Flow Map Learning Models Unknown Nonlocal PDEs from Data

Nonlocal partial differential equations (PDEs) are prevalent in various scientific and engineering applications, yet their inherent nonlocal operators make them notoriously difficult to model and learn. Traditional methods often struggle with the complexity of these operators. This research presents a novel approach called Flow Map Learning (FML) designed to overcome these challenges by directly learning from solution data. The FML framework focuses on learning the finite-time evolution operator, rather than attempting to approximate or learn the underlying nonlocal operators themselves. It offers two complementary formulations, catering to both spectral and grid-based solution representations. Through numerical experiments on one- and two-dimensional fractional diffusion and wave equations, the method has demonstrated its capability for accurate and stable long-time predictions, even when trained with only short observation windows. This data-driven framework provides an effective means to model unknown nonlocal dynamics without the need for explicit evaluation of complex nonlocal operators.