Statistical closure modeling for reduced-order models of stationary systems by the ROMES method

TL;DR

Numerical experiments performed on both linear and nonlinear stationary systems illustrate the ability of the technique to improve ROM prediction accuracy by an order of magnitude, to statistically quantify the error in arbitrary quantities of interest, and to realize a more cost-effective methodology for reducing the error than a ROM-only approach in the case of nonlinear systems.

Abstract

This work proposes a technique for constructing a statistical closure model for reduced-order models (ROMs) applied to stationary systems modeled as parameterized systems of algebraic equations. The proposed technique extends the reduced-order-model error surrogates (ROMES) method to closure modeling. The original ROMES method applied Gaussian-process regression to construct a statistical model that maps cheaply computable error indicators (e.g., residual norm, dual-weighted residuals) to a random variable for either (1) the norm of the state error or (2) the error in a scalar-valued quantity of interest. Rather than target these two types of errors, this work proposes to construct a statistical model for the state error itself; it achieves this by constructing statistical models for the generalized coordinates characterizing both the in-plane error (i.e., the error in the trial subspace) and a low-dimensional approximation of the out-of-plane error. The former can be considered a statistical closure model, as it quantifies the error in the ROM generalized coordinates. Because any quantity of interest can be computed as a functional of the state, the proposed approach enables any quantity-of-interest error to be statistically quantified a posteriori, as the state-error model can be propagated through the associated quantity-of-interest functional. Numerical experiments performed on both linear and nonlinear stationary systems illustrate the ability of the technique (1) to improve (expected) ROM prediction accuracy by an order of magnitude, (2) to statistically quantify the error in arbitrary quantities of interest, and (3) to realize a more cost-effective methodology for reducing the error than a ROM-only approach in the case of nonlinear systems.

Figures (20)

• Figure 1: Graphical depiction of the decomposition of the state error ${\boldsymbol \delta}_\mathbf{x}\in\mathbb{R}^{{N}}$ into the in-plane error $\boldsymbol\delta^{\mathbin{\|}}\in\mathcal{V}$ and the out-of-plane error $\boldsymbol\delta^{\perp}\in\mathcal{V}^\perp$.
• Figure 1: Test case 1. Schematic representation of the computational domain and finite-element solutions for different values of the system parameters $\boldsymbol{\mu}$.
• Figure 2: \newlabelfig:GPMSE0 Test case 1. ROMES models constructed for the first two error generalized coordinates. The solid line represents the GP mean; the dashed lines represent the limits of the 99% prediction interval; the grey diamonds represent data related to online points $\boldsymbol{\mu}\in\mathcal{D}_\mathrm{online}$, while the blue crosses represent training data related to training points $\boldsymbol{\mu}\in\mathcal{D}_\mathrm{ROMES}$. We have employed $n=2$, $n^\perp = 0$, $n_{p}=10$ and we have selected hyperparameters according to Eq. \ref{['eq:hyperparamsOpt']} with $L_{i,j}(\boldsymbol{\theta}) = L_{\text{likelihood},i,j}(\boldsymbol{\theta})$, and a training set with $|\mathcal{D}_\mathrm{ROMES}| = 1000$.
• Figure 3: \newlabelfig:histGPMSE0 Test case 1. Histogram of the standardized data $\{\hat{\delta}_{i}(\boldsymbol{\mu}) - \nu_{i}(\rho_{i}(\boldsymbol{\mu})))/\bar{\sigma}_{i}(\rho_{i}(\boldsymbol{\mu}))\}_{\boldsymbol{\mu}\in\mathcal{D}_\mathrm{online}}$, $i=1,2$ (blue bar plot) as compared to the PDF of the standard Gaussian distribution $\mathcal{N} \left(0,1 \right)$ (red curve). We have employed $n=2$, $n^\perp = 0$, $n_{p}=10$, and have selected hyperparameters according to Eq. \ref{['eq:hyperparamsOpt']} with $L_{i,j}(\boldsymbol{\theta}) = L_{\text{likelihood},i,j}(\boldsymbol{\theta})$. The number of training-parameter instances is $|\mathcal{D}_\mathrm{ROMES}|=1000$.
• Figure 4: \newlabelfig:inplane_plot0 Test case 1. Mean relative ROM error $e_\mathbf{x}$ (red), mean relative ROM error after applying the in-plane ROMES correction $\tilde{e}_\mathbf{x}^{\mathbin{\|}}$ (yellow), and mean relative projection error $e_\mathbf{x}^{\mathbin{\|}}$ (blue) for a varying reduced-subspace dimension $n$ and dual-subspace dimension $n_{p}$. Here, we set $n^\perp = 0$, $L_{i,j}(\boldsymbol{\theta}) = L_{\text{C},i,j}(\boldsymbol{\theta})$, $|\mathcal{D}_\mathrm{ROMES}| = 1000$ and $|\mathcal{D}_\mathrm{online}|=1500$.

Theorems & Definitions (6)

• Remark 3.1: Necessary conditions for zero in-plane error
• Remark 3.2: Out-of-plane basis matrix construction
• Remark 3.3: Unique v. shared dual bases
• Remark 4.1: Offline stage: training cost
• Remark 4.2: Offline stage: specifying quantities of interest not required
• Remark 4.3: Online stage: comparison with a 'ROM-only' approach