Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

• URL: http://arxiv.org/abs/2608.00400v1
• Date: Sat, 01 Aug 2026 02:46:52 GMT
• Title: Modeling Unknown Nonlocal PDE Systems via Flow Map Learning
• Abstract summary: Nonlocal partial differential equations arise in many applications but are often difficult to model and learn because of the presence of nonlocal operators.<n>We present a flow-map learning framework for modeling unknown nonlocal PDEs directly from solution data.
• Score: 3.5831215265763987
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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