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

arXiv: https://arxiv.org/abs/2406.16740 · HTML: https://mlsanity.com/paper/2406.16740/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 10 references that are in the map.
- First posted on arXiv 24 Jun 2024.

## Builds on (most cited first, 7 of 7)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)

## Built on by (most cited first, 4 of 4)

- [Imposing Boundary Conditions on Neural Operators via Learned Function Extensions](https://mlsanity.com/paper/2602.04923/) (2026)
- [Plane-Wave Decomposition and Randomised Training; a Novel Path to Generalised PINNs for SHM](https://mlsanity.com/paper/2504.00249/) (2025)
- [A Boundary Integral-based Neural Operator for Mesh Deformation](https://mlsanity.com/paper/2602.23703/) (2026)
- [Physics-Guided Dual-Stream Heterogeneous Graph Neural Network for Predicting Full-Field Structural Response of Stiffened Panels](https://mlsanity.com/paper/2606.20916/) (2026)

## Abstract (arXiv)

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.