  The data-driven recovery of the unknown governing equations of dynamical
systems has recently received an increasing interest. However, the
identification of governing equations remains challenging when dealing with
noisy and partial observations. Here, we address this challenge and investigate
variational deep learning schemes. Within the proposed framework, we jointly
learn an inference model to reconstruct the true states of the system and the
governing laws of these states from series of noisy and partial data. In doing
so, this framework bridges classical data assimilation and state-of-the-art
machine learning techniques. We also demonstrate that it generalises
state-of-the-art methods. Importantly, both the inference model and the
governing model embed stochastic components to account for stochastic
variabilities, model errors, and reconstruction uncertainties. Various
experiments on chaotic and stochastic dynamical systems support the relevance
of our scheme w.r.t. state-of-the-art approaches.
