Analyzing multimodal probability measures with autoencoders

TL;DR

This paper explores neural network autoencoders for identifying collective variables in molecular dynamics, analyzing their mathematical properties, physical interpretations, and extensions to improve description of physical systems.

Contribution

It introduces new mathematical insights and extensions for autoencoder-based methods to better capture transition states and multiple pathways in physical systems.

Findings

Autoencoders can effectively identify collective variables in toy and molecular systems.

Extensions incorporating transition states improve the physical relevance of the learned variables.

Mathematical analysis clarifies the loss function's role in capturing system features.

Abstract

Finding collective variables to describe some important coarse-grained information on physical systems, in particular metastable states, remains a key issue in molecular dynamics. Recently, machine learning techniques have been intensively used to complement and possibly bypass expert knowledge in order to construct collective variables. Our focus here is on neural network approaches based on autoencoders. We study some relevant mathematical properties of the loss function considered for training autoencoders, and provide physical interpretations based on conditional variances and minimum energy paths. We also consider various extensions in order to better describe physical systems, by incorporating more information on transition states at saddle points, and/or allowing for multiple decoders in order to describe several transition paths. Our results are illustrated on toy two dimensional systems and on alanine dipeptide.