PlasmaNet: a framework to study and solve elliptic
differential equations using neural networks inplasma fluid simulations

1Lionel Cheng
CERFACS
cheng@cerfacs.fr

Ekhi Ajuria Illarramendi
ISAE-SUPAERO /CERFACS
ekhi.ajuria@cerfacs.fr

Michaël Bauerheim
ISAE-SUPAERO
bauerheim@isae-supaero.fr

Bénédicte Cuenot
CERFACS
cuenot@cerfacs.fr

Introduction

Elliptic partial differential equations (PDEs) are common in many areas of physics, from the Poisson
equation in plasmas and incompressible flows to the Helmholtz equation in electromagnetism. Theirnumerical solution requires to solve a linear system and many libraries have been developed for thistask [1]. Solving a linear system efficiently requires preconditioning the system which is a difficult
task. It can become a bottleneck for performance when the number of nodes increases.
The rise of computational power and inherent speed of GPUs offers exciting opportunities to solve
PDEs by recasting them in terms of optimization problems. Since the major introduction of Physics
informed neural networks (PINN) [2], other architectures [3] and frameworks [4] have been created.
In plasma fluid simulations, the Poisson equation is solved, coupled to the charged species transport
equations [5]. A pioneer work [6] has shown significant speedup using neural networks to solve thePoisson equation compared to classical linear system solvers on this problem. Coupling the neuralnetwork Poisson solver to plasma transport equations has shown promising results and the neuralnetwork can be considered as a viable option in terms of accuracy.
This work extends [6] and introduces PlasmaNet (https://gitlab.com/cerfacs/plasmanet),
an open-source library written to study neural networks in plasma simulations. The optimal network
parameters are first recalled and metrics to help design appropriate network architectures for solving
elliptic differential equations are presented. We then attempt to solve a new class of elliptic differentialequations, the screened Poisson equations using neural networks. These equations are used to infer thephotoionization source term from the ionization rate in streamer discharges [7]. Finally a simulation
running with three neural networks, coupled to plasma transport equations, to solve both the Poissonand the photoionization equations, is performed to assess the accuracy of neural networks predictions.

2Network architectures for elliptic differential equations

Laplace and Poisson equations form the basis of elliptic PDEs. Studying them can give insights
on how to solve all elliptic PDEs. From a given charge density ρq the Poisson equation yields theelectromagnetic potential φ from which we can compute the electric field E = −∇φ so that in theFourth Workshop on Machine Learning and the Physical Sciences (NeurIPS 2021).

end E = f (ρq ). In numerical simulations the physical domain is finite so that the Poisson equation issupplemented by Dirichlet and Neumann boundary conditions:∇2 φ = −ρqε0inΩ̊withφ = 0 on ∂ΩDE · n = EN on ∂ΩN(1)The analytical solution of the Poisson equation with boundary conditions depends on the Greenfunction G of the chosen configuration [8, Chap. 1.10]:
φ(x) =

14π0Zρ(x0 )G(x, x0 ) dV 0 +14πZ G∂G∂φ−φ 00∂n∂ndS 0(2)Expressions of the Green functions [8, Chap. 3] in series for cartesian, cylindrical and sphericalcoordinates highlight the importance of multiple scales in the Poisson equation which must beincorporated in the neural network architecture. The UNet architecture [9] has proved to work best toinfer E = NN(ρq ) in [6] compared to the Multi-Scale architecture [10]. In UNets, the local branch b
is defined as the power of 2 by which the initial resolution is divided by and the number of branches
nb is the number of scales involved. Local db and global D depths are defined as the number ofsuccessive convolutional layers in branch b and across the whole network, respectively.
Concerning losses, the use of a physical loss in the form of a LaplacianLoss inspired from PINNs[2] is critical to yield stable trainings and stable simulations when coupled to transport equations [6].
Elliptic PDEs, having no real characteristic curves, need the information of the whole domain at everypoint. This is highlighted by the analytical solution Eq. (2) which incorporates a domain integral.
To quantify the information propagation across the neural network the global receptive field RF isdefined as the size of the domain of influence of the input center point in number of points in theoriginal scale np . The receptive field can be splitted into local receptive fields per branch RFb so that

RF =nXb −1

RFbwith RFb =b=0

1 + db (ks − 1)2b if b = 0db (ks − 1)2b otherwise(3)

where ks is the kernel size assumed constant in the network.
In [11], a theoretical receptive field is defined as the size of the input domain of influence on theoutput center point. Tests carried in PlasmaNet on the studied UNets show that this definition matches
Eq. (3) so that both formulations are equivalent. However, the importance of the domain influence
is not uniform, as points closer to the studied pixel will have more paths to influence the output,resulting in a gaussian-like distribution [12]. A parametric study in [6, Sec. 5] across multiple UNets
showed that the optimal global parameters of the network for a given number of pixels np should be:
RF = 2np

and nb = max{b ∈ N|bnp /2b c > ks } + 1(4)With these parameters the receptive field fills the entire computational domain for all the input points
and convolution is relevant in the downscaled branches [6, Sec. 5].

3Photoionization in plasma discharges

We model plasma discharges in air using the chemistry from [13]. It consists of electrons (ne ),positive ions (np ) and negative ions (nn ). Those three species are modeled in a drift-diffusion
approximation where the ions are considered not moving [7]:∂ne+ ∇ · (ne We − De ∇ne ) = ne α|We | − ne η|We | − ne np β + Sph∂t∂np∂nn= ne α|We | − ne np β − nn np β + Sph= ne η|We | − nn np β∂t∂t2(5)(6)where Sph is the photoionization source term, α = α(E/N ) is the ionization coefficient, η =η(E/N ) the attachment coefficient, E the electric field magnitude, N the neutral gas density, β therecombination rate, We = −µe E the drift-velocity of the electrons and µe = µe (E/N ) the electronmobility. These transport equations are coupled to the Poisson equation as ρq = e(np − ne − nn ).
Such a coupling with the Poisson neural network solver has already been studied in [6]. Howevera rather high background density of charged species was used so that the photoionization source
term Sph could be neglected in previous work. Here, similar cases where photoionization cannot beneglected are considered, highlighting the benefits of the proposed open-source framework PlasmaNet
to incorporate new physics for CNN-based fluid plasma simulations.
The plasma discharge is a double headed streamer [7], initialized with a neutral Gaussian profile at
x = 2 mm and r = 0 mm with a background density in a rectangular domain of Lx × Lr = 4 × 1mm2 , corresponding to an azimuthal cut of the cylindrical geometry.
In plasma air discharges, the electron-impact reactions produce excited states of N2 . Radiative
relaxations of these states are absorbed by O2 causing ionization [14]. Integral models have beendeveloped to model this photoionization source term but are very costly in numerical simulations[14]. Simplifiying assumptions allow to recast this photoionization source term in terms of screened
Poisson equations [7]:Sph (r) =jm
X
j=1

jjjSph(r) ∀j ∈ J1, jm K ∇2 Sph(r) − (λj pO2 )2 Sph(r) = −Aj p2O2 I(r)(7)

where I(r) = f (E/p)ne α|We | is an effective ionization term, pO2 the oxygen pressure, λj , Aj
are fitting parameters and jm = 2 or 3 depending on the level of precision wanted. Note that jm
resolutions of linear systems must be performed inside a numerical time iteration when solving
Sph so that the computational cost of photoionization is high. Here only jm = 2 is considered forsimplicity. Each component of the photoionoization source term obeys a screened Poisson equation:∇2 φ − λ2 φ = −R

(8)

where λ controls the amount of diffusion of the solution, the higher the value of λ, the lower thediffusion (λ = 0 is the maximum diffusion and reduces to the Poisson equation). We apply a neuralnetwork to solve each component of the photoionization source term using the optimal parametersin Eq. (4). The same network architecure as the Poisson neural network solver (UNet5-RFx 800RFy 200) is thus applied for each component of the photoionization source term. We introduce aphysical loss associated to the screened Poisson equation called PhotoLoss:
LP (φout ; λ) =i2
XhL2x L2yb,j,i
2 b,j,i∇2 φb,j,iout − λ φout + Rin
bs (nx − 1)(ny − 1)

(9)

b,j,i

Training is done using random datasets of 10 000 snapshots introduced in [15] and already used in [6,2Sec. 4]. A snapshot of such a dataset is shown in Fig. 1. It can be seen that Sphis less diffusive than1
Sphwhich behaves closer to the Poisson equation due to λ1  λ2 .

Figure 1: Example of random_12 source term input in a 4 × 1 mm2 cylindrical domain.

34Results

First, snapshots of the propagation of the double headed streamer without photoionization using alinear system solver and a neural network solver for the Poisson equation are shown in Fig 2. Electrondensity and electric field profiles tend to be more diffusive when the Poisson equation is solved by theneural network. However the speed of propagation is well-captured as well as the discharge energy[6, Sec. 7].

Figure 2: Electron density ne and norm of electric field |E| of a double headed streamer using
UNet5-RFx800-RFy200 (top-half) and a linear system solver (bottom-half) at 1.2, 2.0 and 2.8 ns.
E = NN1 (ρq )Sph = NN2 (I) + NN3 (I)Un+1 = Un + ∆t[Rt (E) + Rc (E, Sph )]Figure 3: Interaction of the neural network with plasma transport equations at each iteration. U =(ne , np , nn ), Rt and Rc are transport and chemistry residuals.
1
Second, simulations with photoionization have been performed with two networks inferring Sphand2
Sph . Another network is used to infer the electric field E (as in [6] and results from Fig. 2) so thatthree networks are coupled to the plasma transport equations to replace linear system solvers. Results
are shown in Fig. 4 with a sketch of coupling in Fig. 3. The interaction of multiple neural networks
solutions seem to yield promising results as the photoionization source term and electric field arecorrectly predicted. The right-propagating positive streamer ia slightly more diffuse when looking
at the electron density than the reference solution whereas the left-propagating negative streamer isbetter captured by the neural networks. We note however negative values in the prediction of Sph by
the neural network (white regions in the snapshots due to clipping), which strictly speaking should
not be allowed as the ionization source term is positive. These values have a relative amplitude of
10−3 and have been clipped to get good streamer propagation.

To prevent the rise of these negative values, a positive-valued dataset can be used so that the network
learns only to infer positive-values. A similar dataset as the one shown in Fig. 1 but without negativevalues for the ionization rate I is used to train the networks and results are shown in Fig. 5. The
negative values are indeed removed but the overall shape of the electron density is more diffused
especially for the right-propagating positive streamer: the network struggles to predict very low
values of Sph .

5Conclusion

We have introduced PlasmaNet and shown its ability to couple neural network solvers to plasmatransport equations. The range of applicability of the method developed in [6] has been extended tothe more general screened Poisson equations, which highlights the flexibility of the present framework
PlasmaNet to incorporate new complex physics simulated by neural networks. Future work will try
to integrate the screening length λ directly inside the network so that one network and not three will
be necessary to solve both the Poisson equation and the photoionization source term. Dedicatedregularized terms, such as penalty on negative Sph could also be tested. Interaction of the neuralnetwork with other plasma test cases such as Hall effect thrusters are also planned.
4Figure 4: Electron density ne , norm of electric field |E| an photoionization source term Sph of adouble headed streamer using three neural networks (top-half) and a linear system solver (bottom-half)
at 1.6, 2.2 and 2.8 ns.

Figure 5: Electron density ne and photoionization source term Sph of a double headed streamer using
three neural networks (top-half) and a linear system solver (bottom-half) at 1.6, 2.2 and 2.8 ns using
a positive-valued training dataset.

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