Domain Decomposition Method

Domain decomposition methods (DDMs) partition complex problems into smaller, more manageable subproblems, improving computational efficiency and enabling parallel processing for solving partial differential equations (PDEs). Current research focuses on integrating DDMs with machine learning, particularly neural networks like Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs), to enhance accuracy and speed, often employing techniques like non-overlapping subdomains and optimized interface conditions. This hybrid approach shows promise for accelerating solutions to large-scale PDEs across diverse scientific and engineering fields, including fluid dynamics, electromagnetics, and imaging. The development of efficient algorithms, such as those incorporating multigrid methods and graph neural networks, is a key area of ongoing investigation.