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Scientists use a mathematical subject called topology to study the shapes of objects. An important part of topology is counting the number of pieces and the number of holes in an object, and researchers use this information to group objects into different types. For example, a doughnut has the same number of holes and the same number of pieces as a teacup with one handle, but it is different from a ball. In studies that resemble activities like “connect-the-dots,” scientists use ideas from topology to study the “shape” of data. Ideas and methods from topology have been used to study the branching structures of veins in leaves, voting in elections, flight patterns in models of bird flocking, and more. 

Abstract We report on an unpublished and previously unknown manuscript of John von Neumann and contextualize it within the development of the theory of shock waves and detonations during the nineteenth and twentieth centuries. Von Neumann studies bombs comprising a primary explosive charge along with explosive booster material. His goal is to calculate the minimal amount of booster needed to create a sustainable detonation, presumably because booster material is often more expensive and more volatile. In service of this goal, he formulates and analyzes a partial differential equation-based model describing a moving shock wave at the interface of detonated and undetonated material. We provide a complete transcription of von Neumann’s work and give our own accompanying explanations and analyses, including the correction of two small errors in his calculations. Today, detonations are typically modeled using a combination of experimental results and numerical simulations particular to the shape and materials of the explosive, as the complex three dimensional dynamics of detonations are analytically intractable. Although von Neumann’s manuscript will not revolutionize our modern understanding of detonations, the document is a valuable historical record of the state of hydrodynamics research during and after World War II. 

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. 

In a complex system, the interactions between individual agents often lead to emergent collective behavior such as spontaneous synchronization, swarming, and pattern formation. Beyond the intrinsic properties of the agents, the topology of the network of interactions can have a dramatic influence over the dynamics. In many studies, researchers start with a specific model for both the intrinsic dynamics of each agent and the interaction network and attempt to learn about the dynamics of the model. Here, we consider the inverse problem: given data from a system, can one learn about the model and the underlying network? We investigate arbitrary networks of coupled phase oscillators that can exhibit both synchronous and asynchronous dynamics. We demonstrate that, given sufficient observational data on the transient evolution of each oscillator, machine learning can reconstruct the interaction network and identify the intrinsic dynamics.