How much can one learn a partial differential equation from its solution?

TL;DR

This paper investigates the extent to which solution data can reveal the underlying PDE operator, proposing a data-driven approach for stable PDE identification and validating it through numerical experiments.

Contribution

It introduces a novel data-driven, adaptive method combining local regression and global consistency for PDE learning from solution data.

Findings

Solution data can reveal PDE operators depending on the operator and initial data

The proposed method achieves stable PDE identification

Numerical experiments confirm the effectiveness of the approach

Abstract

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.