Hessian Reparametrization for Coarse-grained Energy Minimization

Abstract

Energy minimization problems are highly non-convex problems at the heart of physical sciences. These problems often suffer from slow convergence due to sharply falling potentials, leading to small gradients. To make them tractable, we often resort to coarse-graining (CG), a type of lossy compression. We introduce a new way to perform CG using reparametrization, which can avoid some of the costly steps of traditional CG, such as force-matching and back-mapping. We focus on improving the slow dynamics by using CG to projecting onto slow modes. We show that in many physical systems slow modes can remain robust under dynamics and hence can be pre-computed from the Hessian of random configurations. We show the advantage of our CG method on some difficult synthetic problems inspired by molecular dynamics (MD). We also test our method on molecular dynamics for folding of small proteins, showing modest improvements. We observe that our method either reaches lower energies or runs in shorter time than the baseline non-CG simulations.