# Neural PDE Solvers for Irregular Domains

arXiv: https://arxiv.org/abs/2211.03241 · HTML: https://mlsanity.com/paper/2211.03241/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 68 references that are in the map.
- First posted on arXiv 7 Nov 2022.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Loss landscapes and optimization in over-parameterized non-linear systems and neural networks](https://mlsanity.com/paper/2003.00307/) (2020)
- [SGD: General Analysis and Improved Rates](https://mlsanity.com/paper/1901.09401/) (2019)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (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)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [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)
- [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)
- [Phase Transitions, Distance Functions, and Implicit Neural Representations](https://mlsanity.com/paper/2106.07689/) (2021)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Solving Irregular and Data-enriched Differential Equations using Deep Neural Networks](https://mlsanity.com/paper/1905.04351/) (2019)
- [Deep Generative Models that Solve PDEs: Distributed Computing for Training Large Data-Free Models](https://mlsanity.com/paper/2007.12792/) (2020)
- [Distributed Multigrid Neural Solvers on Megavoxel Domains](https://mlsanity.com/paper/2104.14538/) (2021)
- [Algorithmically-Consistent Deep Learning Frameworks for Structural Topology Optimization](https://mlsanity.com/paper/2012.05359/) (2020)
- [Industrial scale large eddy simulations (LES) with adaptive octree meshes using immersogeometric analysis](https://mlsanity.com/paper/2009.00706/) (2020)

## Built on by (most cited first, 6 of 6)

- [Geometry Matters: Benchmarking Scientific ML Approaches for Flow Prediction around Complex Geometries](https://mlsanity.com/paper/2501.01453/) (2024)
- [3D Neural Operator-Based Flow Surrogates around 3D geometries: Signed Distance Functions and Derivative Constraints](https://mlsanity.com/paper/2503.17289/) (2025)
- [A note on the error analysis of data-driven closure models for large eddy simulations of turbulence](https://mlsanity.com/paper/2405.17612/) (2024)
- [Physics informed cell representations for variational formulation of multiscale problems](https://mlsanity.com/paper/2405.16770/) (2024)
- [Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networks](https://mlsanity.com/paper/2512.04434/) (2025)
- [Feature Space Adaptation for Effortless Gaussian Process Flows](https://mlsanity.com/paper/2610.11459/) (2026)

## 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.