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 multibody terms that emerge from the dimensionality reduction.

Brief Background This codebook provides information on the EQI survey data, which is intended to provide scholars and policy makers with metrics about citizens' perceptions and experiences with governance in Europe. The survey has been thus far done in two rounds-2010 and 2013. The data file here provides the combined microdata for both years. The EQI survey data was originally funded by the EU Commission (REGIO) and published in a report by Charron, Lapuente and Rothstein (2010) and later by Charron, Dijkstra and Lapuente (2014) in Regional Studies. In 2013, the survey was redone ; this time funded by the EU Commission via ANTICORRP, a large collaborative research group of scholars across Europe 1. The survey provides unique data for researchers and policy makers in that it is mainly concerned with governance of public sector institutions at the sub-national level. Questions are posed to respondents about perceived and experience with corruption, impartiality of services and quality of public services in several public service sectors. As opposed to many other survey in Europe that allow crosscountry or regional comparisons, the EQI Sample and Method The survey, part of a European-wide anti-corruption research project, was conducted in both rounds by Efficience 3 (E3), a French market-research, survey company specializing in public opinion throughout Europe for researchers, politicians and advertising firms. E3 conducted the interviews

Coarse graining enables the investigation of molecular dynamics for larger systems and at longer timescales than is possible at an atomic resolution. However, a coarse graining model must be formulated such that the conclusions we draw from it are consistent with the conclusions we would draw from a model at a finer level of detail. It has been proved that a force matching scheme defines a thermodynamically consistent coarse-grained model for an atomistic system in the variational limit. Wang et al. [ACS Cent. Sci. 5, 755 (2019)] demonstrated that the existence of such a variational limit enables the use of a supervised machine learning framework to generate a coarse-grained force field, which can then be used for simulation in the coarse-grained space. Their framework, however, requires the manual input of molecular features to machine learn the force field. In the present contribution, we build upon the advance of Wang et al. and introduce a hybrid architecture for the machine learning of coarse-grained force fields that learn their own features via a subnetwork that leverages continuous filter convolutions on a graph neural network architecture. We demonstrate that this framework succeeds at reproducing the thermodynamics for small biomolecular systems. Since the learned molecular representations are inherently transferable, the architecture presented here sets the stage for the development of machine-learned, coarse-grained force fields that are transferable across molecular systems.