We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning (DL). The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple residual convolutional neural network, while assuming the dynamics to be known. Experiments are performed with the Lorenz 96 dynamics, which display spatiotemporal chaos and for which solid benchmarks for DA performance exist. The accuracy of the states obtained from the learned analysis approaches that of the best possibly tuned ensemble Kalman filter (EnKF), and is far better than that of variational DA alternatives. Critically, this can be achieved while propagating even just a single state in the forecast step. We investigate the reason for achieving ensemble filtering accuracy without an ensemble. We diagnose that the analysis scheme actually identifies key dynamical perturbations, mildly aligned with the unstable subspace, from the forecast state alone, without any ensemble-based covariances representation. This reveals that the analysis scheme has learned some multiplicative ergodic theorem associated to the DA process seen as a non-autonomous random dynamical system.

翻译：我们研究了利用深度学习（DL）发现适用于混沌动力学的数据同化（DA）方案的能力。重点在于，在假设动力学已知的前提下，使用简单的残差卷积神经网络，从状态轨迹及其观测中学习序列DA的分析步骤。实验采用Lorenz 96动力学进行，该模型展示了时空混沌特性，并且存在坚实的DA性能基准。通过习得的分析步骤所获得的状态精度，接近最佳可能调优的集合卡尔曼滤波器（EnKF），并且远优于变分DA替代方案。关键的是，即使在预报步骤中仅传播单个状态，也能实现这一目标。我们探究了在没有集合的情况下实现集合滤波精度的原因。我们诊断发现，分析方案实际上仅从预报状态本身就能识别出关键的动力学扰动，这些扰动与不稳定子空间轻微对齐，而无需任何基于集合的协方差表示。这表明分析方案已经习得了与DA过程相关的某种乘法遍历定理，该过程被视为一个非自治随机动力系统。