Front Transport Reduction for Complex Moving Fronts

TL;DR

This paper introduces a novel decomposition and hyper-reduction method for efficiently modeling complex moving fronts in advection and reaction-diffusion systems, handling topological changes and providing accurate reduced-order solutions.

Contribution

It proposes a new approach combining level-set parameterization and neural network-inspired activation functions for model order reduction of complex fronts, addressing limitations of existing methods.

Findings

Achieves less than 1% error in representative examples.

Maintains computational complexity comparable to POD-Galerkin methods.

Successfully applied to real-life 2D Bunsen flame simulations.

Abstract

This work addresses model order reduction for complex moving fronts, which are transported by advection or through a reaction-diffusion process. Such systems are especially challenging for model order reduction since the transport cannot be captured by linear reduction methods. Moreover, topological changes, such as splitting or merging of fronts pose difficulties for many nonlinear reduction methods and the small non-vanishing support of the underlying partial differential equations dynamics makes most nonlinear hyper-reduction methods infeasible. We propose a new decomposition method together with a hyper-reduction scheme that addresses these shortcomings. The decomposition uses a level-set function to parameterize the transport and a nonlinear activation function that captures the structure of the front. This approach is similar to autoencoder artificial neural networks, but additionally provides insights into the system, which can be used for efficient reduced order models. We make use of this property and are thus able to solve the advection equation with the same complexity as the POD-Galerkin approach while obtaining errors of less than one percent for representative examples. Furthermore, we outline a special hyper-reduction method for more complicated advection-reaction-diffusion systems. The capability of the approach is illustrated by various numerical examples in one and two spatial dimensions, including real-life applications to a two-dimensional Bunsen flame.