# RBF-MGN:Solving spatiotemporal PDEs with Physics-informed Graph Neural Network

arXiv: https://arxiv.org/abs/2212.02861 · HTML: https://mlsanity.com/paper/2212.02861/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 6 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2202.12358/) (2022)
- [PhyGNNet: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network](https://mlsanity.com/paper/2208.04319/) (2022)
- [Physics-Informed Deep Learning: A Promising Technique for System Reliability Assessment](https://mlsanity.com/paper/2108.10828/) (2021)

## Built on by (most cited first, 4 of 4)

- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Physics-Informed Graph Convolutional Networks: Towards a generalized framework for complex geometries](https://mlsanity.com/paper/2310.14948/) (2023)
- [TG-PhyNN: An Enhanced Physically-Aware Graph Neural Network framework for forecasting Spatio-Temporal Data](https://mlsanity.com/paper/2408.16379/) (2024)
- [Physics-Informed Graph-Mesh Networks for PDEs: A hybrid approach for complex problems](https://mlsanity.com/paper/2410.02819/) (2024)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have lately received significant attention as a representative deep learning-based technique for solving partial differential equations (PDEs). Most fully connected network-based PINNs use automatic differentiation to construct loss functions that suffer from slow convergence and difficult boundary enforcement. In addition, although convolutional neural network (CNN)-based PINNs can significantly improve training efficiency, CNNs have difficulty in dealing with irregular geometries with unstructured meshes. Therefore, we propose a novel framework based on graph neural networks (GNNs) and radial basis function finite difference (RBF-FD). We introduce GNNs into physics-informed learning to better handle irregular domains with unstructured meshes. RBF-FD is used to construct a high-precision difference format of the differential equations to guide model training. Finally, we perform numerical experiments on Poisson and wave equations on irregular domains. We illustrate the generalizability, accuracy, and efficiency of the proposed algorithms on different PDE parameters, numbers of collection points, and several types of RBFs.