Research Progress on Solving Partial Differential Equations Using Physics-Informed Neural Networks

Abstract

Physics-Informed Neural Networks (PINNs) are a new method which, over the past few years, due to the development of the deep learning technology, has demonstrated significant potential in solving Partial Differential Equations (PDEs). Directly incorporating physical laws into the model training process does not only increase the generalization capability of a model but also minimizes its reliance on data as well as enhances the interpretability of a model. Nevertheless, PINNs continue to encounter many issues with PDEs with high dimensionality and nonlinear, chaotic dynamics including error propagation, absence of initial conditions, and big data. Thus, the aim of the review is to summarize and discuss the existing research developments in the field of solving PDEs with the help of PINNs, touching upon various components of the field, such as basic theories, as well as particular applications, and results of the research concerning the development of various optimization algorithms, network optimization, and boundary conditions.