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We use topological data analysis and machine learning to study a seminal model of collective motion in biology [DOrsogna et al., Phys. Rev. Lett. 96 (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 in 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.
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Collective motion in animal groups, such as swarms of insects, flocks of birds, and schools of fish, are some of the most visually striking examples of emergent behavior. Empirical analysis of these behaviors in experiment or computational simulation primarily involves the use of swarm-averaged metrics or order parameters such as velocity alignment and angular momentum. Recently, tools from computational topology have been applied to the analysis of swarms to further understand and automate the detection of fundamentally different swarm structures evolving in space and time. Here, we show how the field of graph signal processing can be used to fuse these two approaches by collectively analyzing swarm properties using graph Fourier harmonics that respect the topological structure of the swarm. This graph Fourier analysis reveals hidden structure in a number of common swarming states and forms the basis of a flexible analysis framework for collective motion.
The study of topological bandstructures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine-learning, another rising topic of interest. Specifically, we introduce an unsupervised machine-learning approach that searches for and retrieves paths of adiabatic deformations between Hamiltonians, thereby clustering them according to their topological properties. The algorithm is general as it does not rely on a specific parameterization of the Hamiltonian and is readily applicable to any symmetry class. We demonstrate the approach using several different models in both one and two spatial dimensions and for different symmetry classes with and without crystalline symmetries. Accordingly, it is also shown how trivial and topological phases can be diagnosed upon comparing with a generally designated set of trivial atomic insulators.
Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.
We explore the possibility of inferring the properties of the Galactic neutron star population through machine learning. In particular, in this paper we focus on their dynamical characteristics and show that an artificial neural network is able to estimate with high accuracy the parameters which control the current positions of a mock population of pulsars. For this purpose, we implement a simplified population-synthesis framework (where selection biases are neglected at this stage) and concentrate on the natal kick-velocity distribution and the distribution of birth distances from the Galactic plane. By varying these and evolving the pulsar trajectories in time, we generate a series of simulations that are used to train and validate a suitably structured convolutional neural network. We demonstrate that our network is able to recover the parameters governing the kick-velocity and Galactic height distribution with a mean relative error of about $10^{-2}$. We discuss the limitations of our idealized approach and study a toy problem to introduce selection effects in a phenomenological way by incorporating the observed proper motions of 216 isolated pulsars. Our analysis highlights that increasing the sample of pulsars with accurate proper motion measurements by a factor of $sim$10, one of the future breakthroughs of the Square Kilometer Array, we might succeed in constraining the birth spatial and kick-velocity distribution of the neutron stars in the Milky Way with high precision through machine learning.
Microfluidic techniques have been extensively developed to realize micro-total analysis systems in a small chip. For microanalysis, electro-magnetic forces have generally been utilized for the trapping of objects, but hydrodynamics has been little explored despite its relevance to pattern formation. Here, we report that water-in-oil (W/O) droplets can be transported in the grid of an array of other large W/O droplets. As each droplet approaches an interspace of the large droplet array, while exhibiting persistent back-and-forth motion, it is conveyed at a velocity equal to the droplet array. We confirm the appearance of closed streamlines in a numerical simulation, suggesting that a vortex-like stream is involved in trapping the droplet. Furthermore, more than one droplet is also conveyed as an ordered cluster with dynamic reposition.
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