{"slug": "modeling-unknown-nonlocal-pde-systems-via-flow-map-learning", "title": "Modeling Unknown Nonlocal PDE Systems via Flow Map Learning", "summary": "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.", "body_md": "arXiv:2608.00400v1 Announce Type: new\nAbstract: 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.", "url": "https://wpnews.pro/news/modeling-unknown-nonlocal-pde-systems-via-flow-map-learning", "canonical_source": "https://arxiv.org/abs/2608.00400", "published_at": "2026-08-04 04:00:00+00:00", "updated_at": "2026-08-04 04:34:47.353695+00:00", "lang": "en", "topics": ["machine-learning"], "entities": [], "alternates": {"html": "https://wpnews.pro/news/modeling-unknown-nonlocal-pde-systems-via-flow-map-learning", "markdown": "https://wpnews.pro/news/modeling-unknown-nonlocal-pde-systems-via-flow-map-learning.md", "text": "https://wpnews.pro/news/modeling-unknown-nonlocal-pde-systems-via-flow-map-learning.txt", "jsonld": "https://wpnews.pro/news/modeling-unknown-nonlocal-pde-systems-via-flow-map-learning.jsonld"}}