Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being advanced by machine learning, including: 1) the discovery of new governing PDEs and coarse-grained approximations for complex natural and engineered systems, 2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and 3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.

翻译：偏微分方程（PDEs）是对自然物理定律最普遍且最简洁的描述之一，能通过紧凑的符号表征捕捉丰富的现象学与多尺度物理特性。本综述将探讨机器学习推动的若干富有前景的PDE研究方向，包括：1）发现新的控制方程以及用于复杂自然与工程系统的粗粒化近似；2）学习有效坐标系与降阶模型以增强PDE的可分析性；3）表征解算子并改进传统数值算法。在每一研究领域，我们总结了关键进展、当前挑战以及未来发展的机遇。