# Finite-difference-informed graph network for solving steady-state incompressible flows on block-structured grids

arXiv: https://arxiv.org/abs/2406.10534 · HTML: https://mlsanity.com/paper/2406.10534/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 48 references that are in the map.
- First posted on arXiv 15 Jun 2024.

## Builds on (most cited first, 26 of 26)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [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)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)

## Built on by (most cited first, 1 of 1)

- [Scale-PINN: Learning Efficient Physics-Informed Neural Networks Through Sequential Correction](https://mlsanity.com/paper/2602.19475/) (2026)

## Abstract (arXiv)

Advances in deep learning have enabled physics-informed neural networks to solve partial differential equations. Numerical differentiation using the finite-difference (FD) method is efficient in physics-constrained designs, even in parameterized settings. In traditional computational fluid dynamics(CFD), body-fitted block-structured grids are often employed for complex flow cases when obtaining FD solutions. However, convolution operators in convolutional neural networks for FD are typically limited to single-block grids. To address this issue, blueTextgraphs and graph networks are used to learn flow representations across multi-block-structured grids. blueTextA graph convolution-based FD method (GC-FDM) is proposed to train graph networks in a label-free physics-constrained manner, enabling differentiable FD operations on unstructured graph outputs. To demonstrate model performance from single- to multi-block-structured grids, blueTextthe parameterized steady incompressible Navier-Stokes equations are solved for a lid-driven cavity flow and the flows around single and double circular cylinder configurations. When compared to a CFD solver under various boundary conditions, the proposed method achieves a relative error in velocity field predictions on the order of 10-3. Furthermore, the proposed method reduces training costs by approximately 20% compared to a physics-informed neural network. blueTextTo further verify the effectiveness of GC-FDM in multi-block processing, blueTexta 30P30N airfoil geometry is considered and the blueTextpredicted results are reasonable compared with those given by CFD. blueTextFinally, the applicability of GC-FDM to three-dimensional (3D) case is tested using a 3D cavity geometry.