# On the Relationships between Graph Neural Networks for the Simulation of Physical Systems and Classical Numerical Methods

arXiv: https://arxiv.org/abs/2304.00146 · HTML: https://mlsanity.com/paper/2304.00146/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 53 references that are in the map.
- First posted on arXiv 31 Mar 2023.

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

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [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)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [What Can Neural Networks Reason About?](https://mlsanity.com/paper/1905.13211/) (2019)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Neural Execution of Graph Algorithms](https://mlsanity.com/paper/1910.10593/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Coarse Graining Molecular Dynamics with Graph Neural Networks](https://mlsanity.com/paper/2007.11412/) (2020)
- [Simulating Continuum Mechanics with Multi-Scale Graph Neural Networks](https://mlsanity.com/paper/2106.04900/) (2021)
- [Learning the Dynamics of Physical Systems from Sparse Observations with Finite Element Networks](https://mlsanity.com/paper/2203.08852/) (2022)
- [Generative Coarse-Graining of Molecular Conformations](https://mlsanity.com/paper/2201.12176/) (2022)
- [Finite Volume Neural Network: Modeling Subsurface Contaminant Transport](https://mlsanity.com/paper/2104.06010/) (2021)
- [Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients](https://mlsanity.com/paper/2201.13240/) (2022)
- [Simulating Liquids with Graph Networks](https://mlsanity.com/paper/2203.07895/) (2022)
- [Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators](https://mlsanity.com/paper/2206.14687/) (2022)

## Built on by (most cited first, 3 of 3)

- [Neural SPH: Improved Neural Modeling of Lagrangian Fluid Dynamics](https://mlsanity.com/paper/2402.06275/) (2024)
- [JAX-SPH: A Differentiable Smoothed Particle Hydrodynamics Framework](https://mlsanity.com/paper/2403.04750/) (2024)
- [From Numerical Simulators of PDEs to Neural Emulators and Back](https://mlsanity.com/paper/2608.24547/) (2026)

## Abstract (arXiv)

Recent developments in Machine Learning approaches for modelling physical systems have begun to mirror the past development of numerical methods in the computational sciences. In this survey, we begin by providing an example of this with the parallels between the development trajectories of graph neural network acceleration for physical simulations and particle-based approaches. We then give an overview of simulation approaches, which have not yet found their way into state-of-the-art Machine Learning methods and hold the potential to make Machine Learning approaches more accurate and more efficient. We conclude by presenting an outlook on the potential of these approaches for making Machine Learning models for science more efficient.