Data assimilation refers to a set of algorithms designed to compute the
optimal estimate of a system's state by refining the prior prediction (known as
background states) using observed data. Variational assimilation methods rely
on the maximum likelihood approach to formulate a variational cost, with the
optimal state estimate derived by minimizing this cost. Although traditional
variational methods have achieved great success and have been widely used in
many numerical weather prediction centers, they generally assume Gaussian
errors in the background states, which limits the accuracy of these algorithms
due to the inherent inaccuracies of this assumption. In this paper, we
introduce VAE-Var, a novel variational algorithm that leverages a variational
autoencoder (VAE) to model a non-Gaussian estimate of the background error
distribution. We theoretically derive the variational cost under the VAE
estimation and present the general formulation of VAE-Var; we implement VAE-Var
on low-dimensional chaotic systems and demonstrate through experimental results
that VAE-Var consistently outperforms traditional variational assimilation
methods in terms of accuracy across various observational settings.

VAE-Var 是一种新的变分算法，利用变分自动编码器（VAE）对背景误差分布进行建模，理论推导了 VAE 估计下的变分成本，并提出了 VAE-Var 的一般公式；我们在低维混沌系统上实现了 VAE-Var，并通过实验结果证明了在不同观测设置下 VAE-Var 始终优于传统的变分同化方法。