Data-driven modeling of multi-timescale systems by mixtures of neural ordinary differential equations

• Accurately and efficiently predicting complex fluid dynamics, especially in systems prone to extreme events, is a critical challenge across various scientific and engineering disciplines. Here we present a data-driven approach to modeling systems with multiple timescales, with focus on systems exhibiting phenomena such as bursting or quasi-laminarization. In recent works, it has been shown that decomposing a data-manifold into an atlas of charts can provide benefits in developing reduced order models, as it allows for the separation of the state space into dynamically distinct regions, on which local dynamical models can be obtained. These methods typically rely on pre-processing the dataset using spatial-clustering techniques to identify an appropriate set of charts. Here we present a method that learns such a domain decomposition concurrently with learning the dynamics, allowing for the representation of multiple dynamic regimes and and the transitions between them. This is achieved by modeling the dynamics as a weighted sum of neural ODEs, where different vector fields are learned in different regions of the state space. We demonstrate our method on low-dimensional dynamical systems exhibiting distinct timescales and extreme events, as well as on a Kuramoto-Sivashinsky system exhibiting complex, non-periodic bursting dynamics, where the method is paired with an autoencoder for dimension reduction.