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Figure 1: Schematic structure of the neural network model used for approximating the ICE results. The input layer and the output layer are linear, while the hidden layer is assumed to be non-linear, with an hyperbolic-tangent activation function. The input layer consists of three neurons for the temperature, hydrogen number density and electronic number density. The output neuron gives the partial pressure of element i for the physical conditions in the input layer.

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In the text

Figure 2: Relative error obtained after the training of the neural network for hydrogen, oxygen, carbon and iron. The distribution of relative errors are shown for the learning set and for a verification set, which aids at detecting overtraining. We indicate the value of the standard deviation of the distribution and the number of points below 1, 2 and 3. We also indicate these values in the plots as horizontal lines.

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In the text

	Figure 3: Histograms of relative errors for the 2.55105 points of the three-dimensional convection simulation of Asplund et al. (2000). We indicate the position of 1, 2 and 3 of the distribution. Note that they are similar to those obtained for the verification set during the training of the networks.
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In the text

	Figure 4: Number density of carbon and oxygen obtained with the simplified model and with the neural network for two different combinations of hydrogen and electron densities and for different values of the temperature. Note that this kind of investigation is greatly simplified due to the analytical character of the neural network.
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In the text

	Figure 5: Contribution to the total pressure of carbon and oxygen in two different cases. The curves have been obtained using the neural networks. The addition of all the contribution closely matches the fictitious pressure of carbon and oxygen. This is the reason why the very simplified model for C and O works very well.
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In the text