  Machine-learned coarse-grained (CG) models have the potential for simulating
large molecular complexes beyond what is possible with atomistic molecular
dynamics. However, training accurate CG models remains a challenge. A widely
used methodology for learning CG force-fields maps forces from all-atom
molecular dynamics to the CG representation and matches them with a CG
force-field on average. We show that there is flexibility in how to map
all-atom forces to the CG representation, and that the most commonly used
mapping methods are statistically inefficient and potentially even incorrect in
the presence of constraints in the all-atom simulation. We define an
optimization statement for force mappings and demonstrate that substantially
improved CG force-fields can be learned from the same simulation data when
using optimized force maps. The method is demonstrated on the miniproteins
Chignolin and Tryptophan Cage and published as open-source code.
