Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method

TL;DR

This work introduces OpInf-Schwarz, a minimally intrusive domain-decomposition approach that couples subdomain-local OpInf ROMs with subdomain-local FOMs using the overlapping Schwarz alternating method for PDEs. It combines POD-based operator inference with Schwarz data exchange to create FOM-ROM and ROM-ROM couplings, demonstrated on a 2D unsteady heat equation, and shows that the method can achieve accurate solutions with substantial online speedups in several configurations. The study also analyzes the influence of subdomain geometry, overlap, and training data on accuracy and convergence, and discusses stability challenges inherent to learned operators and data-driven coupling. The results indicate strong potential for modular, scalable simulations and highlight directions for extending to 3D, nonlinear, and parametric problems, including parallel Additive Schwarz variants to improve performance.

Abstract

This paper presents and evaluates an approach for coupling together subdomain-local reduced order models (ROMs) constructed via non-intrusive operator inference (OpInf) with each other and with subdomain-local full order models (FOMs), following a domain decomposition of the spatial geometry on which a given partial differential equation (PDE) is posed. Joining subdomain-local models is accomplished using the overlapping Schwarz alternating method, a minimally-intrusive multiscale coupling technique that works by transforming a monolithic problem into a sequence of subdomain-local problems, which communicate through transmission boundary conditions imposed on the subdomain interfaces. After formulating the overlapping Schwarz alternating method for OpInf ROMs, termed OpInf-Schwarz, we evaluate the method's accuracy and efficiency on several test cases involving the heat equation in two spatial dimensions. We demonstrate that the method is capable of coupling together arbitrary combinations of OpInf ROMs and FOMs, and that speed-ups over a monolithic FOM are possible when performing OpInf ROM coupling.

Figures (5)

• Figure 2.1: Illustration showing an overlapping domain decomposition of a 2D domain $\Omega$ for the application of the Schwarz alternating method. Note $\Gamma_1, \Gamma_2 \not\subset\partial\Omega$.
• Figure 4.1: Overlapping domain decomposition configurations
• Figure 4.2: OpInf-Schwarz vs. monolithic simulation. OpInf parameters: $r = 6$, Overlap = 10, Data = 30.
• Figure 4.3: OpInf-Schwarz solutions and Monolithic FOM at $t = 1$.
• Figure 4.4: Pareto plot for time-varying BCs, $\mathbf{E}_{\ell^2}^{\text{avg}}$ vs Time (s). Fixed parameters: $r = 6$, Overlap $=10$. Plotted points are "Data" as it ranges over 30, 40, 50, and 60 marked by $\times$, $\blacklozenge$, $\boldsymbol{+}$, and $\bullet$ respectively as in Table \ref{['tab:Time_Varying_Squares']}. Values in bottom left are preferred.