In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.

中文翻译：

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一个人可以从偏微分方程的解中学到多少知识？

在这项工作中，我们研究了从解数据中学习偏微分方程（PDE）的问题。各种类型的偏微分方程用于说明解数据可以在多大程度上揭示偏微分方程算子，具体取决于基础算子和初始数据。提出了一种基于局部回归和全局一致性的数据驱动和数据自适应方法，用于稳定的偏微分方程识别。提供数值实验来验证我们的分析并证明所提出算法的性能。