Multiscale Elliptic

Multiscale elliptic problems, characterized by solutions exhibiting variations across vastly different length scales, are a significant challenge in scientific computing. Current research focuses on developing data-driven methods, such as neural operators (including Fourier and convolutional variants) and radial basis function neural networks, to efficiently approximate solutions without explicitly resolving all scales. These approaches leverage machine learning to learn the mapping between input parameters and solutions, offering potential for faster and more scalable solutions compared to traditional numerical methods. The ability to efficiently solve multiscale elliptic problems has broad implications for diverse fields, including material science, fluid dynamics, and biomedical engineering.