# Delta-PINNs: physics-informed neural networks on complex geometries

arXiv: https://arxiv.org/abs/2209.03984 · HTML: https://mlsanity.com/paper/2209.03984/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 8 Sep 2022.

## Builds on (most cited first, 15 of 15)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [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)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical maps](https://mlsanity.com/paper/2201.12362/) (2022)
- [Fast characterization of inducible regions of atrial fibrillation models with multi-fidelity Gaussian process classification](https://mlsanity.com/paper/2112.08075/) (2021)

## Built on by (most cited first, 28 of 28)

- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learning](https://mlsanity.com/paper/2511.07262/) (2025)
- [Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Models](https://mlsanity.com/paper/2312.09787/) (2023)
- [Residual-based Attention Physics-informed Neural Networks for Spatio-Temporal Ageing Assessment of Transformers Operated in Renewable Power Plants](https://mlsanity.com/paper/2405.06443/) (2024)
- [Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space](https://mlsanity.com/paper/2501.12116/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.13717/) (2025)
- [Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features](https://mlsanity.com/paper/2510.19399/) (2025)
- [Physics-informed neural networks for transformed geometries and manifolds](https://mlsanity.com/paper/2311.15940/) (2023)
- [Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEs](https://mlsanity.com/paper/2506.00947/) (2025)
- [Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometries](https://mlsanity.com/paper/2512.14941/) (2025)
- [Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings](https://mlsanity.com/paper/2602.16193/) (2026)
- [DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEs](https://mlsanity.com/paper/2603.01762/) (2026)
- [Ensemble learning of the atrial fiber orientation with physics-informed neural networks](https://mlsanity.com/paper/2410.23388/) (2024)
- [Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics](https://mlsanity.com/paper/2412.09453/) (2024)
- [Solved in Unit Domain: JacobiNet for Differentiable Coordinate-Transformed PINNs](https://mlsanity.com/paper/2508.02537/) (2025)
- [Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamics](https://mlsanity.com/paper/2510.13601/) (2025)
- [Learning Null Geodesics for Gravitational Lensing Rendering in General Relativity](https://mlsanity.com/paper/2507.15775/) (2025)
- [Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Sampling](https://mlsanity.com/paper/2512.10873/) (2025)
- [Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs](https://mlsanity.com/paper/2601.00672/) (2026)
- [Non-Invasive Reconstruction of Cardiac Activation Dynamics Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2603.03832/) (2026)
- [An adaptive wavelet-based PINN for problems with localized high-magnitude source](https://mlsanity.com/paper/2604.28180/) (2026)
- [Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs](https://mlsanity.com/paper/2605.24774/) (2026)
- [Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields](https://mlsanity.com/paper/2605.28551/) (2026)
- [Wall Shear Stress Reconstruction from Concentration: Differentiable Physics and Physics-Informed Neural Networks](https://mlsanity.com/paper/2606.06313/) (2026)
- [P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolution](https://mlsanity.com/paper/2606.19303/) (2026)
- [Hierarchical rank-evolving representation for physics-informed neural networks](https://mlsanity.com/paper/2608.09483/) (2026)
- [Neural networks for spectral optimization](https://mlsanity.com/paper/2609.36047/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. Despite recent progress on expanding the class of problems that can be tackled by PINNs, most of existing use-cases involve simple geometric domains. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. In this work, we propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator. This technique allows to create an input space for the neural network that represents the geometry of a given object. We approximate the eigenfunctions as well as the operators involved in the partial differential equations with finite elements. We extensively test and compare the proposed methodology against traditional PINNs in complex shapes, such as a coil, a heat sink and a bunny, with different physics, such as the Eikonal equation and heat transfer. We also study the sensitivity of our method to the number of eigenfunctions used, as well as the discretization used for the eigenfunctions and the underlying operators. Our results show excellent agreement with the ground truth data in cases where traditional PINNs fail to produce a meaningful solution. We envision this new technique will expand the effectiveness of PINNs to more realistic applications.