  Atomistic or ab-initio molecular dynamics simulations are widely used to
predict thermodynamics and kinetics and relate them to molecular structure. A
common approach to go beyond the time- and length-scales accessible with such
computationally expensive simulations is the definition of coarse-grained
molecular models. Existing coarse-graining approaches define an effective
interaction potential to match defined properties of high-resolution models or
experimental data. In this paper, we reformulate coarse-graining as a
supervised machine learning problem. We use statistical learning theory to
decompose the coarse-graining error and cross-validation to select and compare
the performance of different models. We introduce CGnets, a deep learning
approach, that learns coarse-grained free energy functions and can be trained
by a force matching scheme. CGnets maintain all physically relevant invariances
and allow one to incorporate prior physics knowledge to avoid sampling of
unphysical structures. We show that CGnets can capture all-atom
explicit-solvent free energy surfaces with models using only a few
coarse-grained beads and no solvent, while classical coarse-graining methods
fail to capture crucial features of the free energy surface. Thus, CGnets are
able to capture multi-body terms that emerge from the dimensionality reduction.
