# Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction

arXiv: https://arxiv.org/abs/2007.04439 · HTML: https://mlsanity.com/paper/2007.04439/

## Key facts

- Cited by 138 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2020.

## Built on by (most cited first, 30 of 138)

- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
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- [Continuous PDE Dynamics Forecasting with Implicit Neural Representations](https://mlsanity.com/paper/2209.14855/) (2022)
- [Benchmarking Autoregressive Conditional Diffusion Models for Turbulent Flow Simulation](https://mlsanity.com/paper/2309.01745/) (2023)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
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- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
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- [Differentiable Simulation of Soft Multi-body Systems](https://mlsanity.com/paper/2205.01758/) (2022)
- [Learning Neural PDE Solvers with Parameter-Guided Channel Attention](https://mlsanity.com/paper/2304.14118/) (2023)
- [Physical Design using Differentiable Learned Simulators](https://mlsanity.com/paper/2202.00728/) (2022)
- [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)
- [Graph Neural Networks Accelerated Molecular Dynamics](https://mlsanity.com/paper/2112.03383/) (2021)
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- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)
- [Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks](https://mlsanity.com/paper/2504.02843/) (2025)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [PhysGNN: A Physics-Driven Graph Neural Network Based Model for Predicting Soft Tissue Deformation in Image-Guided Neurosurgery](https://mlsanity.com/paper/2109.04352/) (2021)

## Abstract (arXiv)

Solving large complex partial differential equations (PDEs), such as those that arise in computational fluid dynamics (CFD), is a computationally expensive process. This has motivated the use of deep learning approaches to approximate the PDE solutions, yet the simulation results predicted from these approaches typically do not generalize well to truly novel scenarios. In this work, we develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. By combining an actual CFD simulator (run on a much coarser resolution representation of the problem) with the graph network, we show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions, while also substantially outperforming the coarse CFD simulation alone.