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.

翻译：诸如傅里叶神经算子（FNO）之类的神经算子已被证明能够提供与分辨率无关的深度学习模型，这些模型可以学习函数空间之间的映射。例如，一个初始条件可以通过神经算子映射到偏微分方程（PDE）在未来某个时间步的解。尽管神经算子很流行，但仅给定边界上的数据（例如空间变化的狄利克雷边界条件）来预测整个域上的解函数，其应用仍未得到探索。在本文中，我们将此类问题称为边界到域问题；它们在流体力学、固体力学、传热学等领域有广泛的应用。我们提出了一种新颖的基于FNO的架构，命名为提升积FNO（或LP-FNO），它可以将定义在低维边界上的任意边界函数映射到整个域中的解。具体来说，两个定义在低维边界上的FNO通过我们提出的提升积层被提升到高维域中。我们针对二维泊松方程，证明了所提出的LP-FNO的有效性和分辨率无关性。