# Learning Lagrangian Fluid Mechanics with E(3)-Equivariant Graph Neural Networks

arXiv: https://arxiv.org/abs/2305.15603 · HTML: https://mlsanity.com/paper/2305.15603/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 23 of 23)

- [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)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [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)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [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)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Boundary Graph Neural Networks for 3D Simulations](https://mlsanity.com/paper/2106.11299/) (2021)

## Built on by (most cited first, 12 of 12)

- [LagrangeBench: A Lagrangian Fluid Mechanics Benchmarking Suite](https://mlsanity.com/paper/2309.16342/) (2023)
- [Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators](https://mlsanity.com/paper/2402.12365/) (2024)
- [Neural SPH: Improved Neural Modeling of Lagrangian Fluid Dynamics](https://mlsanity.com/paper/2402.06275/) (2024)
- [Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogates](https://mlsanity.com/paper/2506.05513/) (2025)
- [JAX-SPH: A Differentiable Smoothed Particle Hydrodynamics Framework](https://mlsanity.com/paper/2403.04750/) (2024)
- [Mean flow data assimilation using physics-constrained Graph Neural Networks](https://mlsanity.com/paper/2411.09476/) (2024)
- [Bridging Data and Physics: A Graph Neural Network-Based Hybrid Twin Framework](https://mlsanity.com/paper/2512.15767/) (2025)
- [Reversible GNS for Dissipative Fluids with Consistent Bidirectional Dynamics](https://mlsanity.com/paper/2509.22207/) (2025)
- [Learning Mesh-Free Discrete Differential Operators with Self-Supervised Graph Neural Networks](https://mlsanity.com/paper/2603.24641/) (2026)
- [FlowSymm: Physics Aware, Symmetry Preserving Graph Attention for Network Flow Completion](https://mlsanity.com/paper/2601.22317/) (2026)
- [Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation](https://mlsanity.com/paper/2608.01164/) (2026)
- [Phase-aware video generation for physics-grounded dynamics and interactions](https://mlsanity.com/paper/2610.11791/) (2026)

## Abstract (arXiv)

We contribute to the vastly growing field of machine learning for engineering systems by demonstrating that equivariant graph neural networks have the potential to learn more accurate dynamic-interaction models than their non-equivariant counterparts. We benchmark two well-studied fluid-flow systems, namely 3D decaying Taylor-Green vortex and 3D reverse Poiseuille flow, and evaluate the models based on different performance measures, such as kinetic energy or Sinkhorn distance. In addition, we investigate different embedding methods of physical-information histories for equivariant models. We find that while currently being rather slow to train and evaluate, equivariant models with our proposed history embeddings learn more accurate physical interactions.