Hierarchical Machine Learning of Low-Resolution Coarse-Grained Free Energy Potentials

Abstract

A force-matching-based method for supervised machine learning (ML) of coarse-grained (CG) free energy (FE) potentials�known as multiscale coarse-graining via force-matching (MSCG/FM)�is an efficient method to develop microscopically informed CG models that are thermodynamically and statistically equivalent to the reference microscopic models. For low-resolution models, when the coarse-graining is at supramolecular scales, objective-oriented clustering of nonbonded particles is required and the reduced description becomes a function of the clustering algorithm. In the present work, we explore the dependence of the ML of the CG Helmholtz FE potential on the clustering algorithm. We consider coarse-graining based on partitional (k-means, leading to Voronoi diagram) and hierarchical agglomerative (bottom-up) clustering algorithms common in unsupervised ML and develop theory connecting the MSCG/FM learned CG Helmholtz potential and the clustering statistics. By combining the agglomerative clustering and the MSCG/FM learning in a recursive manner, we propose an efficient ML methodology to develop the fine-to-low resolution hierarchies of the CG models. The methodology does not suffer from degrading accuracy or increased computational cost to construct larger hierarchies and as such does not impose an upper size limitation of the CG particles resulting from the extended hierarchies. The utility of the methodology is demonstrated by obtaining the bottom-up agglomerative hierarchy for liquid nitromethane from all-atom molecular dynamics (MD) simulations. For agglomerative hierarchies, we prove the existence of renormalization group transformations that indicate self-similarity and allow for learning the low-resolution MSCG/FM potentials at low computational cost by rescaling and renormalizing the certain finer-resolution members of the hierarchy. The hierarchies of the CG models can be used to carry out simulations under constant-pressure conditions.

OpenMSCG: A Software Tool for Bottom-Up Coarse-Graining

“…To further increase the spatiotemporal scales, models with less granularity (lower CG resolution) in which a single CG site represents multiple functional groups are needed. We note that several methods have been proposed toward optimal spatial mappings, and this remains an active area of research. − To this end, two systematic and quantitative approaches, the Essential Dynamics Coarse-Graining (EDCG) and HeteroENM methods, have been successfully applied in various CG studies of complex biomolecular systems. Both methods aim to retain and reproduce the most significant atomistic fluctuations (also known as essential dynamics) from the CG models based on analysis from all-atom simulation trajectories.…”

Maximum Entropy Theory of Multiscale Coarse-Graining via Matching Thermodynamic Forces: Application to a Molecular Crystal (TATB)

“…for equal-sized clusters of the same composition). We note that, in accordance to eq , the variables λ e 0 can be also viewed as the state variables for the ( R N , P N )-restricted ensemble on the phase space of irrelevant coordinates ( x < , p x < ) . In eq , we took into account that at equilibrium, the set λ e 0 describes the unrestricted microscopic ensemble as well.…”

“…The coordinates ( x < , p x < ) are averaged out in the MSCG/FM learning. To describe the statistics, for example, in the NVT ensemble, we may select the remaining λ I 0 parameters to be volume, λ 2 N +1 0 = V (the conjugate is A 2 N + 1 = β scriptP , where scriptP is the system instantaneous pressure for which ⟨ P ⟩ = P is the thermodynamic pressure) and the inverse temperature, λ 2 N +2 0 = β (the conjugate to A 2 N + 2 = H n ). ,,, The respective potential β –1 Φ(λ 0 ) is the CG Helmholtz FE A ( R N , P N , V , β). Similar considerations hold for the other ensembles, and if we denote λ e 0 as the ensemble state variables other than { R N , P N }, e.g., λ e 0 = { V , β} for the NVT ensemble (correspondingly, we denote the conjugate forces as A e ), then the function H CG ( R N , P N ) = −β –1 Φ( R N , P N , λ e 0 ) plays the role of an effective Hamiltonian for the CG system.…”

“…The coarse-graining of molecular compounds at submolecular resolution must properly handle the intramolecular (bonded) and nonbonded interactions at the relevant scales. The potential U appears to be essentially a many-body PMF, even for a microscopic system with pure pair interactions . Therefore, we can always write an arbitrary FE potential U in eq involving at most k -body terms as the following sum of the s -body terms U s , s ≤ k U ( R N , λ normale 0 ) = ∑ I U 1 ( R I , λ normale 0 ) + ∑ { I 1 , I 2 } U 2 ( R I 1 , R I 2 , λ normale 0 ) + · · · + ∑ { I 1 , · · · , I k } U k ( R I 1 , · · · , R I k , λ normale 0 ) The two-body contribution U 2 can be separated into purely bonded U 2 b and nonbonded U 2 nb terms, although in general, the higher order contributions cannot be similarly separated.…”

“…Considering the importance of the many-body contributions to A ( R N , ε̂, T ) at lower resolutions even for microscopic systems with purely pair interactions, the pair nonparametric (spline) representation of the thermodynamic forces adopted in conventional MSCG force-matching might be a major decrement to the quality of the CG models. Nevertheless, it has been demonstrated on many occasions that the pairwise spline hypothesis for nonbonded interactions performs fairly well in approximating the CG PESs of many complex condensed matter systems in which the many-body effects are expected to be significant. ,,,,, For submolecular scales, as considered in the present work, coarse-graining requires matching the microscopic intramolecular PESs which often include three- and four-body terms.…”