Leveraging time and parameters for nonlinear model reduction methods

TL;DR

This paper explores how extending the system and data in nonlinear model reduction enables replacing nonlinear encoders with linear ones, simplifying training while maintaining accuracy for challenging problems.

Contribution

It introduces a method to replace nonlinear encoders with linear ones in autoencoders for model reduction, reducing hyperparameter tuning complexity.

Findings

Linear encoders achieve comparable accuracy to nonlinear ones.

Training complexity is roughly halved.

Applicable to problems with slow Kolmogorov decay.

Abstract

In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov $n$-widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.