Artificial intelligence and deep learning methods for solving differential equations: the state of the art and prospects - Jingwei LU; Xiang CHENG; Fei-Yue WANG (智能科学与技术学报)
With the rapid advancement of fundamental theories and computing capacity, deep learning techniques have made impressive achievements in many fields.Differential equations, as an important tool for describing the physical world, have long been a focus of interest for researchers in various fields.Combining the two methods has gained popularity as a study issue in recent years.Since deep learning can efficiently extract features from large amounts of data and differential equations can reflect objective physical laws, the combination of the two can effectively improve the generalization ability of deep learning and enhance the interpretability of deep learning.Firstly, the problem of solving differential equations by deep learning was briefly introduced.Then, two types of deep learning methods for solving differential equations were introduced: data-driven and physical-informed methods.Furthermore, the applications of relevant deep learning-based solving methods were discussed.Meanwhile, DeDAO (differential equations DAO), a foundation model for artificial intelligence for science, was proposed to address existing challenges.Finally, conclusions of deep learning methods for solving differential equations were presented.