Online interpolation point refinement for reduced order models using a genetic algorithm

TL;DR

This paper introduces a genetic algorithm-based method to refine interpolation points in reduced order models, improving accuracy and enabling online adaptation for nonlinear PDEs.

Contribution

The paper presents a novel genetic algorithm approach for online refinement of interpolation points in DEIM for reduced order modeling of nonlinear PDEs.

Findings

Nearly optimal interpolation points achieved with few generations

Significant reduction in reconstruction error, nearly an order of magnitude

Demonstrated effectiveness on Ginzburg-Landau and Navier-Stokes equations

Abstract

A genetic algorithm procedure is demonstrated that refines the selection of interpolation points of the discrete empirical interpolation method (DEIM) when used for constructing reduced order models for time dependent and/or parametrized nonlinear partial differential equations (PDEs) with proper orthogonal decomposition. The method achieves nearly optimal interpolation points with only a few generations of the search, making it potentially useful for {\em online} refinement of the sparse sampling used to construct a projection of the nonlinear terms. With the genetic algorithm, points are optimized to jointly minimize reconstruction error and enable dynamic regime classification. The efficiency of the method is demonstrated on two canonical nonlinear PDEs: the cubic-quintic Ginzburg-Landau equation and the Navier-Stokes equation for flow around a cylinder. Using the former model, the procedure can be compared to the ground-truth optimal interpolation points, showing that the genetic algorithm quickly achieves nearly optimal performance and reduced the reconstruction error by nearly an order of magnitude.