Dynamical systems are found in innumerable forms across the physical and biological sciences, yet all these systems fall naturally into universal equivalence classes: conservative or dissipative, stable or unstable, compressible or incompressible. Predicting these classes from data remains an essential open challenge in computational physics at which existing time-series classification methods struggle. Here, we propose, \texttt{phase2vec}, an embedding method that learns high-quality, physically-meaningful representations of 2D dynamical systems without supervision. Our embeddings are produced by a convolutional backbone that extracts geometric features from flow data and minimizes a physically-informed vector field reconstruction loss. In an auxiliary training period, embeddings are optimized so that they robustly encode the equations of unseen data over and above the performance of a per-equation fitting method. The trained architecture can not only predict the equations of unseen data, but also, crucially, learns embeddings that respect the underlying semantics of the embedded physical systems. We validate the quality of learned embeddings investigating the extent to which physical categories of input data can be decoded from embeddings compared to standard blackbox classifiers and state-of-the-art time series classification techniques. We find that our embeddings encode important physical properties of the underlying data, including the stability of fixed points, conservation of energy, and the incompressibility of flows, with greater fidelity than competing methods. We finally apply our embeddings to the analysis of meteorological data, showing we can detect climatically meaningful features. Collectively, our results demonstrate the viability of embedding approaches for the discovery of dynamical features in physical systems.

翻译：动力系统在物理与生物科学中以无数形式存在，但这些系统自然归属于普适的等价类：保守或耗散、稳定或不稳定、可压缩或不可压缩。如何从数据中预测这些类别仍是计算物理学中一项重要的开放性挑战，现有的时间序列分类方法难以胜任。本文提出 phase2vec 嵌入方法，该方法无需监督即可学习二维动力系统的高质量、具有物理意义的表征。其嵌入结果由卷积主干网络生成，该网络从流数据中提取几何特征，并最小化基于物理信息的向量场重建损失。在辅助训练阶段，嵌入结果经过优化，使其能够稳健地编码未见数据的方程，且性能超越基于逐方程拟合的方法。训练后的架构不仅能预测未见数据的方程，更重要的是，其习得的嵌入结果尊重所嵌入物理系统的基础语义。我们通过评估从嵌入结果中解码输入数据物理类别的能力（与标准黑箱分类器及最先进的时间序列分类技术对比），验证了所习得嵌入结果的质量。研究发现，我们的嵌入结果能更忠实地编码底层数据的重要物理属性（包括不动点稳定性、能量守恒及流的不可压缩性），且保真度优于竞争方法。最后，我们将所提出的嵌入方法应用于气象数据分析，证明了其可检测具有气候意义的特征。综上，我们的结果证明了嵌入方法在物理系统动力学特征发现中的可行性。