We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatiotemporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and subspace clustering to robustly identify translational invariances that can be leveraged to build improved reduced-order models (ROMs). Invariances, whether translational or rotational, are well known to compromise the ability of ROMs to produce accurate and/or low-rank representations of the spatiotemporal dynamics. However, by discovering translations in a principled way, data can be shifted into a coordinate systems where quality, low-dimensional ROMs can be constructed. This approach can be used on either numerical or experimental data with or without knowledge of the governing equations. We demonstrate our method on a variety of PDEs of increasing difficulty, taken from the field of fluid dynamics, showing the efficacy and robustness of the proposed approach.

中文翻译：

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行波物理的降维和降阶建模

我们开发了一种无监督机器学习算法，用于自动发现和识别由偏微分方程 (PDE) 控制的时空系统中的行波。我们的方法使用稀疏回归和子空间聚类来稳健地识别可用于构建改进的降阶模型 (ROM) 的平移不变性。众所周知，不变性，无论是平移的还是旋转的，都会损害 ROM 产生时空动态的准确和/或低秩表示的能力。然而，通过以一种有原则的方式发现翻译，数据可以转移到一个坐标系统中，在那里可以构建高质量的低维 ROM。无论是否了解控制方程，此方法均可用于数值或实验数据。