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This paper integrates nonlinear-manifold reduced order models (NM-ROMs) withdomain decomposition (DD). NM-ROMs approximate the FOM state in anonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshotdata. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) forproblems with slowly decaying Kolmogorov n-width. However, the number of NM-ROMparameters that need to be trained scales with the size of the FOM. Moreover,for "extreme-scale" problems, the storage of high-dimensional FOM snapshotsalone can make ROM training expensive. To alleviate the training cost, thispaper applies DD to the FOM, computes NM-ROMs on each subdomain, and couplesthem to obtain a global NM-ROM. This approach has several advantages: SubdomainNM-ROMs can be trained in parallel, involve fewer parameters to be trained thanglobal NM-ROMs, require smaller subdomain FOM dimensional training data, andcan be tailored to subdomain-specific features of the FOM. The shallow, sparsearchitecture of the autoencoder used in each subdomain NM-ROM allowsapplication of hyper-reduction (HR), reducing the complexity caused bynonlinearity and yielding computational speedup of the NM-ROM. This paperprovides the first application of NM-ROM (with HR) to a DD problem. Inparticular, it details an algebraic DD reformulation of the FOM, trains aNM-ROM with HR for each subdomain, and develops a sequential quadraticprogramming (SQP) solver to evaluate the coupled global NM-ROM. Theoreticalconvergence results for the SQP method and a priori and a posteriori errorestimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM withHR approach is numerically compared to a DD LS-ROM with HR on the 2Dsteady-state Burgers' equation, showing an order of magnitude improvement inaccuracy of the proposed DD NM-ROM over the DD LS-ROM.

A fast and accurate domain-decomposition nonlinear manifold reduced order model