[论文解读] Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications

Zhongkai Hao, Songming Liu+5 more|arXiv (Cornell University)|Nov 15, 2022

Model Reduction and Neural Networks被引用 88

一句话总结

本综述回顾了物理信息先验与数据驱动模型的整合，涵盖 PIML 的表示、方法及应用。

ABSTRACT

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.

研究动机与目标

• 阐明将物理定律与数据驱动学习相结合以提高鲁棒性和泛化能力的必要性。
• 提供一个将物理先验表示并融入 ML 模型的正式框架。
• 综述神经仿真方法（神经求解器和神经算子）以及 PIML 中的反问题。
• 识别待解决的挑战与未来方向，以加速 PIML 的跨学科研究。

提出的方法

• 将来自 PDEs/ODEs/SDEs、对称性和直观物理的物理先验定义为归纳偏好。
• 解释物理先验如何嵌入数据、模型结构、损失函数、优化器和推理中。
• 介绍 Physics-Informed Neural Networks (PINNs) 框架及其损失函数形式。
• 讨论神经求解器和神经算子作为神经仿真工具。
• 概述将先验整合到计算机视觉和强化学习任务的策略。

实验结果

研究问题

• RQ1 哪些形式的物理先验（从强到弱）对引导 ML 模型最有效？
• RQ2 如何将物理先验融入 ML 工作流的数据、架构、损失、优化和推理组件？
• RQ3 对于 PINNs、DeepONet 及相关方法的关键进展、局限性及理论保障是什么？
• RQ4 哪些未解决的问题和未来方向将推动科学、工程和科学 AI 领域的 PIML 发展？

主要发现

• 物理信息先验通过将学习约束在物理上合理的解上，可以提高鲁棒性、可解释性和泛化性。
• PIML 通过像 PINNs 这样的神经求解器和神经算子将数据与物理定律结合，支持正问题和反问题。
• 对称性、守恒定律和直观物理提供比 PDEs/ODEs/SDEs 更灵活但较弱的归纳偏好。
• 广泛的应用领域遍布科学领域、计算机视觉和强化学习，凸显跨学科潜力。