TY - JOUR
T1 - Blending neural operators and relaxation methods in PDE numerical solvers
AU - Zhang, Enrui
AU - Kahana, Adar
AU - Kopaničáková, Alena
AU - Turkel, Eli
AU - Ranade, Rishikesh
AU - Pathak, Jay
AU - Karniadakis, George Em
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Nature Limited 2024.
PY - 2024/11
Y1 - 2024/11
N2 - Neural networks suffer from spectral bias and have difficulty representing the high-frequency components of a function, whereas 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 behaviour 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; and it can also be used to precondition Krylov methods.
AB - Neural networks suffer from spectral bias and have difficulty representing the high-frequency components of a function, whereas 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 behaviour 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; and it can also be used to precondition Krylov methods.
UR - http://www.scopus.com/inward/record.url?scp=85207239106&partnerID=8YFLogxK
U2 - 10.1038/s42256-024-00910-x
DO - 10.1038/s42256-024-00910-x
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AN - SCOPUS:85207239106
SN - 2522-5839
VL - 6
SP - 1303
EP - 1313
JO - Nature Machine Intelligence
JF - Nature Machine Intelligence
IS - 11
ER -