Operator Learning

Operator learning models mappings between functions, such as PDE solution maps, dynamical-system evolution, and inverse or control operators. DeepONets, Fourier neural operators, encoder–decoder networks, kernels, and transformers learn these mappings from examples, often aiming to work across different grids, geometries, or input conditions. Current methods increasingly combine multiscale or low-rank representations with physical constraints, uncertainty estimates, and specialized structure for tasks such as fluid prediction and sparse-field reconstruction. A central challenge is making resolution transfer and out-of-distribution predictions reliable: approximation theory and statistical guarantees are advancing, but success depends on the data, regularity, and structure of each problem.