Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

2026/08/01 by Zhongshu Xu, Ying Li, Yanzhi Zhang +1
Computer Science · Mathematics · #cs.LG #cs.NA #math.DS #math.NA

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arxiv created 2026/08/01 · arxiv updated 2026/08/04

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.

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