# BINN: A deep learning approach for computational mechanics problems based on boundary integral equations

arXiv: https://arxiv.org/abs/2301.04480 · HTML: https://mlsanity.com/paper/2301.04480/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 73 references that are in the map.
- First posted on arXiv 11 Jan 2023.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture](https://mlsanity.com/paper/2011.13511/) (2020)
- [PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometries](https://mlsanity.com/paper/2004.06490/) (2020)
- [A Physics-Guided Neural Network Framework for Elastic Plates: Comparison of Governing Equations-Based and Energy-Based Approaches](https://mlsanity.com/paper/2010.06050/) (2020)
- [Analysis of three dimensional potential problems in non-homogeneous media with physics-informed deep collocation method using material transfer learning and sensitivity analysis](https://mlsanity.com/paper/2010.12060/) (2020)
- [CENN: Conservative energy method based on neural networks with subdomains for solving variational problems involving heterogeneous and complex geometries](https://mlsanity.com/paper/2110.01359/) (2021)
- [Stochastic analysis of heterogeneous porous material with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning](https://mlsanity.com/paper/2010.12344/) (2020)

## Built on by (most cited first, 11 of 11)

- [Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks](https://mlsanity.com/paper/2406.11045/) (2024)
- [Energy-based physics-informed neural network for frictionless contact problems under large deformation](https://mlsanity.com/paper/2411.03671/) (2024)
- [Physics-Informed Boundary Integral Networks (PIBI-Nets): A Data-Driven Approach for Solving Partial Differential Equations](https://mlsanity.com/paper/2308.09571/) (2023)
- [Green Multigrid Network](https://mlsanity.com/paper/2407.03593/) (2024)
- [PINNIES: An Efficient Physics-Informed Neural Network Framework to Integral Operator Problems](https://mlsanity.com/paper/2409.01899/) (2024)
- [A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials](https://mlsanity.com/paper/2605.31231/) (2026)
- [PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs](https://mlsanity.com/paper/2607.20378/) (2026)
- [Virtual boundary integral neural network for three-dimensional exterior acoustic problems](https://mlsanity.com/paper/2604.18636/) (2026)
- [Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates](https://mlsanity.com/paper/2605.02280/) (2026)
- [Neural Harmonic Measure Operator](https://mlsanity.com/paper/2609.35752/) (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)

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are approximated using neural networks and solved through a training process. The loss function is chosen as the residuals of the boundary integral equations. Regularization techniques are adopted to efficiently evaluate the weakly singular and Cauchy principle integrals in boundary integral equations. Potential problems and elastostatic problems are mainly concerned in this article as a demonstration. The proposed method has several outstanding advantages: First, the dimensions of the original problem are reduced by one, thus the freedoms are greatly reduced. Second, the proposed method does not require any extra treatment to introduce the boundary conditions, since they are naturally considered through the boundary integral equations. Therefore, the method is suitable for complex geometries. Third, BINN is suitable for problems on the infinite or semi-infinite domains. Moreover, BINN can easily handle heterogeneous problems with a single neural network without domain decomposition.