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.

翻译：摘要：非线性流形降阶模型（NM-ROM）是一种将物理基本原理融入基于神经网络的数据驱动方法的有效途径。我们结合NM-ROM与区域分解（DD）实现高效计算。相较于线性子空间降阶模型（LS-ROM），NM-ROM具有优势，但因其参数规模随全阶模型（FOM）尺寸扩展，训练成本较高。为此，我们对FOM进行区域分解，分别计算各子区域的NM-ROM，再将其合并为全局NM-ROM。该方法具有多重优势：子区域NM-ROM可并行训练、参数量少于全局NM-ROM，且能适应子区域FOM的特定特征。每个子区域NM-ROM采用浅层稀疏自编码器，支持超缩减（HR）以提高计算速度。本文详述了针对FOM的代数区域分解形式，训练了配备HR的子区域NM-ROM，并将其与采用HR的区域分解LS-ROM进行数值对比。结果表明，在求解二维稳态Burgers方程时，本文提出的区域分解NM-ROM相比区域分解LS-ROM实现了数量级的精度提升。