Scale Bridging

Across scales, from the atomic to the continuum, the free energy occupies a central role in mechanisms of phase transformations and, at larger scales, in microstructure evolution. It is in the free energy that the coupling between different fields, such as mechanics and chemistry, is encoded within continuum descriptions of far-from-equilibrium materials phenomena. In material systems that demonstrate mechanochemical interactions, even treatments that are restricted to compositions, order parameters, and strains yield a free energy description that lives in a high-dimensional space. One approach to bridging the scales between the electronic structure of a solid and continuum descriptions of its nonequilibrium behavior in this high-dimensional setting is via a class of neural networks that we have called integrable deep neural networks (IDNNs). They can be trained to free energy derivative data that are typically obtained from first principles statistical mechanics simulations, and because of their construction based on the fundamental theorem of calculus, they can then be analytically integrated to recover a free energy density function. In this chapter we will note how the combination of the IDNN with an active learning workflow can help attain well-distributed sampling of the data on free energy derivatives and their arguments in high-dimensional input spaces. This allows bridging of the scales between first principles statistical mechanics and continuum phase field models. As in other chapters, the machine learning methods are demonstrated on continuum materials systems of interest—here on a Nickel–Aluminum alloy.