LEARNING DYNAMICAL CHARACTERISTICS WITH NEURAL OPERATORS FOR DATA ASSIMILATION

Abstract

Data assimilation refers to a group of algorithms that combine numerical models of a system with observations to obtain an optimal estimation of the system's states. In domains like earth science, numerical models are usually formulated by differential equations, also known as prior dynamics. It is a great challenge for neural networks to properly exploit the dynamical characteristics for data assimilation, because first, it is difficult to represent complicated dynamical characteristics in neural networks, and second, the dynamics are likely to be biased. The state-of-the-art neural networks emulate traditional approaches to introduce dynamical characteristics by optimizing an objective function in which the dynamics are inherently quantified, but the iterative optimization process leads to high computational cost. In this paper, we develop a novel deep learning framework with neural operators for data assimilation. The key novelty of our proposed approach is that we design a so-called flow operator to explicitly learn dynamical characteristics for reconstructing sequences of physical states. Numerical experiments on the Lorenz-63 and Lorenz-96 systems, which are the standard benchmarks for data assimilation performance evaluation, show that the proposed method is at least three times faster than state-of-the-art neural networks, and reduces the dynamic loss by two orders of magnitude. It is also demonstrated that our method is well-adapted to biases in the prior dynamics.

1. INTRODUCTION

Data assimilation can be essentially defined as a statistical technique to combine the prior dynamics with a sequence of noisy and irregularly-sampled observations, which plays an important role in utilizing observational data, especially for numerical weather forecasting systems. For instance, the European Centre for Medium-Range Weather Forecasts (ECMWF) employs the variational assimilation algorithms in its operational systems to take full advantage of both in situ and satellite-derived data, as well as state-of-the-art numerical models (Rabier et al., 2000) . Classical Assimilation Methods Classical data assimilation methods range from early empirical analysis (Bergthorsson et al., 1955) and optimal interpolation (Gandin, 1963) to later variationalbased assimilation algorithms (Sasaki, 1970) and the filtering methods based on statistical estimation theory, such as Kalman filter (Welch et al., 1995) , and particle filter (Carpenter et al., 1999) . Although classical methods have been essential tools for improving the capability of major numerical weather prediction centers worldwide (Kalnay, 2003) , they generally do not account for the propagation of observation information along time and therefore are unable to effectively utilize observations at different times (Song et al., 2017) .

Time-dependent Assimilation Method

In recent years, new data assimilation methods have been developed (Evensen, 2003; Lorenc & Rawlins, 2005; Hunt et al., 2007) . Unlike previous methods, they consider the revolution of observation information along time, in other words, the time-dependent dynamical characteristics (Song et al., 2017) . Among them, the four-dimensional variational (4D-Var) assimilation algorithm is a cutting-edge one. Experimental results prove the advantage of 4D-Var assimilation methods in utilizing observational information (Lorenc & Rawlins, 2005) . However, due to the complicated modeling and solving process of the 4D-Var algorithm, it is considered computationally expensive, especially for high resolution cases (Fisher, 1998) .

