  System identification in scenarios where the observed number of variables is
less than the degrees of freedom in the dynamics is an important challenge. In
this work we tackle this problem by using a recognition network to increase the
observed space dimensionality during the reconstruction of the phase space. The
phase space is forced to have approximately Markovian dynamics described by a
Stochastic Differential Equation (SDE), which is also to be discovered. To
enable robust learning from stochastic data we use the Bayesian paradigm and
place priors on the drift and diffusion terms. To handle the complexity of
learning the posteriors, a set of mean field variational approximations to the
true posteriors are introduced, enabling efficient statistical inference.
Finally, a decoder network is used to obtain plausible reconstructions of the
experimental data. The main advantage of this approach is that the resulting
model is interpretable within the paradigm of statistical physics. Our
validation shows that this approach not only recovers a state space that
resembles the original one, but it is also able to synthetize new time series
capturing the main properties of the experimental data.
