REVIEW OF NEURAL NETWORK-BASED METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS

Neural networks are widely used as a powerful information processing tool in the fields of computer vision, biomedicine, and oil-gas engineering, triggering technological changes. Due to the powerful learning ability, deep learning networks can not only discover physical laws but also solve partial differential equations (PDEs). In recent years, PDE solving based on deep learning has been a new research hotspot. Following the terms of traditional PDE analytical solution, this paper calls the method of solving PDE by neural network as PDE intelligent solution or PDE neural-network solution. This paper briefly introduces the development history of PDE intelligent solution, and then discusses the development of recovering unknown PDEs and solving known PDEs. The main focus of this paper is on a neural network solution method for a known PDE. It is divided into three categories according to the way of constructing loss functions. The first is data-driven method, which mainly learns PDEs from partially known data and can be applied to recovering physical equations, discovering unknown equations, parameter inversion, etc. The second is physical-constraint method, i.e., data-driven supplemented by physical constraints, which is manifested by adding physical laws such as governing equation to the loss function, thus reducing the network's reliance on labeled data and improving the generalization ability and application value. The third is physics-driven method (purely physical constraints), which solves PDEs by physical laws without any labeled data. However, such methods are currently only applied to solve simple PDEs and still need to be improved for complex physics. This paper introduces the research progress of intelligent solution of PDEs from these three aspects, involving various network structures such as fully-connected neural networks, convolutional neural networks, recurrent neural networks, etc. Finally, we summarize the research progress of PDE intelligent solutions, and outline the corresponding application scenarios and future research outlook.