  Traditionally, 1D models based on scaling laws have been used to
parameterized convective heat transfer rocks in the interior of terrestrial
planets like Earth, Mars, Mercury and Venus to tackle the computational
bottleneck of high-fidelity forward runs in 2D or 3D. However, these are
limited in the amount of physics they can model (e.g. depth dependent material
properties) and predict only mean quantities such as the mean mantle
temperature. We recently showed that feedforward neural networks (FNN) trained
using a large number of 2D simulations can overcome this limitation and
reliably predict the evolution of entire 1D laterally-averaged temperature
profile in time for complex models. We now extend that approach to predict the
full 2D temperature field, which contains more information in the form of
convection structures such as hot plumes and cold downwellings. Using a dataset
of 10,525 two-dimensional simulations of the thermal evolution of the mantle of
a Mars-like planet, we show that deep learning techniques can produce reliable
parameterized surrogates (i.e. surrogates that predict state variables such as
temperature based only on parameters) of the underlying partial differential
equations. We first use convolutional autoencoders to compress the temperature
fields by a factor of 142 and then use FNN and long-short term memory networks
(LSTM) to predict the compressed fields. On average, the FNN predictions are
99.30% and the LSTM predictions are 99.22% accurate with respect to unseen
simulations. Proper orthogonal decomposition (POD) of the LSTM and FNN
predictions shows that despite a lower mean absolute relative accuracy, LSTMs
capture the flow dynamics better than FNNs. When summed, the POD coefficients
from FNN predictions and from LSTM predictions amount to 96.51% and 97.66%
relative to the coefficients of the original simulations, respectively.
