Advancing domain decomposition methods and entity resolution with graph neural networks

Domain decomposition methods (DDMs) are popular solvers for discretized systems of PDEs, with one-level and multilevel variants. Yet the optimal construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. These solvers rely on several algorithmic and mathematical parameters, prescribing overlap, subdomain boundary conditions, and other properties of the DDM. In the first two sections of this study, we develop methods for learning optimal one and two-level DDMs utilizing graph neural networks (GNNs). Key components of our proposed methods include novel loss functions enjoying theoretical guarantees to converge to global optimum in limits. First, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured- and unstructured-grid problems. Second, we propose multigrid graph neural networks (MG-GNN), a novel GNN architecture for learning optimized parameters in two-level DDMs. We train MG-GNN using a new unsupervised loss function, enabling effective training on small problems that yields robust performance on unstructured grids that are orders of magnitude larger than those in the training set. We show that MG-GNN outperforms popular hierarchical graph network architectures for this optimization and that our proposed loss function is critical to achieving this improved performance. Cross-device user matching is a critical problem in numerous domains, including advertising, recommender systems, and cybersecurity. It involves identifying and linking different devices belonging to the same person, utilizing sequence logs. Previous data mining techniques have struggled to address the long-range dependencies and higher-order connections between the logs. Recently, researchers have modeled this problem as a graph problem and proposed a two-tier graph contextual embedding (TGCE) neural network architecture, which outperforms previous methods. In the last section of this study, we propose a novel hierarchical graph neural network architecture (HGNN), which has a more computationally efficient second level design than TGCE. Furthermore, we introduce a cross-attention (Cross-Att) mechanism in our model, which improves performance by 5% compared to the state-of-the-art TGCE method.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Taghibakhshi, Ali

Contributors dc:contributor

• MacLachlan, Scott
• West, Matthew
• Olson, Luke
• Salapaka, Srinivasa

• Restricted Additive Schwarz
• Optimized Restricted Additive Schwarz
• Amg Algebraic Multigrid
• Ml Machine Learning
• Graph Neural Network
• Multigrid Graph Neural Network
• Hierarchical Graph Neural Network
• Domain Decomposition Method

Statement dc:rights

• Copyright 2023 Ali Taghibakhshi

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/121471

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F121471