Adaptive Physics-informed Neural Networks: A Survey

Advances in machine learning have led to important applications in various fields, such as computer vision (enabling technologies like self-driving cars), natural language processing (powering intelligent agents and chatbots), and image generation (facilitating media creation). Motivated by this success, there has been growing interest in developing Machine Learning (ML) solutions to solve problems in science and engineering. Unlike other fields where data is abundant or easily obtained, however, science and engineering often face data scarcity due to the high costs associated with generating data through expensive experiments or simulations. Therefore, to facilitate the development of ML approaches in these disciplines, AI methods that are data-efficient and computationally efficient need to be created. To this end, other domains have tackled similar problems with techniques such as transfer learning, meta-learning, and few-shot learning, indicating significant potential for applying these techniques in the context of science and engineering. One specific application in science and engineering where these efficient ML models can be particularly beneficial is to determine the approximate solutions of PDEs. PDEs are fundamental for modeling and describing natural phenomena in various scientific and engineering domains. Traditionally, these equations are solved numerically, which can become prohibitively expensive, especially when dealing with nonlinear and high-dimensional problems [Han et al., 2018]. This challenge limits their application in areas where a fast evaluation of a PDE is required.