  In this work we study the problem about learning a partial differential
equation (PDE) from its solution data. PDEs of various types are used as
examples 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.
