Modeling of electron nonlocal transport in plasmas using artificial neural networks

This article presents the use of artificial neural networks (ANN) to predict nonlocal heat flux transport within hydrodynamic simulations. Several cases of laser driven ablation of a plastic target are considered. The database for the ANN training phase is built using the transport module of the hydrodynamic code CHIC. It covers a range of parameters characteristic of laser experiments in the context of high-energy-density physics. Results show that an ANN can efficiently replace a module of nonlocal transport in one- and two-dimensional hydrodynamic simulations, with an error less than 3% in a radius of $0.5\mu \mathrm{m}$ and an average computation gain of a factor 433 in two dimensions.

Figure 1

Scheme of an artificial neuron: The neuron takes the values from neurons at the preceding layer, multiplies them by a weight matrix, adds a bias to the weighted sum, and passes the result through an activation function. This value is then attributed to the neuron

Figure 2

Example of a 1D normalized profile: a heat flux divergence $S_{\mathrm{nl}}$ (solid curve) and the corresponding temperature $T_e$ (dotted curve) and density $n_e$ (dashed curve).

Figure 3

Each time step, CHIC feeds the temperature profile to the ANN, which predicts the heat source term and feeds it back to CHIC.

Figure 4

Dependence of the loss (top) and the 3% accuracy (bottom) criteria on the number of training sessions (epochs) for the 1D database. Training (solid curve) and validation (dashed curve) sets. ANN contains four hidden layers of 120 neurons each with a sigmoid activation, a decaying learning rate (dash-dotted curve, top panel), and batches of size 450.

Figure 5

Relative error histogram on the 1D test set: the 3% accuracy of 99.8% with a maximum of 3.05%.

Figure 6

Close-up on the oscillating portion of the heat source (top), temperature (middle), and density (bottom) profiles at $t$ = 500 ps when the ANN's output is not smoothed. Comparison of SNB (solid curve) to ANN (dashed curve) simulations, quantified by a gamma index (dash-dotted curve).

Figure 7

Close-up on the oscillating portion of the heat source (top), temperature (middle), and density (bottom) profiles at $t$ = 500 ps when the ANN's output is smoothed. Comparison of SNB (solid curve) to ANN (dashed curve) simulations, quantified by a gamma index (dash-dotted curve).

Figure 8

Dependence of the loss (top) and the 3% accuracy (bottom) criteria on the number of training sessions (epochs) for the 2D database. Training (solid curve) and validation (dotted curve) sets. ANN contains four hidden layers of 150 neurons with a sigmoid activation, a decaying learning rate (dash-dotted curve, top panel), and batches of size 60.

Figure 9

Relative error histogram on the 2D test set: the $3\%$ accuracy of $100\%$ with a maximum of $2.04\%$.

Figure 10

Profiles of the heat source (top), temperature (middle), and density (bottom) at $t$ = 500 ps. Comparison of a SNB simulation and a smoothed ANN simulation.

Figure 11

Maps of the gamma index between a SNB simulation and a smoothed ANN simulation at $t$ = 500 ps for the heat source (top), temperature (middle), and density (bottom). Dashed lines define the position of the 1D sections displayed in Fig. 12 (horizontal) and Fig. 13 (vertical).

Figure 12

One-dimensional longitudinal section along the $x$ axis at $t$ = 500 ps for the heat source (top), temperature (middle), and density (bottom). Comparison of SNB (solid curve) to nonsmoothed (dotted curve) and smoothed (triangle marker) ANN.

Figure 13

One-dimensional transversal section along the $y$ axis at $t$ = 500 ps for the heat source (top), temperature (middle), and density (bottom). Comparison of SNB (solid curve) to nonsmoothed (dotted curve) and smoothed (triangle marker) ANN.