Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations

Conference · Mon Jul 01 00:00:00 EDT 2024

OSTI ID:2429817

Kashi, Aditya ^[1]; Daw, Arka ^[1]; Gopalakrishnan Meena, Murali ^[1]; Lu, Hao ^[1]

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1. ORNL

Neural operators such as the Fourier Neural Operator (FNO) have been shown to provide resolution-independent deep learning models that can learn mappings between function spaces. For example, an initial condition can be mapped to the solution of a partial differential equation (PDE) at a future time-step using a neural operator. Despite the popularity of neural operators, their use to predict solution functions over a domain given only data over the boundary (such as a spatially varying Dirichlet boundary condition) remains unexplored. In this paper, we refer to such problems as boundary-to-domain problems; they have a wide range of applications in areas such as fluid mechanics, solid mechanics, heat transfer etc. We present a novel FNO-based architecture, named Lifting Product FNO (or LP-FNO) which can map arbitrary boundary functions defined on the lower-dimensional boundary to a solution in the entire domain. Specifically, two FNOs defined on the lower-dimensional boundary are lifted into the higher dimensional domain using our proposed lifting product layer. We demonstrate the efficacy and resolution independence of the proposed LP-FNO for the 2D Poisson equation.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
• Sponsoring Organization: USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE
• DOE Contract Number: AC05-00OR22725
• OSTI ID: 2429817
• Resource Type: Conference paper/presentation
• Conference Information: International Conference on Machine Learning Workshop on AI for Science: Scaling in AI for Scientific Discovery - Vienna, Austria - 7/26/2024-7/27/2024
• Country of Publication: United States
• Language: English