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

TL;DR

This paper introduces LP-FNO, a novel neural operator architecture based on Fourier Neural Operators, capable of mapping boundary conditions to entire domain solutions for PDEs, demonstrated on the 2D Poisson equation.

Contribution

The paper proposes LP-FNO, a new architecture that extends Fourier Neural Operators to boundary-to-domain problems, enabling solution prediction from boundary data.

Findings

LP-FNO effectively maps boundary functions to domain solutions.

The method demonstrates resolution independence.

Validated on the 2D Poisson equation with promising results.

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.