Neural networks, local minima and computational thermodynamics

Thermodynamics plays an increasingly important role in computational geodynamics, as the community moves towards including chemical reactions in mechanical models of deformation. An example is mineral reactions that induce volume changes which affect the local state of stress that may trigger nonlinear feedbacks. Another example is numerical simulations of magmatic systems where the chemistry of melts changes dramatically as the melt differentiates on its way up through the lithosphere. In order to compare the results of numerical models with geochemical and petrological field data, it is crucial to predict the rock types and major element chemistry from numerical simulations of magmatic systems. Precomputing phase diagrams and including them as a lookup table, which is the current standard approach, works when the chemistry is more or less constant, but is nontrivial for magmatic systems where the chemistry changes drastically and involves a 13-dimensional system (11 oxides, plus pressure and temperature). Parameterizing the main reactions is a possibility, but this renders the comparison of simulation results with available data difficult.

In the context of magmatic systems, significant progress has been made in recent years and we now have thermodynamic melting models that are consistent with experiments and can simulate the full compositional range from mafic to felsic melt compositions. Yet, in order to be useful for geodynamic applications, we also need sufficiently fast Gibbs energy minimization software tools that can automatically determine the most stable assemblage for a given pressure, temperature and chemistry. This is a nonlinear constrained optimization problem, for which we have developed a new, parallel, solution approach. One novelty of our approach is that it can efficiently make use of good initial guesses, for example obtained from the previous timestep of a geodynamic simulation or from nearby points in pressure and temperature space. Yet, as with any gradient-based method, a risk remains to be trapped in a local minimum in the solution space that is not the overall most stable assemblage. It is thus important to explore a sufficiently broad range of starting parameters to ensure convergence toward the global minimum. Whereas this problem is well-known among users of existing thermodynamic software (such as THERMOCALC, where the starting values have to be adjusted manually), it is much less clear how nonlinear the parameter space actually is for real applications. Do we have thousands of local minima, or are there only a few (and if yes, can we precompute some of these)?

Here, we will discuss several examples of melting models and map out the nonlinearity of the parameter space for these cases to get better insights in how to further speed up such calculations. We also discuss how shallow and deep neural networks can be trained and implemented as part of the workflow.