  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.
