Nonintrusive projection-based reduced order modeling using stable learned differential operators

Abstract

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a black-box. This article proposes a learn-then-project approach for nonintrusive model reduction. In the first step of this approach, high-dimensional stable sparse learned differential operators (S-LDOs) are determined using the generated data. In the second step, the ordinary differential equations, comprising these S-LDOs, are used with suitable dimensionality reduction and low-dimensional subspace projection methods to provide equations for the evolution of reduced states. This approach allows easy integration into the existing intrusive ROM framework to enable nonintrusive model reduction while allowing the use of Petrov-Galerkin projections. The applicability of the proposed approach is demonstrated for Galerkin and LSPG projection-based ROMs through three numerical experiments: 1-D advection equation, 1-D Burgers equation and 2-D advection equation. The results indicate that the proposed nonintrusive ROM strategy provides accurate and stable dynamics prediction.

Figures (8)

• Figure 1: 1-D scalar advection problem: Relative error in approximating the ROM operator using S-LDOs compared to intrusive Galerkin projection-based ROM.
• Figure 2: 1-D scalar advection problem: Solution prediction with different ROMs for $r = 20$ modes at (a) $t = 20s$ and (b) $t = 200s$. All the tested ROMs have overlapping solutions.
• Figure 3: 1-D scalar advection problem: Variation of relative error in predicted solution in time for different ROMs with number of modes (a) $r = 10$ and (b) $r = 40$. (c) Total relative error in space and time for different mode selections The red-shaded region in (a) and (b) marks when the data is extracted to learn ROMs. The unshaded region is the time of extrapolation for ROMs.
• Figure 4: 1-D Burgers problem: Relative error in approximating the ROM operator using S-LDOs compared to intrusive Galerkin projection-based ROM.
• Figure 5: 1-D Burgers problem: (a) Spatio-temporal solution prediction for the reference FOM. Errors ($u^m (x,t) - u (x,t)$) in spatio-temporal solution prediction for (b) I-G ROM and (c) NI-G-SLDO ROM for $r = 8$ modes.