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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.

Thermal evolution simulations of planetary mantles in 2D and 3D are computationally intensive. A low-fidelity alternative is to use scaling laws based on boundary-layer theory to express Nusselt Number (Nu) as a function of Rayleigh Number (Ra). Such a Ra-Nu relation can be used to run `0D' parametrized evolution models by solving a simple energy balance equation. Yet scaling relations are available only for simple flows that cannot capture the full complexity of mantle dynamics. We propose leveraging Machine Learning to find a higher-dimensional mapping from five different parameters to the entire 1D temperature profile. The parameters are Ra, internal heating Ra, dissipation number and the maximum viscosity contrast between top and bottom due to temperature and pressure. We train a Neural Network (NN) to take these inputs and predict the resulting steady-state temperature profile. The training data comes from a subset of 20,000 compressible simulations on a 2D cylindrical grid. This results in predictions with an average error of 1.6% on the test set. The NN can potentially be used to build a 1D evolution model by stacking several steady-state temperature profiles together, with each prediction serving as an input at the next time-step.

One of the main goals in exoplanetary science is the interior characterization of observed exoplanets. A common approach to characterize the interior of a known exoplanet is the use of numerical models to compute an interior structure which complies with the measured mass and radius of the planet (Sotin et al. 2017, Seager et al. 2007). With only these two observables, possible solutions tend to be highly degenerate, with multiple, qualitatively different interior compositions that can match the observations equally well. Other potential observables include the Love number k2 (bearing information on the mass concentration in the interior of the planet), and the elemental abundances of the host star, which may be representative of those of the planet. We explore the application of a deep learning neural network to the interior characterization of exoplanets. We employ a simple 1D structure model to construct a large training set of sub-Neptunian exoplanets up to 20 Earth-masses. A model planet consists of five layers: an iron-rich core, a lower and upper silicate mantle, a water ice layer, and a gaseous H/He envelope. The size of each layer is constrained by prescribed mass fractions. Using a feedforward neural network trained on a large dataset of such modelled planets, we show that we can reasonably well predict the original model input parameters (core, mantle, ice layer and atmosphere mass fractions) from just mass, radius and the fluid Love number k2.