Nonlinear Model Order Reduction on Quadratic Manifolds via Greedy Algorithms with Dimension-Dependent Regularization

TL;DR

This paper introduces a novel double-greedy algorithm for quadratic manifold-based reduced-order models, improving efficiency and stability for parametric PDEs with complex solution manifolds.

Contribution

It proposes a new double-greedy algorithm for regularization parameters in quadratic manifold ROMs, enhancing accuracy and computational efficiency.

Findings

Demonstrates improved accuracy on linear transport and wave equations

Achieves stable ROMs with fewer data points

Outperforms existing methods in efficiency and stability

Abstract

Traditional projection-based reduced-order modeling approximates the full-order model by projecting it onto a linear subspace. With a fast-decaying Kolmogorov $n$-width of the solution manifold, the resulting reduced-order model (ROM) can be an efficient and accurate emulator. However, for parametric partial differential equations with slowly decaying Kolmogorov $n$-width, the dimension of the linear subspace required for a reasonable accuracy becomes very large, which undermines computational efficiency. To address this limitation, quadratic manifold methods have recently been proposed. These data-driven methods first identify a quadratic mapping by minimizing the linear projection error over a large set of snapshots, often with the aid of regularization techniques to solve the associated minimization problem, and then use this mapping to construct ROMs. In this paper, we propose and test a novel enhancement to this quadratic manifold approach by introducing a first-of-its-kind double-greedy algorithm on the regularization parameters coupled with a standard greedy algorithm on the physical parameter. Our approach balances the trade-off between the accuracy of the quadratic mapping and the stability of the resulting nonlinear ROM, leading to a highly efficient and data-sparse algorithm. Numerical experiments conducted on equations such as linear transport, acoustic wave, advection-diffusion, and Burgers' demonstrate the accuracy, efficiency, and stability of the proposed algorithm.