We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) solved by the method of lines and their representation in chaotic ordinary differential equations, based on neural ordinary differential equations (NODEs). Solving systems with fine temporal and spatial grid scales is an ongoing computational challenge, and closure models are generally difficult to tune. Machine learning approaches have increased the accuracy and efficiency of computational fluid dynamics solvers. In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. We propose a strategy that uses the NODE and partial knowledge to learn the source dynamics at a continuous level. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers. Numerical results with the two-scale Lorenz 96 ODE, the convection-diffusion PDE, and the viscous Burgers' PDE are used to illustrate this approach.

翻译：我们提出了一种新方法，用于在偏微分方程（PDEs）通过直线法求解及其混沌常微分方程表示中学习子网格尺度模型，该方法基于神经常微分方程（NODEs）。精细时间和空间网格尺度下的系统求解是一个持续的计算挑战，且闭合模型通常难以调整。机器学习方法已提高了计算流体动力学求解器的准确性和效率。该方法中，神经网络用于学习粗网格到细网格的映射，可视为子网格尺度参数化。我们提出了一种策略，利用NODE和部分知识在连续层面上学习源动力学。我们的方法继承了NODEs的优势，可用于参数化子网格尺度、近似耦合算子，并提高低阶求解器的效率。双尺度Lorenz 96 ODE、对流扩散PDE和粘性Burgers' PDE的数值结果用于说明该方法。