A learning-based projection method for model order reduction of transport problems

TL;DR

This paper introduces a novel learning-based projection approach for creating nonlinear adaptive reduced order models of transport problems, combining deep learning with traditional projection techniques to improve efficiency and accuracy.

Contribution

It presents a new neural network-driven method for constructing adaptive reduced bases that depend on time and parameters, enhancing model efficiency for transport-dominated problems.

Findings

More efficient than traditional linear projection methods

Potentially reduces generalization error of ROMs

Does not require neural network derivatives online

Abstract

The Kolmogorov $n$-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue, we propose a new learning-based projection method to construct nonlinear adaptive ROMs for transport problems. The construction follows the offline-online decomposition. In the offline stage, we train a neural network to construct adaptive reduced basis dependent on time and model parameters. In the online stage, we project the solution to the learned reduced manifold. Inheriting the merits from both deep learning and the projection method, the proposed method is more efficient than the conventional linear projection-based methods, and may reduce the generalization error of a solely learning-based ROM. Unlike some learning-based projection methods, the proposed method does not need to take derivatives of the neural network in the online stage.