mlsanityData to 14 Oct 2026

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

arXiv 2209.03984 ↗First posted 8 Sep 2022cs.LGcited by 28

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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.

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

15 of 36 references.

References · 15

  1. 2017Attention Is All You Needcited by 60,361
  2. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  3. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  4. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  6. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  7. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 174
  8. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 140
  9. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 137
  10. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  11. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  12. 2020Mat'ern Gaussian processes on Riemannian manifoldscited by 82
3 more
  1. 2020EikoNet: Solving the Eikonal equation with Deep Neural Networkscited by 50
  2. 2022Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical mapscited by 14
  3. 2021Fast characterization of inducible regions of atrial fibrillation models with multi-fidelity Gaussian process classificationcited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 4; 2025: 10; 2026: 12.

28 papers.

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  1. 2024PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systemscited by 18
  2. 2025AgenticSciML: Collaborative Multi-Agent Systems for Emergent Discovery in Scientific Machine Learningcited by 14
  3. 2023Physics-informed Neural Network Estimation of Material Properties in Soft Tissue Nonlinear Biomechanical Modelscited by 11
  4. 2024Residual-based Attention Physics-informed Neural Networks for Spatio-Temporal Ageing Assessment of Transformers Operated in Renewable Power Plantscited by 11
  5. 2025Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Spacecited by 6
  6. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  7. 2025A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networkscited by 5
  8. 2025Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Featurescited by 4
  9. 2023Physics-informed neural networks for transformed geometries and manifoldscited by 3
  10. 2025Deformable registration and generative modelling of aortic anatomies by auto-decoders and neural ODEscited by 3
  11. 2025Boundary condition enforcement with PINNs: a comparative study and verification on 3D geometriescited by 2
  12. 2026Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappingscited by 2
16 more
  1. 2026DGNet: Discrete Green Networks for Data-Efficient Learning of Spatiotemporal PDEscited by 2
  2. 2024Ensemble learning of the atrial fiber orientation with physics-informed neural networkscited by 1
  3. 2024Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanicscited by 1
  4. 2025Solved in Unit Domain: JacobiNet for Differentiable Coordinate-Transformed PINNscited by 1
  5. 2025Physics-augmented Multi-task Gaussian Process for Modeling Spatiotemporal Dynamicscited by 1
  6. 2025Learning Null Geodesics for Gravitational Lensing Rendering in General Relativitycited by 0
  7. 2025Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Samplingcited by 0
  8. 2026Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEscited by 0
  9. 2026Non-Invasive Reconstruction of Cardiac Activation Dynamics Using Physics-Informed Neural Networkscited by 0
  10. 2026An adaptive wavelet-based PINN for problems with localized high-magnitude sourcecited by 0
  11. 2026Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEscited by 0
  12. 2026Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fieldscited by 0
  13. 2026Wall Shear Stress Reconstruction from Concentration: Differentiable Physics and Physics-Informed Neural Networkscited by 0
  14. 2026P-K-GCN: Physics-augmented Koopman-enhanced Graph Convolutional Network for Deep Spatiotemporal Super-resolutioncited by 0
  15. 2026Hierarchical rank-evolving representation for physics-informed neural networkscited by 0
  16. 2026Neural networks for spectral optimizationcited by 0