A neural network closure for the Euler-Poisson system based on kinetic simulations

Figure 1.  Graph of the composition of the closure. The different operations are (Re)-(Re') resampling, (P) pre-processing, (Sl) slicing, ($ \text{NN}_\theta $) neural network, (R) reconstruction, (P') post-processing and (Sm) smoothing

Figure 2.  Visual representation of the three operations involved in the non-locality: the (Sm) smoothing, the (R) reconstruction and (Sl) slicing mechanism, and the ($ \text{NN}_\theta $) neural network. Highlighted in gray is the part of the data used to compute the output value whose position is indicated by the red line. For better readability, only one of the four vectors of the input $ X $, and only the three windows involved, are represented. These examples use the redundancy parameter $ r = 2 $. Two scenarios are illustrated: on the left, each output value of the neural network depends on relatively few input values, while on the right it depends on relatively many. Also, the position of the output value considered is not the same

Figure 3.  Graph of a 1D V-Net with a window of size $ N = 512 $, a depth $ k = 2 $ and $ \ell = 3 $ levels

Figure 4.  Scheme of the data generation process for one simulation with the kinetic model. With one Knudsen number and one initial condition, we produce 20 entries in the dataset

Figure 5.  Possible functions $ \rho $, $ u $ and $ T $ generated with the process described in Section 4.2

Figure 6.  Scheme of the whole data processing for the training and the prediction processes respectivily. The different operations are (S) standardization, (Sl) slicing, (N) normalization, (I) inverse normalization, (R) reconstruction and (Sm) smoothing

Figure 7.  Slicing of a 1D signal into overlapping windows with different redundancy parameters $ r $. The original signal is represented vertically on the left, and the windows are on the right. The hatched parts of each window are the margins that are ignored when reconstructing the signal. The windows that exceed the span of the original signal are completed by periodicity

Figure 8.  Kernel used for the reconstruction of an entire signal from the windows predicted by the neural network, with the redundancy parameter $ r = 3 $

Figure 9.  Three examples of predictions. Note the different scales of the y-axis: the amplitude of the heat flux tends to increase with the Knudsen number

Figure 10.  Distribution of the relative errors of the neural network and the Navier-Stokes estimation over the test dataset ($ 10, 000 $ predictions)

Figure 11.  Error of the Navier-Stokes estimations and of the network on the test dataset, depending on the Knudsen number and the $ L^2 $ norm of the real heat flux. Each dot corresponds to one entry in the test dataset. For better clarity, only one $ 31^{\text{th}} $ of the data is shown here

Figure 12.  Relative error of the Navier-Stokes estimations and of the network on the test dataset, depending on the Knudsen number. The line represents the median of the error over the 100 entries of the test dataset for each epsilon, and the coloured area the interquartile interval

Figure 13.  Norm of the real heat flux and relative errors of the predicted heat flux and the Navier-Stokes estimation throughout simulations. Median and interquartile interval over 50 simulations

Figure 14.  Median relative error and number of learnable parameters for V-Nets with different hyper-parameters : ($ \ell $) number of levels, ($ d $) depth, and ($ p $) size of the kernels. Points sharing the same color represent networks that differ by only one hyper-parameter

Figure 15.  Example of the evolution of the density, mean velocity, temperature and heat flux during the first time unit with a randomly generated initial condition and $ \varepsilon = 0.1 $, obtained with the kinetic model

Figure 16.  Comparison of the four models on the density and heat flux at $ t = 1 $ with a randomly generated initial condition and $ \varepsilon = 0.1 $

Figure 17.  Examples of the evolution of the electric energy with different initial conditions and different Knudsen numbers

Figure 18.  Distributions over 200 simulations of the relative errors of the three fluid models compared to the kinetic model, measured on the logarithm of the electric energy from $ t = 0 $ to $ t = 8 $. For better readability, the x-axis in in logarithmic scale

Figure 19.  $ L^2 $ relative errors of the fluid models over the kinetic model depending on the Knudsen number. The first plot shows the raw data and the second shows the median and interquartile interval for 20 uniform classes of Knudsen numbers between $ 0.01 $ and $ 1 $

Figure 20.  Example of a heat flux predicted by the neural network, with and without smoothing. The smoothing here uses $ \sigma\simeq0.05 $

Figure 21.  Relative errors of the predictions over the test dataset depending on the quantity $ \sigma $ of smoothing

Figure 22.  Proportion of simulations of the fluid model reaching $ t = 3 $ out of 30 simulations, depending on the quantity $ \sigma $ of smoothing

Figure 23.  Relative errors of the fluid+network model on five different quantities depending on the resolution used. Each point is an average of 10 simulations with always the same 10 different initial conditions and $ \varepsilon = 0.1 $. The network used was trained with data generated with $ N_x = 1\, 024 $. To adapt to different resolutions, option 1 relies on the slicing mechanism to adapt to the different input sizes, while option 2 uses the resampling of the input to $ N_x = 1\, 024 $ and the resampling of the output back to the original resolution. For all resolutions, the errors are computed relative to kinetic simulations with a high resolution ($ N_x = 2\, 048 $, $ N_v = 141 $)

Figure 24.  Example of convergence of the fluid+network model with option 2, towards what seems to be a slightly different solution than the kinetic model

Figure 25.  Distributions over 100 simulations of the relative errors of the three fluid models compared to the kinetic model, measured on the logarithm of the electric energy from $ t = 0 $ to $ t = 8 $, with discontinuous initial conditions. For better readability, the x-axis in in logarithmic scale. It looks a lot like figure 18, which seems to indicate that the fluid+network model is not affected much by the initial discontinuities

Figure 26.  Example of the evolution of the density and temperature up to $ t = 0.5 $ with a randomly generated discontinuous initial condition and $ \varepsilon = 0.1 $, obtained with the kinetic model. The initial discontinuities are quickly smoothed out by the dynamic of the system