Modeling Unknown Nonlocal PDE Systems via Flow Map Learning

2026-07-31

AI Summary

Nonlocal partial differential equations are challenging to model and predict due to the complexity of their nonlocal operators. This work proposes a flow map learning framework that directly learns the finite-time evolution operator from solution data, bypassing the need for explicit modeling or approximation of the nonlocal operator. The approach is compatible with both spectral and grid-based representations and integrates evolution operator learning with spectral and finite difference methodologies. Demonstrated on one- and two-dimensional fractional diffusion and wave equations, the method achieves accurate and stable long-term dynamical predictions using only short-time observational windows, substantially enhancing the capability to model unknown nonlocal systems.

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.

Problem

• nonlocal PDEs
• flow map learning
• data-driven modeling
• nonlocal operators
• unknown dynamics

Innovation

• flow map learning
• nonlocal PDEs
• data-driven modeling
• fractional diffusion
• evolution operator