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Neural Domain Decomposition for Variable Coefficient Poisson Solvers

Authors: Sebastian Barschkis and Zitong Li (University of California, Irvine); Hengjie Wang (Modular Inc); and Aparna Chandramowlishwaran (University of California, Irvine)

Abstract: The computational bottleneck in many fluid simulations arises from solving the variable coefficient Poisson equation. To tackle this challenge, we propose a novel neural domain decomposition algorithm to accelerate its solution. Our approach hinges on two key ideas: first, using neural PDE solvers to approximate the solutions within subdomains, and second, ensuring continuity across subdomain boundaries by solving a Schur complement system derived from the cell-centered discretized Poisson equation. A distinct advantage of our approach lies in generating a large dataset consisting only of small-scale problems to train the subdomain solver. This trained model can subsequently be applied to problems with large and complex geometries. Moreover, by batching the independent subdomain solves, we achieve high GPU utilization with neural solvers compared to state-of-the-art numerical methods. In contrast to neural domain decomposition algorithms that rely on Schwarz overlapping methods, our optimization-based approach, coupled with neural PDE solvers, improves accuracy and performance.

Best Poster Finalist (BP): no

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