Closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.

翻译：封闭问题在多尺度系统模拟中普遍存在，其中某些量值及过程虽影响模拟精度却无法完全描述。近年来，科学机器学习方法被提出以应对封闭问题，该方法结合传统（基于物理的）建模与数据驱动（机器学习）技术，通常通过神经网络丰富微分方程实现。本文综述了不同简化模型形式——根据其对已知物理规律的包含程度区分，以及先验学习与后验学习的差异化目标。探讨了在选择简化模型形式与学习算法时遵循物理定律（如对称性与守恒律）的重要性，同时分析了时空离散化对模型的影响，以及近期向离散化不变模型发展的趋势。此外，本文建立了封闭问题与逆问题、Mori-Zwanzig理论、多保真度方法等若干研究领域的联系。总体而言，科学机器学习方法在解决封闭问题方面已取得重要进展，但仍面临诸多挑战，尤其是学习模型的泛化性与可解释性有待进一步解决。