This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM-ROMs approximate the FOM state in a nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for problems with slowly decaying Kolmogorov n-width. However, the number of NM-ROM parameters that need to be trained scales with the size of the FOM. Moreover, for "extreme-scale" problems, the storage of high-dimensional FOM snapshots alone can make ROM training expensive. To alleviate the training cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and couples them to obtain a global NM-ROM. This approach has several advantages: Subdomain NM-ROMs can be trained in parallel, involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and can be tailored to subdomain-specific features of the FOM. The shallow, sparse architecture of the autoencoder used in each subdomain NM-ROM allows application of hyper-reduction (HR), reducing the complexity caused by nonlinearity and yielding computational speedup of the NM-ROM. This paper provides the first application of NM-ROM (with HR) to a DD problem. In particular, it details an algebraic DD reformulation of the FOM, trains a NM-ROM with HR for each subdomain, and develops a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and a priori and a posteriori error estimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM with HR on the 2D steady-state Burgers' equation, showing an order of magnitude improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM.

翻译：本文整合了非线性流形降阶模型(NM-ROM)与域分解(DD)方法。NM-ROM通过使用全阶模型(FOM)快照数据训练浅层稀疏自编码器，在非线性流形上逼近FOM状态。对于具有慢衰减Kolmogorov n-宽度的问题，此类NM-ROM相较于线性子空间降阶模型(LS-ROM)具有优势。然而，NM-ROM需要训练的参数数量与FOM规模成正比。此外，对于"超大规模"问题，仅高维FOM快照的存储就可能导致ROM训练成本高昂。为降低训练成本，本文对FOM进行域分解，在各子域上计算NM-ROM，并通过耦合得到全局NM-ROM。该方法具有多重优势：子域NM-ROM可并行训练，需要训练的参数少于全局NM-ROM，所需子域FOM维度的训练数据更少，且可针对FOM的子域特定特征进行定制。各子域NM-ROM中使用的浅层稀疏自编码器结构允许应用超简化(HR)技术，从而降低非线性引起的计算复杂度，实现NM-ROM的计算加速。本文首次将NM-ROM（带HR）应用于域分解问题。具体而言，本文对FOM进行代数域分解重构，为每个子域训练带HR的NM-ROM，并开发了序列二次规划(SQP)求解器来评估耦合后的全局NM-ROM。提供了SQP方法的理论收敛性结果，以及带HR的DD NM-ROM的先验误差估计和后验误差估计。通过二维稳态Burgers方程将所提出的带HR的DD NM-ROM与带HR的DD LS-ROM进行数值比较，结果表明所提出的DD NM-ROM的精度比DD LS-ROM提升了一个数量级。