物理信息神经网络新进展，提升微分方程求解效率与精度
From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning
- 基于柯尔莫哥洛夫定理的PIKAN模型替代传统PINN，增强表达能力
- 通过自适应权重与域分解等技术，显著提升训练稳定性与收敛速度
- 适用于生物医学、流体力学等多领域，适合科研与工程应用者参考
物理信息神经网络（PINNs）自2017年提出以来，已成为科学机器学习的重要工具，能够利用稀疏观测数据高效求解常微分和偏微分方程。近年来，PINN在网络结构、自适应细化、域分解、自适应权重与激活函数等方面取得显著进展。近期重要突破是物理信息柯尔莫哥洛夫网络（PIKANs），该方法基于柯尔莫哥洛夫1957年提出的表示理论，为传统PINN提供了有前景的替代方案。本文综述了PINNs最新进展，涵盖网络设计改进、特征扩展、优化技术、不确定性量化及理论洞察。同时梳理其在生物医学、流体与固体力学、地球物理学、动力系统、传热、化学工程等领域的关键应用，并介绍学术界与产业界开发的计算框架与软件工具，助力PINN研究与落地。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) have emerged as a key tool in Scientific Machine Learning since their introduction in 2017, enabling the efficient solution of ordinary and partial differential equations using sparse measurements. Over the past few years, significant advancements have been made in the training and optimization of PINNs, covering aspects such as network architectures, adaptive refinement, domain decomposition, and the use of adaptive weights and activation functions. A notable recent development is the Physics-Informed Kolmogorov-Arnold Networks (PIKANS), which leverage a representation model originally proposed by Kolmogorov in 1957, offering a promising alternative to traditional PINNs. In this review, we provide a comprehensive overview of the latest advancements in PINNs, focusing on improvements in network design, feature expansion, optimization techniques, uncertainty quantification, and theoretical insights. We also survey key applications across a range of fields, including biomedicine, fluid and solid mechanics, geophysics, dynamical systems, heat transfer, chemical engineering, and beyond. Finally, we review computational frameworks and software tools developed by both academia and industry to support PINN research and applications.
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