Nonlinear-manifold reduced order models with domain decomposition

Abstract

A nonlinear-manifold reduced order model (NM-ROM) is a great way of incorporating underlying physics principles into a neural network-based data-driven approach. We combine NM-ROMs with domain decomposition (DD) for efficient computation. NM-ROMs offer benefits over linear-subspace ROMs (LS-ROMs) but can be costly to train due to parameter scaling with the full-order model (FOM) size. To address this, we employ DD on the FOM, compute subdomain NM-ROMs, and then merge them into a global NM-ROM. This approach has multiple advantages: parallel training of subdomain NM-ROMs, fewer parameters than global NM-ROMs, and adaptability to subdomain-specific FOM features. Each subdomain NM-ROM uses a shallow, sparse autoencoder, enabling hyper-reduction (HR) for improved computational speed. In this paper, we detail an algebraic DD formulation for the FOM, train HR-equipped NM-ROMs for subdomains, and numerically compare them to DD LS-ROMs with HR. Results show a significant accuracy boost, on the order of magnitude, for the proposed DD NM-ROMs over DD LS-ROMs in solving the 2D steady-state Burgers' equation.