The general procedure that we developed was to split the problem into two: a static variational problem and a predictive dynamic Monte Carlo problem. We start out the variational problem as the optimization of an error functional that would parameterize 3 coupled analytic functions that are trained according to mainly classically-extrapolated experimental data. The reason we pick the classical extrapolation is that the simulations themselves that employ the analytic functions are purely classical, and thus, cannot capture the quantum effects corresponding to temperatures below the Debye temperature. We refer to this variational problem as a static one, since the optimization is done at the classical 0K (not the quantum 0K). This optimization involves not only elastic properties but also stacking fault and spectral phonon properties at high-symmetry points in the Brillouin Zone.
After the parameterization is found, then a series of Monte Carlo simulations are performed at various different temperatures in order to generate large enough samplings to employ a strain fluctuation formulation to determine thermoelastic properties. The discrepancy between these Monte Carlo results and experimental data then guides how to modify static variational problem in order to remove any effect of bias and over or under-fitting. Although we did not employ a formal Bayesian technique for this step, it would make sense if one had a large amount of computational resources in order to automate the procedure. Instead, we had to employ more heuristic techniques because the modifications to the static variational step in each iteration cycle requires not only a variation in the parameters selected, but also a variation in the function space defining the model. Furthermore, the there was no training data (other than from previous iterations) to help us know the form of the likelihoods or priors. Generalized splines could work, but we still would require variations in the spline orders as well as the number and position of spline nodal points. So there was no straightforward Bayesian or other textbook procedure that (to our knowledge) could have worked. Furthermore, there are many other practical constraints and considerations that prevent straightforward application of machine learning techniques without some modification or extension.
This procedure produced excellent results after 105 iterations. (You can see these in the other presentation). This presentation was given at the day of my departure from my stay and collaboration with the Institute for Shock Physics at Washington State University. It was such a fruitful collaboration because it combined the invaluably rigorous treatment of continuum mechanics from Mike Winey and Yogi Gupta, with my numerical and simulation wizardry.
