Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Xu, Zhongshu, Li, Ying, Zhang, Yanzhi, Xiu, Dongbin
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Awards: G1523888516, G6580055118
OpenAlex ID: W7172425952
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