Hybrid dynamical systems are prevalent in science and engineering to express complex systems with continuous and discrete states. To learn the laws of systems, all previous methods for equation discovery in hybrid systems follow a two-stage paradigm, i.e. they first group time series into small cluster fragments and then discover equations in each fragment separately through methods in non-hybrid systems. Although effective, these methods do not fully take advantage of the commonalities in the shared dynamics of multiple fragments that are driven by the same equations. Besides, the two-stage paradigm breaks the interdependence between categorizing and representing dynamics that jointly form hybrid systems. In this paper, we reformulate the problem and propose an end-to-end learning framework, i.e. Amortized Equation Discovery (AMORE), to jointly categorize modes and discover equations characterizing the dynamics of each mode by all segments of the mode. Experiments on four hybrid and six non-hybrid systems show that our method outperforms previous methods on equation discovery, segmentation, and forecasting.

翻译：混合动力系统在科学与工程中广泛存在，用于表达兼具连续状态与离散状态的复杂系统。为学习系统规律，现有混合系统方程发现方法均遵循两阶段范式：首先将时间序列划分为小簇片段，随后通过非混合系统方法独立发现各片段对应的方程。尽管该方法有效，但未能充分利用受相同方程驱动的多个片段在共享动力学特性上的共性。此外，两阶段范式割裂了分类与表征动力学之间本应相互依存的联合建模关系。本文对该问题进行了重新定义，提出端到端学习框架AMORE（Amortized Equation Discovery），通过联合分类模态并利用模态所有分段数据，发现刻画各模态动力学的方程。在四个混合系统与六个非混合系统上的实验表明，本方法在方程发现、分段预测及序列预测方面均优于现有方法。