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、计算能力以及自动微分和可表达神经网络架构等工具的日益普及，序列数据现在常以数据驱动方式处理，从观测数据中训练动力学模型。尽管神经网络常被视为不可解释的黑箱架构，但它们仍能从数据的物理先验和数学知识中获益。本文利用一种基于Koopman算子理论的神经网络架构，将动力系统嵌入其动力学可线性描述的潜在空间，从而赋予模型诸多优良特性。我们提出了相关方法，使此类模型能够实现长期连续重构，即便在数据以不规则采样时间序列呈现的困难情境下仍能有效训练。本文还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，在时间序列插值和预测等应用中展现出令人期待的效果。