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The authors formulate bottom-up coarse-graining as an implicit score matching problem, directly learning effective energy surfaces from coordinate distributions while circumventing the computationally prohibitive calculation of explicit coarse-grained forces. Standard force-matching frameworks fail at high coarsening ratios because integrating over the statistical ensembles of composite atomic configurations is intractable. By training directly on the score of the marginal configuration density, this framework drastically reduces training data requirements and supports coarse-grained sites representing thousands of atoms, enabling mesoscopic simulations up to the millimeter scale with atomic-level fidelity.
Bypassing intractable force calculations entirely, score matching scales bottom-up coarse-graining to thousands of atoms per interaction site with a fraction of the standard training data.
Molecular dynamics simulations provide a computational microscope for atomic scale processes but remain restricted to relatively small spatial and temporal scales. Coarse-grained models extend their reach by representing groups of atoms as effective interaction sites. The number of particles is systematically reduced by embedding the collective behavior of groups of atoms into coarse-grained sites governed by effective potentials. However, constructing accurate, highly coarsened potentials, even with the assistance of modern machine-learning methods, is a grand challenge because coarse grained forces depend on a statistical distribution of composite atomic configurations making direct calculation computationally prohibitive. Here, we introduce an alternative framework based on implicit score matching that directly avoids the need for explicit coarse-grained forces. This approach both substantially reduces the training data required relative to conventional force-matching methods and enables high-fidelity models in which each coarse-grained site represents hundreds to thousands of atoms. We demonstrate the method's versatility and effectiveness across a diverse range of chemical systems and phenomena, developing bottom-up, machine-learned models. These enable efficient modeling at micrometer and millimeter length scales, while retaining significant amounts of the atomic scale fidelity. These results establish implicit score matching as a practical route towards chemically accurate simulations that bridge molecular and mesoscopic scales.