  The identification of the governing equations of chaotic dynamical systems
from data has recently emerged as a hot topic. While the seminal work by
Brunton et al. reported proof-of-concepts for idealized observation setting for
fully-observed systems, {\em i.e.} large signal-to-noise ratios and
high-frequency sampling of all system variables, we here address the learning
of data-driven representations of chaotic dynamics for partially-observed
systems, including significant noise patterns and possibly lower and irregular
sampling setting. Instead of considering training losses based on short-term
prediction error like state-of-the-art learning-based schemes, we adopt a
Bayesian formulation and state this issue as a data assimilation problem with
unknown model parameters. To solve for the joint inference of the hidden
dynamics and of model parameters, we combine neural-network representations and
state-of-the-art assimilation schemes. Using iterative Expectation-Maximization
(EM)-like procedures, the key feature of the proposed inference schemes is the
derivation of the posterior of the hidden dynamics. Using a
neural-network-based Ordinary Differential Equation (ODE) representation of
these dynamics, we investigate two strategies: their combination to Ensemble
Kalman Smoothers and Long Short-Term Memory (LSTM)-based variational
approximations of the posterior. Through numerical experiments on the Lorenz-63
system with different noise and time sampling settings, we demonstrate the
ability of the proposed schemes to recover and reproduce the hidden chaotic
dynamics, including their Lyapunov characteristic exponents, when classic
machine learning approaches fail.
