  An outstanding challenge in chemical computation is the many-electron problem
where computational methodologies scale prohibitively with system size. The
energy of any molecule can be expressed as a weighted sum of the energies of
two-electron wave functions that are computable from only a two-electron
calculation. Despite the physical elegance of this extended ``aufbau''
principle, the determination of the distribution of weights -- geminal
occupations -- for general molecular systems has remained elusive. Here we
introduce a new paradigm for electronic structure where approximate
geminal-occupation distributions are ``learned'' via a convolutional neural
network. We show that the neural network learns the $N$-representability
conditions, constraints on the distribution for it to represent an $N$-electron
system. By training on hydrocarbon isomers with only 2-7 carbon atoms, we are
able to predict the energies for isomers of octane as well as hydrocarbons with
8-15 carbons. The present work demonstrates that machine learning can be used
to reduce the many-electron problem to an effective two-electron problem,
opening new opportunities for accurately predicting electronic structure.
