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

Aditya Kashi, Arka Daw, Muralikrishnan Gopalakrishnan Meena, Hao Lu

Keywords: neural operator, partial differential equations, boundary conditions, tensor product, lifting product, fourier neural operator

TL;DR: We present a novel neural operator architecture for predicting solutions of PDEs in a domain given spatially-varying boundary functions as input and demonstrate its resolution independence.

Abstract: 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.

Submission Number: 180