Dynamically learning the parameters of a chaotic system using partial observations

TL;DR

This paper introduces a data assimilation algorithm that learns parameters of a chaotic system, specifically the Lorenz system, from partial and noisy observations, with proven convergence and practical effectiveness.

Contribution

It presents a novel algorithm for dynamically learning chaotic system parameters from partial data, with rigorous convergence proof and demonstrated robustness in various conditions.

Findings

Algorithm successfully recovers Lorenz system parameters from partial observations.

Convergence of the algorithm is rigorously established for the Lorenz system.

Effective even with noisy, sparse, and stochastic data.

Abstract

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this algorithm to the correct parameters when the system in question is the classic three-dimensional Lorenz system. Computationally, we demonstrate the efficacy of this algorithm on the Lorenz system by recovering any proper subset of the three non-dimensional parameters of the system, so long as a corresponding subset of the state is observable. We also provide computational evidence that this algorithm works well beyond the hypotheses required in the rigorous analysis, including in the presence of noisy observations, stochastic forcing, and the case where the observations are discrete and sparse in time.