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

Estimating initial conditions for dynamical systems with incomplete information