# Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries

arXiv: https://arxiv.org/abs/2308.12939 · HTML: https://mlsanity.com/paper/2308.12939/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 24 Aug 2023.

## Builds on (most cited first, 17 of 17)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)

## Built on by (most cited first, 13 of 13)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study](https://mlsanity.com/paper/2406.14715/) (2024)
- [Separable Operator Networks](https://mlsanity.com/paper/2407.11253/) (2024)
- [HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEs](https://mlsanity.com/paper/2511.00886/) (2025)
- [Fredholm Integral Equations Neural Operator (FIE-NO) for Data-Driven Boundary Value Problems](https://mlsanity.com/paper/2408.12389/) (2024)
- [PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2512.02049/) (2025)
- [A Boundary Integral-based Neural Operator for Mesh Deformation](https://mlsanity.com/paper/2602.23703/) (2026)
- [UniPINN: A Unified PINN Framework for Multi-task Learning of Diverse Navier-Stokes Equations](https://mlsanity.com/paper/2603.10466/) (2026)
- [Virtual boundary integral neural network for three-dimensional exterior acoustic problems](https://mlsanity.com/paper/2604.18636/) (2026)
- [Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometries](https://mlsanity.com/paper/2605.00760/) (2026)
- [Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acoustics](https://mlsanity.com/paper/2607.04407/) (2026)
- [ScaGNN: a Graph Neural Network for Multiple Scattering Simulations](https://mlsanity.com/paper/2609.37509/) (2026)
- [An Attention-Guided Graph Neural Network Surrogate for Boundary Integral Computation in Physical Field Reconstruction](https://mlsanity.com/paper/2610.15590/) (2026)

## Abstract (arXiv)

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from O(Nd) to O(Nd-1), where d is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.