  Accurately describing properties of challenging problems in physical sciences
often requires complex mathematical models that are unmanageable to tackle
head-on. Therefore, developing reduced dimensionality representations that
encapsulate complex correlation effects in many-body systems is crucial to
advance the understanding of these complicated problems. However, a numerical
evaluation of these predictive models can still be associated with a
significant computational overhead. To address this challenge, in this paper,
we discuss a combined framework that integrates recent advances in the
development of active-space representations of coupled cluster (CC) downfolded
Hamiltonians with neural network approaches. The primary objective of this
effort is to train neural networks to eliminate the computationally expensive
steps required for evaluating hundreds or thousands of Hugenholtz diagrams,
which correspond to multidimensional tensor contractions necessary for
evaluating a many-body form of downfolded/effective Hamiltonians. Using small
molecular systems (the H2O and HF molecules) as examples, we demonstrate that
training neural networks employing effective Hamiltonians for a few nuclear
geometries of molecules can accurately interpolate/ extrapolate their forms to
other geometrical configurations characterized by different intensities of
correlation effects. We also discuss differences between effective interactions
that define CC downfolded Hamiltonians with those of bare Hamiltonians defined
by Coulomb interactions in the active spaces.
