Nonlinear projection-based model order reduction with machine learning regression for closure error modeling in the latent space

TL;DR

This work advances nonlinear projection-based model order reduction by replacing opaque latent-space closure modeled by deep neural networks with two interpretable regression approaches: Gaussian process regression and radial basis function interpolation. Built on the LSPG-PROM and ECSW hyperreduction framework, the methods learn latent-space closures to obtain compact, highly efficient surrogates that are accurate for both shock-dominated and turbulent flows, while reducing the data requirements for training. Across a 2D Burgers problem and a 3D Ahmed body wake CFD case, GPR and RBF closures deliver substantial online speedups (up to tens to thousands of times) and competitive accuracy, with the added benefit of analytic Jacobians and potential for error indicators. The results indicate that latent-space, interpretable closures can extend the reach of nonlinear PMOR, mitigating the Kolmogorov barrier and broadening applicability to high-fidelity engineering simulations.

Abstract

A significant advancement in nonlinear projection-based model order reduction (PMOR) is presented through a highly effective methodology. This methodology employs Gaussian process regression (GPR) and radial basis function (RBF) interpolation for closure error modeling in the latent space, offering notable gains in efficiency and expanding the scope of PMOR. Moving beyond the limitations of deep artificial neural networks (ANNs), previously used for this task, this approach provides crucial advantages in terms of interpretability and a reduced demand for extensive training data. The capabilities of GPR and RBFs are showcased in two demanding applications: a two-dimensional parametric inviscid Burgers problem, featuring propagating shocks across the entire computational domain, and a complex three-dimensional turbulent flow simulation around an Ahmed body. The results demonstrate that this innovative approach preserves accuracy and achieves substantial improvements in efficiency and interpretability when contrasted with traditional PMOR and ANN-based closure modeling.

Figures (8)

• Figure 1: Solution slices predicted for out-of-sample parameter point $\boldsymbol{\mu} = (4.75,0.020)$ at the time steps $t = 0$, 5, 10, 15, 20, and 25 using HDM, HPROM-ANN configured with a single-layer ANN, and HPROM-ANN configured with a multi-layer ANN: $y = 50.2$ (top); $x = 50.2$ (bottom).
• Figure 2: Solution slices predicted for $\boldsymbol{\mu}_3 = (4.75,0.020)$ at the time steps $t = 0$, 5, 10, 15, 20, and 25 using HDM, HPROM, and HPROM-RBF: $y = 50.2$ (top); $x = 50.2$ (bottom).
• Figure 3: Solution slices predicted for $\boldsymbol{\mu}_3 = (4.56,0.019)$ at the time steps $t = 0$, 5, 10, 15, 20, and 25 using HDM, HPROM-ANN, and HPROM-GPR: $y = 50.2$ (top); $x = 50.2$ (bottom).
• Figure 4: Solution slices predicted for $\boldsymbol{\mu}_3 = (5.19,0.026)$ at the time steps $t = 0$, 5, 10, 15, 20, and 25 using HDM, HPROM-ANN, HPROM-GPR, and HPROM-RBF: $y = 50.2$ (top); $x = 50.2$ (bottom).
• Figure 5: Ahmed body turbulent wake flow problem: computational fluid domain.