Surrogate modeling for convection-dominated parametric problems based on error learning

Abstract

Convection-dominated problems are known for their slow Kolmogorov $n$-width decays and are challenging for model order reduction (MOR). In this work, we propose a hybrid surrogate modeling approach and a non-intrusive variant that overcome some drawbacks of linear MOR methods. The proposed hybrid surrogate model is a projection-based reduced-order model (ROM), corrected by the error learned from a deep neural network. With the aid of deep learning, the model component of the surrogate model can be kept in a small reduced dimension. The neural network component and the model component are sequentially but separately built during the offline stage. At the online stage, they are easily coupled to output the solution predictions. Due to the intrusive nature of the hybrid-ROM, the numerically discretized operators of the original model must be available. For problems solved using black-box solvers, where the details of the numerical discretization are not accessible, we further propose a non-intrusive variant of the hybrid surrogate. Compared to the existing MOR methods with nonlinear manifolds, the proposed hybrid ROM is more easily built and is also easily assembled for online prediction. In contrast to the surrogate modeling approaches purely based on deep-learning, the proposed non-intrusive variant has a lighter neural network structure with much fewer parameters to be learned. We test the proposed methods on two nonlinear convection parametric problems. The first is the 1D inviscid Burgers' equation with one parameter, and the second is the 2D inviscid Burgers' equation with two parameters. Since both methods are based on error correction, their online predictions exhibit higher accuracy yet with largely reduced prediction time, compared to state-of-the-art methods.