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

TL;DR

This paper provides the first application of NM-ROM (with HR) to a DD problem, 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.

Abstract

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 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, 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 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.

Figures (12)

• Figure 1: Left plot: Each node in the domain corresponds to an unknown $\boldsymbol{x}$ and an equation in the system (\ref{['eq:fom_residual']}). The domain is subdivided into $n_\Omega =2$ subdomains. Residuals corresponding to nodes marked by filled circles near the bboundary in subdomain 1 depend on nodes marked by filled circles in subdomain 2, and residuals corresponding to nodes marked by filled circles near the boundary in subdomain 2 depend on nodes marked by filled circles in subdomain 1. Right plot: Variables that enter computations of residuals in one or more subdomains are duplicated as interface state variables $\boldsymbol{x}_1^\Gamma$ and $\boldsymbol{x}_2^\Gamma$. Variables that only enter the computations of the residuals in one subdomains are the interior state variables $\boldsymbol{x}_1^\Omega$ and $\boldsymbol{x}_2^\Omega$, respectively. Equality $\boldsymbol{x}_1^\Gamma = \boldsymbol{x}_2^\Gamma$ of interface state variables will be enforced via constraints.
• Figure 2: Left: Residual decomposition using $4$ subdomains. Notice that the residuals do not overlap. Right: State decomposition. The regions without overlap correspond to interior states $\boldsymbol{x}_i^\Omega$ while regions with overlap correspond to interface states $\boldsymbol{x}_i^\Gamma$. The overlapping regions enclosed by black dashed lines represent the ports $P(j)\subset\left\{ 1, \dots, n_\Omega \right\}$.
• Figure 3: Left: Dense autoencoder. The HR nodes are represented by solid blue neurons, and the nodes required to compute the HR nodes are outlined in blue. Notice that each node in the decoder hidden layer are required to compute the HR nodes in the output layer. Right: Sparse autoencoder. The encoder input layer and decoder output layer are sparsely connected, and only the blue-outlined hidden nodes are required to compute the HR nodes. The sparse output layer allows one to only keep track of the blue connections to evaluate $\boldsymbol{g}_i^\Omega$, $\boldsymbol{g}_i^\Gamma$ and their Jacobians, resulting in computational speedup.
• Figure 4: Top left: $u$-component with $(a, \lambda)=(10^4, 5)$; Top right: $v$-component with $(a, \lambda)=(10^4, 5)$; Bottom left: $u$-component with $(a, \lambda)=(1, 25)$; Bottom right: $v$-component with $(a, \lambda)=(1, 25)$
• Figure 5: Three-banded sparsity mask for decoder

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• proof
• Theorem 3
• proof
• Lemma 1
• Lemma 2
• Theorem 4
• Remark 1
• Remark 2