Adaptive Finite Element Method

Adaptive Finite Element Methods (AFEMs) aim to efficiently solve partial differential equations (PDEs) by dynamically refining computational meshes based on local error estimates, focusing computational resources where needed most. Current research emphasizes integrating AFEM principles with machine learning, particularly neural networks (e.g., convolutional neural networks and Physics-Informed Neural Networks), to improve efficiency and accuracy in solving high-dimensional or complex PDEs. This approach leverages the adaptive mesh refinement capabilities of AFEM to reduce the data complexity for training neural networks and improve the accuracy of solutions, impacting fields requiring efficient PDE solutions such as uncertainty quantification and industrial simulations.