Characterizing dynamical systems given limited measurements is a common challenge throughout the physical and biological sciences. However, this task is challenging, especially due to transient variability in systems with equivalent long-term dynamics. We address this by introducing smooth prototype equivalences (SPE), a framework that fits a diffeomorphism using normalizing flows to distinct prototypes - simplified dynamical systems that define equivalence classes of behavior. SPE enables classification by comparing the deformation loss of the observed sparse, high-dimensional measurements to the prototype dynamics. Furthermore, our approach enables estimation of the invariant sets of the observed dynamics through the learned mapping from prototype space to data space. Our method outperforms existing techniques in the classification of oscillatory systems and can efficiently identify invariant structures like limit cycles and fixed points in an equation-free manner, even when only a small, noisy subset of the phase space is observed. Finally, we show how our method can be used for the detection of biological processes like the cell cycle trajectory from high-dimensional single-cell gene expression data.

翻译：在物理和生物科学中，根据有限测量数据刻画动力系统是一个普遍存在的挑战。然而，由于具有相同长期动力学的系统存在瞬态变异性，这一任务尤为困难。我们通过引入光滑原型等价性（SPE）框架来解决这一问题，该框架利用归一化流拟合微分同胚映射，将系统映射到不同的原型——即定义行为等价类的简化动力系统。SPE通过比较观测到的稀疏高维测量数据与原型动力学之间的形变损失来实现分类。此外，我们的方法能够通过学习从原型空间到数据空间的映射，估计观测动力系统的不变集。在振荡系统分类任务中，我们的方法优于现有技术，并且能够以无方程的方式高效识别极限环和不动点等不变结构，即使仅观测到相空间的一个小而嘈杂的子集。最后，我们展示了该方法如何用于检测生物过程，例如从高维单细胞基因表达数据中识别细胞周期轨迹。