Estimating initial conditions for dynamical systems with incomplete information

TL;DR

This paper presents a gradient-based method for inferring initial conditions of dynamical systems from incomplete, noisy, and aggregated observations, validated on Lorenz and Mackey-Glass systems, revealing a critical transition in prediction accuracy.

Contribution

The paper introduces a novel approach for estimating initial states of dynamical systems under challenging observational conditions, demonstrating its effectiveness on complex models.

Findings

Successful inference of initial conditions for Lorenz and Mackey-Glass systems.

Identification of a critical observation threshold for accurate initialization.

Validation through out-of-sample prediction performance.

Abstract

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.