导频波动力学：使用动态模式分解来表征分叉、混沌路径和涌现统计
Pilot-Wave Dynamics: Using Dynamic Mode Decomposition to characterize Bifurcations, Routes to Chaos and Emergent Statistics
摘要：
我们开发了一个数据驱动的表征的先导波流体动力学系统中，一个弹跳液滴自我推进沿表面的振动浴。我们考虑液滴运动在一个有限的一维几何形状，并应用{ \em动态模式分解}（DMD），以表征的波场的演变作为浴的振动加速度逐渐增加。DMD提供了一个回归框架，用于在时空数据的快照上自适应地学习最佳拟合线性动力学模型。波场的DMD表征产生了对弹跳液滴问题的新观点，该观点与量子力学的数学机制建立了有价值的新联系。此外，它提供了一个低秩表征的导频波物理的分岔结构。具体地说，分析表明，随着振动加速度的增加，先导波场经历了一系列的霍普夫分岔，最终导致一个混乱的波场。平均导频波场和液滴统计之间建立的关系，使我们能够表征的紧急统计的演变与增加的振动迫使从导频波场的演变。因此，我们开发了一个数值框架，其基本结构与量子力学相同，特别是预测粒子统计的波动理论。
英文摘要：
We develop a data-driven characterization of the pilot-wave hydrodynamic system in which a bouncing droplet self-propels along the surface of a vibrating bath. We consider drop motion in a confined one-dimensional geometry, and apply the {\em Dynamic mode decomposition} (DMD) in order to characterize the evolution of the wave field as the bath's vibrational acceleration is increased progressively. DMD provides a regression framework for adaptively learning a best-fit linear dynamics model over snapshots of spatio-temporal data. The DMD characterization of the wave field yields a fresh perspective on the bouncing-droplet problem that forges valuable new links with the mathematical machinery of quantum mechanics. Moreover, it provides a low-rank characterization of the bifurcation structure of the pilot wave physics. Specifically, the analysis shows that as the vibrational acceleration is increased, the pilot-wave field undergoes a series of Hopf bifurcations that ultimately lead to a chaotic wave field. The established relation between the mean pilot-wave field and the droplet statistics allows us to characterize the evolution of the emergent statistics with increased vibrational forcing from the evolution of the pilot-wave field. We thus develop a numerical framework with the same basic structure as quantum mechanics, specifically a wave theory that predicts particle statistics.
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