Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

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.

Machine Learning (cs.LG); Dynamical Systems (math.DS); Numerical Analysis (math.NA)

arXiv:2608.00400 [cs.LG]

(or arXiv:2608.00400v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.00400 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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