Solving partial differential equations using large-data models: a literature review

Abstract

Mathematics lies at the heart of engineering science and is very important for capturing and modeling of diverse processes. These processes may be naturally-occurring or man-made. One important engineering problem in this regard is the modeling of advanced mathematical problems and their analysis. Partial differential equations (PDEs) are important and useful tools to this end. However, solving complex PDEs for advanced problems requires extensive computational resources and complex techniques. Neural networks provide a way to solve complex PDEs reliably. In this regard, large-data models are new generation of techniques, which have large dependency capturing capabilities. Hence, they can richly model and accurately solve such complex PDEs. Some common large-data models include Convolutional neural networks (CNNs) and their derivatives, transformers, etc. In this literature survey, the mathematical background is introduced. A gentle introduction to the area of solving PDEs using large-data models is given. Various state-of-the-art large-data models for solving PDEs are discussed. Also, the major issues and future scope of the area are identified. Through this literature survey, it is hoped that readers will gain an insight into the area of solving PDEs using large-data models and pursue future research in this interesting area.