Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers

• URL: http://arxiv.org/abs/2208.13273v2
• Date: Mon, 2 Sep 2024 00:22:45 GMT
• Title: Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers
• Authors: Enrui Zhang, Adar Kahana, Alena Kopaničáková, Eli Turkel, Rishikesh Ranade, Jay Pathak, George Em Karniadakis,
• Abstract summary: HINTS is a hybrid, iterative, numerical, and transferable solver for partial differential equations. It balances the convergence behavior across the spectrum of eigenmodes by utilizing the spectral bias of DeepONet. It is flexible with regards to discretizations, computational domain, and boundary conditions.
• Score: 3.2712166248850685
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Neural networks suffer from spectral bias having difficulty in representing the high frequency components of a function while relaxation methods can resolve high frequencies efficiently but stall at moderate to low frequencies. We exploit the weaknesses of the two approaches by combining them synergistically to develop a fast numerical solver of partial differential equations (PDEs) at scale. Specifically, we propose HINTS, a hybrid, iterative, numerical, and transferable solver by integrating a Deep Operator Network (DeepONet) with standard relaxation methods, leading to parallel efficiency and algorithmic scalability for a wide class of PDEs, not tractable with existing monolithic solvers. HINTS balances the convergence behavior across the spectrum of eigenmodes by utilizing the spectral bias of DeepONet, resulting in a uniform convergence rate and hence exceptional performance of the hybrid solver overall. Moreover, HINTS applies to large-scale, multidimensional systems, it is flexible with regards to discretizations, computational domain, and boundary conditions.

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