Learning the boundary-to-domain mapping using Lifting Product Fourier Neural Operators for partial differential equations
This addresses boundary-to-domain problems in fields like fluid and solid mechanics, offering a novel method for an unexplored task, though it is incremental as it builds on existing neural operator frameworks.
The authors tackled the problem of predicting solutions over a domain given only boundary data for partial differential equations, introducing the Lifting Product Fourier Neural Operator (LP-FNO) which maps boundary functions to domain solutions, achieving resolution independence as demonstrated on the 2D Poisson equation.
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.