无需计算非局部算子，直接从数据学习非局部偏微分方程演化规律。
Modeling Unknown Nonlocal PDE Systems via Flow Map Learning
- 通过流映射学习直接建模解在时间上的演化算子。
- 仅用短时观测数据即可实现一维和二维分数阶扩散与波动方程的长期稳定预测。
- 适用于缺乏解析模型的复杂非局部动力系统，适合科学计算与工程建模场景。
非局部偏微分方程广泛应用于各类场景，但因包含非局部算子而难以建模与学习。本文提出一种流映射学习（FML）框架，直接从解数据中建模未知的非局部PDE。该方法不尝试学习或近似底层非局部算子，而是学习在模态空间或节点空间中的有限时间演化算子。针对谱表示和网格表示，发展了两种互补形式。在一维和二维分数阶扩散与波动方程上的数值实验表明，仅需短时观测窗口，即可实现精确且稳定的长期预测。该方法为无需显式计算非局部算子的未知非局部动力系统提供了有效的数据驱动框架。
原文摘要 · Abstract (English)
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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