Model discovery on the fly using continuous data assimilation

TL;DR

This paper reviews and extends a parameter estimation algorithm within the Continuous Data Assimilation framework, framing it as a root-finding problem and comparing different optimization methods across several dynamical systems.

Contribution

It provides an alternative derivation of the existing algorithm, framing it as a finite dimensional root-finding problem, and explores the effectiveness of Levenberg-Marquardt in multi-parameter estimation.

Findings

Levenberg-Marquardt performs similarly to the original algorithm in single-parameter cases.

The new derivation offers a different perspective on the algorithm's foundation.

The methods are tested successfully on Lorenz and Kuramoto-Sivashinsky models.

Abstract

We review an algorithm developed for parameter estimation within the Continuous Data Assimilation (CDA) approach. We present an alternative derivation for the algorithm presented in a paper by Carlson, Hudson, and Larios (CHL, 2021). This derivation relies on the same assumptions as the previous derivation but frames the problem as a finite dimensional root-finding problem. Within the approach we develop, the algorithm developed in (CHL, 2021) is simply a realization of Newton's method. We then consider implementing other derivative based optimization algorithms; we show that the Levenberg Maqrquardt algorithm has similar performance to the CHL algorithm in the single parameter estimation case and generalizes much better to fitting multiple parameters. We then implement these methods in three example systems: the Lorenz '63 model, the two-layer Lorenz '96 model, and the Kuramoto-Sivashinsky equation.