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 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, each involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and training of subdomain NM-ROMs can tailor them 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 formulation 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 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：本文集成非线性流形降阶模型（NM-ROMs）与区域分解（DD）方法。NM-ROMs通过利用全阶模型（FOM）快照数据训练浅层稀疏自编码器，在非线性流形中逼近FOM状态。对于具有缓慢衰减Kolmogorov $n$-宽度的模型问题，NM-ROMs相较于线性子空间降阶模型（LS-ROMs）更具优势。然而，所需训练的NM-ROM参数规模随FOM规模增长。此外，对于"超大规模"问题，仅存储高维FOM快照就可能导致ROM训练成本高昂。为缓解训练成本，本文将DD应用于FOM，在每个子区域计算NM-ROMs并通过耦合获得全局NM-ROM。该方法具有多项优势：子区域NM-ROMs可并行训练，每个子区域模型所需训练参数少于全局NM-ROM，需要更小的子区域FOM维度训练数据，且子区域NM-ROMs的训练可针对FOM子区域特定特征进行定制。每个子区域NM-ROM中采用的浅层稀疏自编码器结构使得超缩减（HR）技术得以应用，降低了非线性引起的复杂度，从而提升NM-ROM计算速度。本文首次将NM-ROM（含HR）应用于DD问题。具体而言，本文描述了FOM的代数DD公式化方法，为每个子区域训练含HR的NM-ROM，并开发了序列二次规划（SQP）求解器以评估耦合后的全局NM-ROM。文中给出了SQP方法的理论收敛性结果，以及含HR的DD NM-ROM的先验和后验误差估计。通过将含HR的DD NM-ROM与含HR的DD LS-ROM在二维稳态Burgers方程上进行数值对比，证明本文提出的含HR的DD NM-ROM相比DD LS-ROM在精度上提升了一个数量级。