How much can one learn from a single solution of a PDE?

TL;DR

This paper investigates whether a single solution snapshot of a linear evolution PDE can be used to reconstruct any solution, revealing dependence on eigenvalue growth and proposing a data-driven model reduction method.

Contribution

It demonstrates the conditions under which a single PDE solution snapshot can represent all solutions and introduces a new data-driven approach for PDE approximation without explicit PDE knowledge.

Findings

Reconstruction feasibility depends on eigenvalue growth rate.

Proposed a simple data-driven model reduction method.

Numerical experiments validate theoretical analysis.

Abstract

Linear evolution PDE $\partial_t u(x,t) = -\mathcal{L} u$, where $\mathcal{L}$ is a strongly elliptic operator independent of time, is studied as an example to show if one can superpose snapshots of a single (or a finite number of) solution(s) to construct an arbitrary solution. Our study shows that it depends on the growth rate of the eigenvalues, $\mu_n$, of $\mathcal{L}$ in terms of $n$. When the statement is true, a simple data-driven approach for model reduction and approximation of an arbitrary solution of a PDE without knowing the underlying PDE is designed. Numerical experiments are presented to corroborate our analysis.