Neural networks for solving partial differential equations, a comprehensive review of recent methods and applications

Neural networks have emerged as powerful tools for constructing numerical solution methods for partial differential equations (PDEs). This review article provides an accessible introduction to recent developments in the field of scientific machine learning, focusing on methods such as Physics-Informed Neural Networks (PINNs), Deep Galerkin Methods (DGM), Deep Ritz Methods, and Neural Operator Methods. We compare these approaches, highlighting their strengths, limitations, and potential areas for improvement. Furthermore, we explore a variety of real-world applications where these neural networkbased PDE solvers have been successfully implemented. Finally, we discuss future directions and the ongoing challenges in this rapidly evolving research area.