  We present a physically-motivated topology of a deep neural network that can
efficiently infer extensive parameters (such as energy, entropy, or number of
particles) of arbitrarily large systems, doing so with O(N) scaling. We use a
form of domain decomposition for training and inference, where each sub-domain
(tile) is comprised of a non-overlapping focus region surrounded by an
overlapping context region. The size of these regions is motivated by the
physical interaction length scales of the problem. We demonstrate the
application of EDNNs to three physical systems: the Ising model and two
hexagonal/graphene-like datasets. In the latter, an EDNN was able to make total
energy predictions of a 60 atoms system, with comparable accuracy to density
functional theory (DFT), in 57 milliseconds. Additionally EDNNs are well suited
for massively parallel evaluation, as no communication is necessary during
neural network evaluation. We demonstrate that EDNNs can be used to make an
energy prediction of a two-dimensional 35.2 million atom system, over 1 square
micrometer of material, at an accuracy comparable to DFT, in under 25 minutes.
Such a system exists on a length scale visible with optical microscopy and
larger than some living organisms.
