  The free energy plays a fundamental role in descriptions of many systems in
continuum physics. Notably, in multiphysics applications, it encodes
thermodynamic coupling between different fields. It thereby gives rise to
driving forces on the dynamics of interaction between the constituent
phenomena. In mechano-chemically interacting materials systems, even
consideration of only compositions, order parameters and strains can render the
free energy to be reasonably high-dimensional. In proposing the free energy as
a paradigm for scale bridging, we have previously exploited neural networks for
their representation of such high-dimensional functions. Specifically, we have
developed an integrable deep neural network (IDNN) that can be trained to free
energy derivative data obtained from atomic scale models and statistical
mechanics, then analytically integrated to recover a free energy density
function. The motivation comes from the statistical mechanics formalism, in
which certain free energy derivatives are accessible for control of the system,
rather than the free energy itself. Our current work combines the IDNN with an
active learning workflow to improve sampling of the free energy derivative data
in a high-dimensional input space. Treated as input-output maps, machine
learning accommodates role reversals between independent and dependent
quantities as the mathematical descriptions change with scale bridging. As a
prototypical system we focus on Ni-Al. Phase field simulations using the
resulting IDNN representation for the free energy density of Ni-Al demonstrate
that the appropriate physics of the material have been learned. To the best of
our knowledge, this represents the most complete treatment of scale bridging,
using the free energy for a practical materials system, that starts with
electronic structure calculations and proceeds through statistical mechanics to
continuum physics.
