Abstract
Abstract. Recent progress in machine learning has shown how to forecast and, to some extent, learn the dynamics of a model from its output, resorting in particular to neural networks and deep learning techniques. We will show how the same goal can be directly achieved using data assimilation techniques without leveraging on machine learning software libraries, with a view to high-dimensional models. The dynamics of a model are learned from its observation and an ordinary differential equation (ODE) representation of this model is inferred using a recursive nonlinear regression. Because the method is embedded in a Bayesian data assimilation framework, it can learn from partial and noisy observations of a state trajectory of the physical model. Moreover, a space-wise local representation of the ODE system is introduced and is key to coping with high-dimensional models. It has recently been suggested that neural network architectures could be interpreted as dynamical systems. Reciprocally, we show that our ODE representations are reminiscent of deep learning architectures. Furthermore, numerical analysis considerations of stability shed light on the assets and limitations of the method. The method is illustrated on several chaotic discrete and continuous models of various dimensions, with or without noisy observations, with the goal of identifying or improving the model dynamics, building a surrogate or reduced model, or producing forecasts solely from observations of the physical model.
Highlights
1.1 Data assimilation and model errorData assimilation aims at estimating the state of a physical system from its observation and a numerical dynamical model for it
Examples of such an approach applied to the forecasting of low-order chaotic geophysical models are Park and Zhu (1994), who used a bilinear recurrent neural network and applied it to the three-variable Lorenz model (Lorenz, 1963, hereafter L63), Pathak et al (2017, 2018), who use reservoir network techniques on the L63 model and on the Kuramoto– Sivashinski model (Kuramoto and Tsuzuki, 1976; Sivashinsky, 1977, hereafter KS), and Dueben and Bauer (2018), who use a neural network on a low-order Lorenz three-scale model and on coarse two-dimensional geopotential height maps at 500 hPa
We introduce a surrogate model defined by a set of ordinary differential equations (ODEs): dx
Summary
Data assimilation aims at estimating the state of a physical system from its observation and a numerical dynamical model for it. It has been successfully applied to numerical weather and ocean prediction, where it often consisted in estimating the initial conditions for the state trajectory of chaotic geofluids (Kalnay, 2002; Asch et al, 2016; Carrassi et al, 2018) This objective is impeded by the deficiencies of the numerical model Other solutions are based on a weakly parametrised form of model error, for instance when it is assumed to be additive noise Such techniques have been developed for variational data assimilation (e.g. Trémolet, 2006; Carrassi and Vannitsem, 2010), for ensemble Kalman filters and Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union. This account is very far from exhaustive as this is a vast, multiform, and very active subject of research These approaches essentially seek to correct, calibrate, or improve an existing model using observations. They all primarily make use of data assimilation techniques
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