Model discovery on the fly using continuous data assimilation

TL;DR

This work considers implementing other derivative based optimization algorithms and shows 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.

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.

Figures (8)

• Figure 1: Time independence of long time error for the Lorenz '63 system while varying across an approximate parameter $c_1$, where $\epsilon = \frac{\Delta c_1}{c_1}$. We see that the nudging converges up to some long term error proportional to the parameter error.
• Figure 2: Error between the true state and the assimilated state for 50 randomly chosen initial conditions for the assimilated state when the parameter error is $\epsilon = 0.5$ for the Lorenz '63 model. Note that despite the differences in the initial conditions, after a sufficiently long time (approximately $1.0$ here) the error collapses to a value dictated by the parameter/model error.
• Figure 3: Comparison of DS and OTF methods for the Lorenz '63 model. Gradient descent is implemented with a learning rate of $r=30$, and the smoothing parameter in Levenberg-Marquardt is $\lambda = 10^{-6}$ for both the OTF and DS methods. The default value of $\mu = 100$ is also used for both, with parameter updates occurring at every $\Delta t= 0.5$ time units.
• Figure 4: DS vs. OTF Levenberg-Marquardt method for various values of $\mu$. All other parameters are the same as those in Figure \ref{['fig:OTFvsDS_lorenz']}. Note the similar behavior for larger values of $\mu$ in agreement with perturbative results.
• Figure 5: Comparison between the DS and OTF methods for the Lorenz '96 model. Note that in this case the asymptotic approximation utilized in the OTF model appears to outperforming the DS approach.