Bhaskar, D; Manhart, A; Milzman, J; Nardini, JT; Storey, KM; Topaz, CM; Ziegelmeier, L; (2019) Analyzing collective motion with machine learning and topology. Chaos: An Interdisciplinary Journal of Nonlinear Science , 29 (12) , Article 123125. 10.1063/1.5125493.

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Abstract

We use topological data analysis and machine learning to study a seminal model of collective motion in biology [M. R. D’Orsogna et al., Phys. Rev. Lett. 96, 104302 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based on topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters. A fundamental goal in the study of complex, nonlinear systems is to understand the link between local rules and collective behaviors. For instance, what influence does coupling between oscillators have on the ability of a network to synchronize? How does policing affect hotspots of crime in urban areas? Why do the movement decisions that fish make lead to schooling structures? We examine such links by bringing together tools from applied topology and machine learning to study a seminal model of collective motion that replicates behavior observed in biological swarming, flocking, and milling. Studies of collective motion often focus on the so-called forward problem: given a particular mathematical model, what dynamics are observed for different parameters input to the model? In contrast, we study an inverse problem: given observed data, what model parameters could have produced it? Also, given observed data, what paradigmatic dynamics are being exhibited? To answer these questions, we use machine learning techniques and find that they achieve higher accuracy when applied to topological summaries of the data—called crockers—as compared to more traditional summaries of the data that are commonly used in biology and physics to characterize collective motion. Ours is the first study to use crockers for nonlinear dynamics classification and parameter recovery.

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