Proper orthogonal decomposition (POD) allows reduced-order modeling of complex dynamical systems at a substantial level, while maintaining a high degree of accuracy in modeling the underlying dynamical systems. Advances in machine learning algorithms enable learning POD-based dynamics from data and making accurate and fast predictions of dynamical systems. In this paper, we leverage the recently proposed heavy-ball neural ODEs (HBNODEs) [Xia et al. NeurIPS, 2021] for learning data-driven reduced-order models (ROMs) in the POD context, in particular, for learning dynamics of time-varying coefficients generated by the POD analysis on training snapshots generated from solving full order models. HBNODE enjoys several practical advantages for learning POD-based ROMs with theoretical guarantees, including 1) HBNODE can learn long-term dependencies effectively from sequential observations and 2) HBNODE is computationally efficient in both training and testing. We compare HBNODE with other popular ROMs on several complex dynamical systems, including the von K\'{a}rm\'{a}n Street flow, the Kurganov-Petrova-Popov equation, and the one-dimensional Euler equations for fluids modeling.

翻译：本征正交分解（POD）能够在高度精确建模底层动力学系统的前提下，实现复杂动力系统的大幅降阶建模。机器学习算法的进步使得从数据中学习基于POD的动力学模型，并对动力系统做出快速准确的预测成为可能。本文利用近期提出的重球神经ODE（HBNODEs）[Xia等人，NeurIPS，2021]，在POD框架下学习数据驱动的降阶模型（ROMs），特别是针对通过求解全阶模型生成的训练快照进行POD分析后产生的时间变化系数动力学。HBNODE在具有理论保证的学习基于POD的ROM方面展现出多项实用优势，包括：1）HBNODE能有效从序列观测中学习长期依赖性；2）HBNODE在训练和测试阶段均具有计算高效性。我们将HBNODE与其他主流ROMs在多个复杂动力学系统上进行了对比，包括冯·卡门涡街流动、Kurganov-Petrova-Popov方程以及用于流体建模的一维欧拉方程。