mlsanityData to 7 Oct 2026

Neural PDE Solvers for Irregular Domains

arXiv 2211.03241 ↗First posted 7 Nov 2022cs.LG cs.NA math.NAcited by 6

Not on the map. Why?

Abstract (arXiv)

Neural network-based approaches for solving partial differential equations (PDEs) have recently received special attention. However, the large majority of neural PDE solvers only apply to rectilinear domains, and do not systematically address the imposition of Dirichlet/Neumann boundary conditions over irregular domain boundaries. In this paper, we present a framework to neurally solve partial differential equations over domains with irregularly shaped (non-rectilinear) geometric boundaries. Our network takes in the shape of the domain as an input (represented using an unstructured point cloud, or any other parametric representation such as Non-Uniform Rational B-Splines) and is able to generalize to novel (unseen) irregular domains; the key technical ingredient to realizing this model is a novel approach for identifying the interior and exterior of the computational grid in a differentiable manner. We also perform a careful error analysis which reveals theoretical insights into several sources of error incurred in the model-building process. Finally, we showcase a wide variety of applications, along with favorable comparisons with ground truth solutions.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 31 of its 68 references that are in the map. First posted on arXiv 7 Nov 2022.

Builds on

31 of 68 references.

References · 31

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 20163D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotationcited by 1,471
  4. 2016Optimization Methods for Large-Scale Machine Learningcited by 1,140
  5. 2017Spectrally-normalized margin bounds for neural networkscited by 681
  6. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  7. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  8. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  9. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  10. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  11. 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 526
  12. 2018Lipschitz regularity of deep neural networks: analysis and efficient estimationcited by 413
19 more
  1. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  2. 2020Loss landscapes and optimization in over-parameterized non-linear systems and neural networkscited by 231
  3. 2019SGD: General Analysis and Improved Ratescited by 206
  4. 2020PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domaincited by 173
  5. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  6. 2021Exact imposition of boundary conditions with distance functions in physics-informed deep neural networkscited by 135
  7. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  8. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  9. 2019An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applicationscited by 118
  10. 2020PFNN: A Penalty-Free Neural Network Method for Solving a Class of Second-Order Boundary-Value Problems on Complex Geometriescited by 38
  11. 2021Phase Transitions, Distance Functions, and Implicit Neural Representationscited by 38
  12. 2021Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNscited by 38
  13. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  14. 2019Solving Irregular and Data-enriched Differential Equations using Deep Neural Networkscited by 5
  15. 2020Deep Generative Models that Solve PDEs: Distributed Computing for Training Large Data-Free Modelscited by 5
  16. 2021Distributed Multigrid Neural Solvers on Megavoxel Domainscited by 5
  17. 2020Algorithmically-Consistent Deep Learning Frameworks for Structural Topology Optimizationcited by 3
  18. 2020Industrial scale large eddy simulations (LES) with adaptive octree meshes using immersogeometric analysiscited by 2
  19. 2021Scalable adaptive PDE solvers in arbitrary domainscited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 2; 2026: 1.

6 papers.

  1. 2024A note on the error analysis of data-driven closure models for large eddy simulations of turbulencecited by 2
  2. 2024Geometry Matters: Benchmarking Scientific ML Approaches for Flow Prediction around Complex Geometriescited by 2
  3. 20253D Neural Operator-Based Flow Surrogates around 3D geometries: Signed Distance Functions and Derivative Constraintscited by 2
  4. 2024Physics informed cell representations for variational formulation of multiscale problemscited by 1
  5. 2025Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networkscited by 0
  6. 2026Feature Space Adaptation for Effortless Gaussian Process Flowscited by 0