Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

Researchers introduced a flow-map learning (FML) framework to model unknown nonlocal partial differential equations (PDEs) directly from solution data, bypassing explicit evaluation of nonlocal operators. The method learns finite-time evolution operators in modal or nodal space and demonstrated accurate, stable long-time predictions on one- and two-dimensional fractional diffusion and wave equations using only short observation windows.

arXiv:2608.00400v1 Announce Type: new Abstract: Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators. We present a flow-map learning FML framework for modeling unknown nonlocal PDEs directly from solution data. Rather than learning or approximating the underlying nonlocal operators, the proposed approach learns the finite-time evolution operator in either modal or nodal space. Two complementary formulations are developed for spectral and grid-based solution representations. Numerical experiments on one- and two-dimensional fractional diffusion and wave equations demonstrate accurate and stable long-time prediction using only short observation windows. The proposed approach provides an effective data-driven framework for learning unknown nonlocal dynamics without explicit evaluation of nonlocal operators.