The residual neural network is used to carry out machine learning on the steady-state flow field of the hemisphere-on-cylinder laser turret model in the range of Ma=0.3~0.8, and the subsonic/transonic flow field under any incoming flow conditions in this range is established. The prediction accuracy of this model is evaluated for beam wavefront distortion under different view-of-field angles. The learning model reproduces flow characteristics such as boundary layers, flow separation, and separated shear layers in turret flows, including in particular unanchored shock discontinuities in transonic flow. The wavefront distribution based on the predicted flow field under different viewing angles is basically consistent with that calculated based on the flow field of CFD. This machine learning method provides a strategy for adaptive correction of laser turret aero-optical effects in the engineering field.

Physics-Informed Neural Networks (PINN) have provided a new way of numerically solving forward and inverse problems of partial differential equations with promising applications. This paper focuses on the diffusion coefficient inverse problem of the diffusion equation. A systematic study is carried out for the problems of fixed coefficients, anisotropic coefficients, spatial dependence coefficients, spatio-temporal dependence coefficients, and nonlinear diffusion coefficients, and the neural network structure and solution method required for solving each type of problem are proposed. Numerical experiments show that the PINN method can reconstruct the unknown coefficients accurately with less data and is robust under a certain noise level.

In recent years, the use of machine learning methods to solve differential equations has attracted increasing attention from researchers in different fields. However, with the deepening of the research, researchers have begun to identify numerous challenges associated with the use of machine learning methods for solving time development equations. This paper presents a summary of the machine learning methods for solving the evolution equation. First, we present a summary of data-driven methods and deep learning methods based on equation learning. Then we introduce targeted algorithms for solving the problem under different neural network architectures. Finally, this paper presents a summary of the training features and recent work on the use of PINN method for solving the evolution equation. It also provides an outlook for future work.