Estimating initial conditions for dynamical systems with incomplete information

In this paper, we study the problem of inferring the latent initial conditions of a dynamical system under incomplete information, i.e., we assume we observe aggregate statistics of the system rather than its state variables directly. Studying several model systems, we infer the 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.