This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM-ROMs approximate the full order model (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 -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, this paper details an algebraic DD reformulation of the FOM, training a NM-ROM with HR for each subdomain, and a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and and 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.

中文翻译：

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快速准确的域分解非线性流形降阶模型

本文将非线性流形降阶模型（NM-ROM）与域分解（DD）相结合。 NM-ROM 通过使用 FOM 快照数据训练浅层稀疏自动编码器来近似非线性流形中的全阶模型 (FOM) 状态。对于缓慢衰减 Kolmogorov 宽度的问题，这些 NM-ROM 比线性子空间 ROM (LS-ROM) 更有优势。然而，需要训练的 NM-ROM 参数的数量与 FOM 的大小成比例。此外，对于“极端规模”的问题，仅存储高维 FOM 快照就可能导致 ROM 训练成本高昂。为了减轻训练成本，本文将DD应用于FOM，计算每个子域上的NM-ROM，并将它们耦合以获得全局NM-ROM。这种方法有几个优点：子域 NM-ROM 可以并行训练，比全局 NM-ROM 需要训练更少的参数，需要更小的子域 FOM 维度训练数据，并且可以针对 FOM 的子域特定特征进行定制。每个子域 NM-ROM 中使用的自动编码器的浅层稀疏架构允许应用超缩减 (HR)，从而降低非线性引起的复杂性并提高 NM-ROM 的计算速度。本文首次将 NM-ROM（带有 HR）应用于 DD 问题。特别是，本文详细介绍了 FOM 的代数 DD 重构、为每个子域训练带有 HR 的 NM-ROM，以及用于评估耦合全局 NM-ROM 的顺序二次规划 (SQP) 求解器。提供了 SQP 方法的理论收敛结果以及带有 HR 的 DD NM-ROM 的误差估计。所提出的采用 HR 方法的 DD NM-ROM 与采用 HR 的 DD LS-ROM 在 2D 稳态 Burgers 方程上进行了数值比较，表明所提出的 DD NM-ROM 的精度比 DD LS-ROM 提高了一个数量级。只读存储器。