Project Details
Description
Overview
This project develops computational algorithms that improve the efficiency and scalability of graph neural networks (GNNs) and creates New approaches for GNNs for solving nonlinear partial differential equations (PDEs) on unstructured meshes. To improve the scalability of GNNs to graphs with millions of nodes, we develop spatial smoothing operators, coarsening operators, and multilevel training schemes. To accelerate PDE simulations on unstructured meshes, we traill GNNs to produce effective prolongation, restriction, and coarse mesh operators in multigrid methods and preconditioners in Krylov methods. We seek to demonstrate that the resulting hybrid schemes accelerate computations and are provably convergent. To this end, we will consider challenging problems in computational fluid dynamics (CFD) and show the broad applicability of the improvements of generic GNNs on established geometric learning benchmark tasks such as shape and node classification. Both themes draw from algebraic multigrid (AMG) techniques as they motivate GNN design and training and are used in the PDE solvers.
Intellectual Merit
The project activities advance geometric machine learning and numerical PDEs and strengthen the connections between both fields. The first theme seeks to increase the scalability of GNNS to graphs with millions of nodes by leveraging a continuous interpretation of graph data, which renders techniques from algebraic multigrid and numerical PDEs applicable. The second theme aims at accelerating multigrid and other iterative solvers for solving large, unstructured, sparse linear systems arising in nonlinear PDEs by embedding scalable GNNs. Our learned non—Galerkin AMG schemes combat the severe stencil growth that limits existing algorithms and our learned preconditioners aim to improve flexible Krylov methods. In both cases, we combine GNNs with traditional solvers to retain their convergence properties. ‘We test our methods on challenging CFD problems involving high Reynolds numbers and irregular meshes. The cross—cut effort connects both themes and advances data structures and parallel algorithms to accelerate graph operators.
Broader Impacts
This US—Israeli collaboration emphasizes the trailling of students and postdocs, will lead to publications in journals and conferences, and produce computational methods that apply to various geometric deep learning problems and numerical PDES. The US PI will support a Ph.D. student and partially support a postdoc, and the Israeli PI will partially support two Ph.D. students. The Pls will jointly mentor the early career researchers, host weekly virtual meetings, and annual in—person meetings at Emory. Besides accelerating PDE simulations, this project also has the potential to impact various fields that involve large unstructured datasets, e.g., computer graphics, computer vision, bioinformatics, social network analysis, and protein folding. We will demonstrate the potential of these methods using example datasets from those areas. Science and engineering applications that involve CFD simulations on unstructured meshes arise in a broad range of fields, including cardiology and aerospace, and our examples of increasingly more challenging flows as the research progresses. The project will lead to open-source implementations of the developed algorithms under permissible licenses to realize the potential broader impacts.
Status | Active |
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Effective start/end date | 1/01/23 → … |
Links | https://www.bsf.org.il/search-grant/ |
Funding
- United States-Israel Binational Science Foundation (BSF)