Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

TL;DR

This paper introduces a ResNet-based closure modeling framework for reduced order models of nonlinear systems, improving accuracy and efficiency by combining physical filtering with deep learning.

Contribution

The novel ResNet-ROM framework integrates deep residual neural networks with physical filtering to enhance nonlinear ROM closure modeling without relying on phenomenological assumptions.

Findings

ResNet-ROM outperforms standard projection ROM in accuracy.

ResNet-ROM is more accurate and efficient than other closure models.

Numerical experiments on 1D Burgers equation validate the approach.

Abstract

Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear PDE and construct a filtered ROM. This filtered ROM is low-dimensional, but is not closed (because of the PDE nonlinearity). (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models.