  In this paper we study the problem of inferring the initial conditions of a
dynamical system under incomplete information. Studying several model systems,
we infer the latent microstates that best reproduce an observed time series
when the observations are sparse,noisy and aggregated under a (possibly)
nonlinear observation operator. This is done by minimizing the least-squares
distance between the observed time series and a model-simulated time series
using gradient-based methods. We validate this method for the Lorenz and
Mackey-Glass systems by making out-of-sample predictions. Finally, we analyze
the predicting power of our method as a function of the number of observations
available. We find a critical transition for the Mackey-Glass system, beyond
which it can be initialized with arbitrary precision.
