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

Extensive deep neural networks for transferring small scale learning to large scale systems