Blending neural operators and relaxation methods in PDE numerical solvers

Enrui Zhang, Adar Kahana, Alena Kopaničáková, Eli Turkel, Rishikesh Ranade, Jay Pathak, George Em Karniadakis*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)1303-1313
Number of pages11
JournalNature Machine Intelligence
Volume6
Issue number11
DOIs
StatePublished - Nov 2024

Funding

FundersFunder number
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
DOE PhILMsde-sc0019453
MURI-AFOSRFA9550-20-1-0358
ONR Vannevar Bush Faculty FellowshipN00014-22-1-2795
School Nutrition Foundation206745

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