- Deutsches Zentrum für Luft- und Raumfahrt (DLR)

Mantle convection plays a fundamental role in the long-term thermal evolution of terrestrial planets like Earth, Mars, Mercury and Venus. The buoyancy-driven creeping flow of silicate rocks in the mantle is modeled as a highly viscous fluid over geological time scales and quantified using partial differential equations (PDEs) for conservation of mass, momentum and energy. Yet, key parameters and initial conditions to these PDEs are poorly constrained and often require a large sampling of the parameter space to find constraints from observational data.

Since it is not computationally feasible to solve hundreds of thousands of forward models in 2D or 3D, scaling laws have been the go-to alternative. These are computationally efficient, but ultimately limited in the amount of physics they can model (e.g., depth-dependent material properties). More recently, machine learning techniques have been used for advanced surrogate modeling. For example, Agarwal et al. (2020) used feedforward neural networks to predict the evolution of entire 1D laterally averaged temperature profile in time from five parameters: reference viscosity, enrichment factor for the crust in heat producing elements, initial mantle temperature, activation energy and activation volume of the diffusion creep. In Agarwal et al. (2021), we extended that study to predict the full 2D temperature field of a Mars-like planet using convolutional autoencoders and long-short-term memory networks.

Despite producing reasonably accurate and realistic looking temperature fields, these data-driven surrogates require a lot of simulations, are not as accurate as traditional numerical solvers and do not predict the remaining state variables (velocity and pressure). Thus, we are keen on exploring physics-based machine learning algorithms that embed a few or all of the underlying PDEs into the loss function of a learning algorithm and thus, might require none (e.g. as in Wandel et. al 2021) to little data. We discretize the PDEs in space using finite differences, which are implemented as convolutional operators. Preliminary results from an analytical benchmark (Trubitsyn, 2006) show that for a given temperature field, the corresponding velocities can be calculated with a mean absolute error of approximately 5x10^-4. The next step is to see if these velocities can be used to advect the temperature field, solve for the new velocities and so on and so forth to advance the flow in time. This will lead to the next benchmark (Blankenbach et al., 1989), where the flow should converge to a steady-state.