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

TL;DR

This work addresses learning PDE solutions from boundary data by introducing LP-FNO, an operator-based approach that maps boundary functions $g:\Gamma\to\mathbb{R}^m$ to domain solutions $u:\Omega\to\mathbb{R}$ via a lifting product of two boundary-FNOs. The method lifts boundary representations into a higher-dimensional domain using a lifting product $c=a\otimes b$, enabling a boundary-to-domain mapping for problems such as the 2D Poisson equation with Dirichlet BC. Empirically, LP-FNO achieves competitive in-distribution accuracy to FNO2d, while demonstrating superior resolution-independence and non-native-resolution generalization, including some zero-shot super-resolution capabilities and artefacts that warrant further refinement. Overall, LP-FNO advances neural operator surrogates for boundary-driven PDEs and suggests a viable path toward resolution-agnostic B2D modeling with potential broad applicability in physics-informed simulations.

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.

Figures (17)

• Figure 1: A schematic representation of the LP-FNO architecture example for the 1D to 2D case
• Figure 2: Comparison of the predicted solutions on in-distribution examples for models trained on 64x64 resolution.
• Figure 3: Comparison of the absolute error on in-distribution examples for models trained on 64x64 resolution.
• Figure 4: Comparison of the predicted solutions on out-of-distribution exponential examples for models trained on 32x32 resolution and evaluated on 64x64.
• Figure 5: Comparison of the absolute error on out-of-distribution exponential examples for models trained on 32x32 resolution and evaluated on 64x64.