Recent advances of data-driven machine learning have revolutionized fields like computer vision, reinforcement learning, and many scientific and engineering domains. In many real-world and scientific problems, systems that generate data are governed by physical laws. Recent work shows that it provides potential benefits for machine learning models by incorporating the physical prior and collected data, which makes the intersection of machine learning and physics become a prevailing paradigm. By integrating the data and mathematical physics models seamlessly, it can guide the machine learning model towards solutions that are physically plausible, improving accuracy and efficiency even in uncertain and high-dimensional contexts. In this survey, we present this learning paradigm called Physics-Informed Machine Learning (PIML) which is to build a model that leverages empirical data and available physical prior knowledge to improve performance on a set of tasks that involve a physical mechanism. We systematically review the recent development of physics-informed machine learning from three perspectives of machine learning tasks, representation of physical prior, and methods for incorporating physical prior. We also propose several important open research problems based on the current trends in the field. We argue that encoding different forms of physical prior into model architectures, optimizers, inference algorithms, and significant domain-specific applications like inverse engineering design and robotic control is far from being fully explored in the field of physics-informed machine learning. We believe that the interdisciplinary research of physics-informed machine learning will significantly propel research progress, foster the creation of more effective machine learning models, and also offer invaluable assistance in addressing long-standing problems in related disciplines.

翻译：数据驱动机器学习的近期进展已彻底革新计算机视觉、强化学习以及众多科学与工程领域。在许多现实世界和科学问题中，生成数据的系统受物理定律支配。近期研究表明，通过将物理先验知识与收集到的数据相结合，可为机器学习模型带来潜在优势，这使得机器学习与物理学的交叉成为主流范式。通过无缝整合数据与数学物理模型，该方法能引导机器学习模型趋向物理上合理的解，即使在不确定性和高维背景下也能提升精度与效率。本综述中，我们提出这一学习范式——物理启发式机器学习（Physics-Informed Machine Learning, PIML），其核心是构建一个利用经验数据及可用物理先验知识，以提升涉及物理机制的各项任务性能的模型。我们从机器学习任务、物理先验表示以及融合物理先验的方法三个视角，系统梳理了物理启发式机器学习的最新进展。基于该领域当前趋势，我们提出若干重要的开放性研究问题。我们认为，将不同形式的物理先验编码进模型架构、优化器、推理算法以及诸如逆向工程设计、机器人控制等重大领域专用应用中，在物理启发式机器学习领域仍远未得到充分探索。我们相信，物理启发式机器学习的跨学科研究将显著推动科研进展，催生更有效的机器学习模型，并为解决相关学科中的长期难题提供宝贵助力。