How Much Can One Learn a PDE From Its Solution Data?

In this work, we study a few basic questions for partial differential equations (PDE) learning from observed solution data. Using various types of PDEs, we show how the approximate dimension (richness) of the data space spanned by all snapshots along a solution trajectory depends on the differential operator and initial data, and identifiability of a differential operator from solution data on local patches. Then we propose a consistent and sparse local regression method for general PDE identification. Our method requires minimal amount of local measurements in space and time from a single solution trajectory by enforcing global consistency and sparsity.