发表机构
Department of Mathematics, The Ohio State University; Department of Mathematics and Statistics, Missouri University of Science and Technology（俄亥俄州立大学数学系; 密苏里科技大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出流映射学习（FML）框架，无需显式处理非局部算子，可从解数据建模未知非局部偏微分方程，经数值实验验证能实现准确稳定的长时间预测，为非局部动力学学习提供有效数据驱动方法。
AI 中文摘要
非局部偏微分方程在诸多应用中出现，但因非局部算子的存在，建模与学习常面临困难。本文提出流映射学习（FML）框架，可直接从解数据中建模未知非局部偏微分方程。该方法不学习或近似底层非局部算子，而是学习模态空间或节点空间中的有限时间演化算子，针对谱型与网格型解表示开发了两种互补形式。对一维、二维分数阶扩散方程和波动方程的数值实验表明，仅利用短观测窗口即可实现准确且稳定的长时间预测，为学习未知非局部动力学提供了无需显式计算非局部算子的有效数据驱动框架。
英文摘要
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.
