points under global random sampling.

Table A2 .
Relative errors of the predicted electric field and equation coefficients using different numbers of data points under global random sampling.

Table A3 .
Relative errors of the predicted electric field and equation coefficients using different numbers of data points under local random sampling.