## 2 Description of Methodology
### Description of DNN  
In this section, a DNN is constructed to assist in modeling the RF CO\({}_{2}\) discharges, as shown in Figure 1. Firstly, a training dataset for the DNN can be obtained from numerical simulations or experimental measurements, and in this study, the simulation data from the validated fluid model were used to train the DNN. Generally the training dataset is required to cover all the information in the sample space, i.e., the entire range of discharge parameters. The testing dataset is used to judge the prediction performance of the DNN model. If the prediction result of the DNN within the testing dataset meets the requirements (i.e., the average relative error between the prediction result and the simulation result is less than 0.5%), then it indicates that the DNN has good generalization performance within the given parameter range. The DNN constructed in this paper is a fully connected multilayer back-propagation neural network with one input layer, three hidden layers, and one output layer. From Figure 1, the circles in the schematic of the DNN denote the neurons that can accept the activations of the neurons from the previous layer as the input and perform a linear transformation, followed by an elementwise nonlinear transformation, and then, propagate the results to the next layer. Up to now, several studies have reported some experience in selecting the number and size of the hidden layers [51; 52], but there are no rigorous rules for selecting these quantities. The appropriate network configuration is usually obtained by extensive experimentation. In general, more hidden layers and neurons are usually adopted to accurately predict the properties of complex systems, but an excessive number of hidden layers and neurons may lead to the overfitting of DNNs [53; 54]. Based on the characteristics of RF CO\({}_{2}\) discharges under Martian pressure, three hidden layers with 40 neurons per layer are adopted in the DNN constructed in Figure 1, and the tanh function and the sigmoid function are used as activation functions in the three hidden layer species in turn. The dimensions of the input layer and output layer determine the size of the input layer and output layer of the DNN. In this paper, the input of the DNN is the driving frequency and power density, and the output is the current-voltage characteristics, electric field, electron temperature, and product particle density. In addition, the selection of training data is related to the training efficiency and prediction performance of the DNN. If the amount of training data is small, it will reduce the training efficiency and lead to the prediction results of DNNs that do not meet the expectations. In contrast, using a large amount of training data may bring a huge computational load to the fluid simulation and DNN training.  
Figure 1: Schematic diagram of DNN-assisted RF CO\({}_{2}\) discharge modeling method.  
In this work, 24 sets of simulation data obtained from the fluid model with a driving frequency between 12 and 75 MHz and 26 sets of simulation data with the power density between 35 and 85 W/cm\({}^{2}\) were used as the training data. For the discharge current and voltage, 2730 time points were sampled uniformly per set of simulated data. For the electric field, 300 space points in each set of the simulated dataset were sampled uniformly, and the same is true for each product particle density. The fluid simulation results for driving frequencies of 13.56, 27.12, 40.68, 54.24, and 67.80 MHz and power densities of 40, 50, 60, 70, and 80 W/cm\({}^{2}\) were selected as the testing dataset.