  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.
