With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

翻译：随着大规模数据集的日益普及，以及自动微分、高性能神经网络架构等工具与计算能力的提升，序列数据如今常以数据驱动方式处理，即基于观测数据训练动力学模型。尽管神经网络常被视为不可解释的黑箱架构，它们仍能受益于数据中的物理先验及数学知识。本文采用一种基于长期存在的库普曼算子理论的神经网络架构，将动力系统嵌入潜在空间，使其动力学过程可线性描述，从而具备多项优异特性。我们提出的方法能够训练此类模型实现长时间连续重构，即使面对数据呈非均匀采样时间序列的困难场景仍有效。同时，本文展示了自监督学习的潜力：将训练好的动力学模型作为变分数据同化技术的先验，在时间序列插值与预测等应用中取得具有前景的效果。