Representations of energy landscapes by sublevelset persistent homology: An example with n-alkanes

Interpretation of autoencoder-learned collective variables using Morse-Smale complex and sublevelset persistent homology: An application on molecular trajectories

Dimensionality reduction often serves as the first step toward a minimalist understanding of physical systems as well as the accelerated simulations of them. In particular, neural network-based nonlinear dimensionality reduction methods, such as autoencoders, have shown promising outcomes in uncovering collective variables (CVs). However, the physical meaning of these CVs remains largely elusive. In this work, we constructed a framework that (1) determines the optimal number of CVs needed to capture the essential molecular motions using an ensemble of hierarchical autoencoders and (2) provides topology-based interpretations to the autoencoder-learned CVs with Morse-Smale complex and sublevelset persistent homology. This approach was exemplified using a series of n-alkanes and can be regarded as a general, explainable nonlinear dimensionality reduction method.

Reproducing the Reaction Route Map on the Shape Space from Its Quotient by the Complete Nuclear Permutation-Inversion Group

This study develops an algorithm to reproduce reaction route maps (RRMs) in the shape space from the outputs of potential search algorithms. To demonstrate the algorithm, global reaction route mapping is utilized as a potential search algorithm, but the proposed algorithm should work with other potential search algorithms in principle. The proposed algorithm does not require any encoding of the molecular configurations and is thus applicable to complicated realistic molecules for which efficient encoding is not readily available. We show that subgraphs of an RRM mapped to each other by the action of the symmetry group are isomorphic and also provide an algorithm to compute the set of feasible transformations in the sense of Longuet-Higgins. We demonstrate the proposed algorithm in toy models and in more realistic molecules. Finally, we remark on absolute rate theory from our perspective.

Topological measures of order for imperfect two-dimensional Bravais lattices

Motivated by patterns with defects in natural and laboratory systems, we develop two quantitative measures of order for imperfect Bravais lattices in the plane. A tool from topological data analysis called persistent homology combined with the sliced Wasserstein distance, a metric on point distributions, are the key components for defining these measures. The measures generalize previous measures of order using persistent homology that were applicable only to imperfect hexagonal lattices in two dimensions. We illustrate the sensitivities of these measures to the degree of perturbation of perfect hexagonal, square, and rhombic Bravais lattices. We also study imperfect hexagonal, square, and rhombic lattices produced by numerical simulations of pattern-forming partial differential equations. These numerical experiments serve to compare the measures of lattice order and reveal differences in the evolution of the patterns in various partial differential equations.

Reactive Vortexes in a Naturally Activated Process: Non-Diffusive Rotational Fluxes at Transition State Uncovered by Persistent Homology

The dynamics of reaction coordinates during barrier-crossing are key to understanding activated processes in complex systems such as proteins. The default assumption from Kramers' physical intuition is that of a diffusion process. However, the dynamics of barrier-crossing in natural complex molecules are largely unexplored. Here we investigate the transition dynamics of alanine dipeptide isomerization, the simplest complex system with a large number of non-reaction coordinates that can serve as an adequate thermal bath feeding energy into the reaction coordinates. We separate conformations along the time axis and construct the dynamic probability surface of reaction. We quantify its topological structure and rotational flux using persistent homology and differential form. Our results uncovered a region with a strong reactive vortex in the configuration-time space, where the highest probability peak and the transition state ensemble are located. This reactive region contains strong rotational fluxes: Most reactive trajectories swirl multiple times around this region in the subspace of the two most important reaction coordinates. Furthermore, the rotational fluxes result from cooperative movement along the isocommitter surfaces and orthogonal barrier-crossing. Overall, our findings offer a first glimpse into the reactive vortex regions that characterize the non-diffusive dynamics of barrier-crossing of a naturally occurring activation process.

The Middle Science: Traversing Scale In Complex Many-Body Systems

A roadmap is developed that integrates simulation methodology and data science methods to target new theories that traverse the multiple length- and time-scale features of many-body phenomena.

Topology Applied to Machine Learning: From Global to Local

Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.