# Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations

arXiv: https://arxiv.org/abs/2406.02298 · HTML: https://mlsanity.com/paper/2406.02298/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 49 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 13 of 13)

- [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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)

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

- [Fredholm Neural Networks for inverse problems in elliptic PDEs](https://mlsanity.com/paper/2507.06038/) (2025)
- [Deep Micro Solvers for Rough-Wall Stokes Flow in a Heterogeneous Multiscale Method](https://mlsanity.com/paper/2507.13902/) (2025)
- [PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2512.02049/) (2025)
- [ScaGNN: a Graph Neural Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2609.37509/) (2026)

## Abstract (arXiv)

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs): the Boundary Integral Type Deep Operator Network (BI-DeepONet) and the Boundary Integral Trigonometric Deep Operator Neural Network (BI-TDONet), which are crafted to address PDEs across diverse domains. Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training. BI-TDONet notably enhances its performance by employing the singular value decomposition (SVD) of bounded linear operators, allowing for the efficient distribution of input functions across its modules. Furthermore, to tackle the issue of function sampling values that do not effectively capture oscillatory and impulse signal characteristics, trigonometric coefficients are utilized as both inputs and outputs in BI-TDONet. Our numerical experiments robustly support and confirm the efficacy of this theoretical framework.