# The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models

arXiv: https://arxiv.org/abs/2410.04668 · HTML: https://mlsanity.com/paper/2410.04668/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 82 references that are in the map.
- First posted on arXiv 7 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [D3M: A deep domain decomposition method for partial differential equations](https://mlsanity.com/paper/1909.12236/) (2019)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method](https://mlsanity.com/paper/2311.00224/) (2023)
- [Train Small, Model Big: Scalable Physics Simulators via Reduced Order Modeling and Domain Decomposition](https://mlsanity.com/paper/2401.10245/) (2023)
- [On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows](https://mlsanity.com/paper/2001.10110/) (2020)
- [Domain decomposition for data-driven reduced modeling of large-scale systems](https://mlsanity.com/paper/2311.00883/) (2023)
- [Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method](https://mlsanity.com/paper/2409.01433/) (2024)
- [Fundamentally New Coupled Approach to Contact Mechanics via the Dirichlet-Neumann Schwarz Alternating Method](https://mlsanity.com/paper/2311.05643/) (2023)
- [A Novel Partitioned Approach for Reduced Order Model -- Finite Element Model (ROM-FEM) and ROM-ROM Coupling](https://mlsanity.com/paper/2206.04736/) (2022)
- [An optimisation-based domain-decomposition reduced order model for the incompressible Navier-Stokes equations](https://mlsanity.com/paper/2211.14528/) (2022)
- [Dynamic flux surrogate-based partitioned methods for interface problems](https://mlsanity.com/paper/2402.03560/) (2024)
- [An optimization-based coupling of reduced order models with efficient reduced adjoint basis generation approach](https://mlsanity.com/paper/2408.14450/) (2024)

## Built on by (most cited first, 4 of 4)

- [Defining Foundation Models for Computational Science: A Call for Clarity and Rigor](https://mlsanity.com/paper/2505.22904/) (2025)
- [Latent Space Element Method](https://mlsanity.com/paper/2601.01741/) (2026)
- [Adaptive hybrid coupling with operator inference, the overlapping Schwarz alternating method and reinforcement learning](https://mlsanity.com/paper/2609.17837/) (2026)
- [Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method](https://mlsanity.com/paper/2609.17841/) (2026)

## Abstract (arXiv)

This paper presents and evaluates a framework for the coupling of subdomain-local projection-based reduced order models (PROMs) using the Schwarz alternating method following a domain decomposition (DD) of the spatial domain on which a given problem of interest is posed. In this approach, the solution on the full domain is obtained via an iterative process in which a sequence of subdomain-local problems are solved, with information propagating between subdomains through transmission boundary conditions (BCs). We explore several new directions involving the Schwarz alternating method aimed at maximizing the method's efficiency and flexibility, and demonstrate it on three challenging two-dimensional nonlinear hyperbolic problems: the shallow water equations, Burgers' equation, and the compressible Euler equations. We demonstrate that, for a cell-centered finite volume discretization and a non-overlapping DD, it is possible to obtain a stable and accurate coupled model utilizing Dirichlet-Dirichlet (rather than Robin-Robin or alternating Dirichlet-Neumann) transmission BCs on the subdomain boundaries. We additionally explore the impact of boundary sampling when utilizing the Schwarz alternating method to couple subdomain-local hyper-reduced PROMs. Our numerical results suggest that the proposed methodology has the potential to improve PROM accuracy by enabling the spatial localization of these models via domain decomposition, and achieve up to two orders of magnitude speedup over equivalent coupled full order model solutions and moderate speedups over analogous monolithic solutions.