Gradient-domain machine learning (GDML) is an accurate and efficient approach to learn a molecular potential and associated force field based on the kernel ridge regression algorithm. Here, we demonstrate its application to learn an effective coarse-grained (CG) model from all-atom simulation data in a sample efficient manner. The CG force field is learned by following the thermodynamic consistency principle, here by minimizing the error between the predicted CG force and the all-atom mean force in the CG coordinates. Solving this problem by GDML directly is impossible because coarse-graining requires averaging over many training data points, resulting in impractical memory requirements for storing the kernel matrices. In this work, we propose a data-efficient and memory-saving alternative. Using ensemble learning and stratified sampling, we propose a 2-layer training scheme that enables GDML to learn an effective CG model. We illustrate our method on a simple biomolecular system, alanine dipeptide, by reconstructing the free energy landscape of a CG variant of this molecule. Our novel GDML training scheme yields a smaller free energy error than neural networks when the training set is small, and a comparably high accuracy when the training set is sufficiently large.
