On the convergence of PINNs for inverse source problem in the complex Ginzburg-Landau equation

TL;DR

This paper investigates the unique recovery of source terms in the complex Ginzburg-Landau equation using regional data, establishing stability conditions and proposing PINN-based algorithms validated by numerical experiments.

Contribution

It provides new theoretical conditions for source recovery from partial data and introduces PINN algorithms for practical inverse problem solutions.

Findings

Unique source recovery from full data at one time point.

Stability estimates from local data at two time points.

PINN algorithms demonstrate high accuracy and efficiency.

Abstract

This paper addresses the problem of recovering the spatial profile of the source in the complex Ginzburg-Landau equation from regional observation data at fixed times. We establish two types of sufficient measurements for the unique solvability of the inverse problem. The first is to determine the source term by using whole data at one fixed instant. Conditional stability is established by using the eigenfunction expansion argument. Next, using the analytic continuation method, both uniqueness and a stability estimate for recovering the unknown source can be established from local data at two instants. Finally, algorithms based on the physics-informed neural networks (PINNs) are proposed, and several numerical experiments are presented to show the accuracy and efficiency of the algorithm.