# Enhancing lattice kinetic schemes for fluid dynamics with Lattice-Equivariant Neural Networks

arXiv: https://arxiv.org/abs/2405.13850 · HTML: https://mlsanity.com/paper/2405.13850/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 35 references that are in the map.
- First posted on arXiv 22 May 2024.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [AutoAugment: Learning Augmentation Policies from Data](https://mlsanity.com/paper/1805.09501/) (2018)
- [Optimizing Millions of Hyperparameters by Implicit Differentiation](https://mlsanity.com/paper/1911.02590/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [TI-POOLING: transformation-invariant pooling for feature learning in Convolutional Neural Networks](https://mlsanity.com/paper/1604.06318/) (2016)
- [Machine Learning for Condensed Matter Physics](https://mlsanity.com/paper/2005.14228/) (2020)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Lat-Net: Compressing Lattice Boltzmann Flow Simulations using Deep Neural Networks](https://mlsanity.com/paper/1705.09036/) (2017)
- [Lettuce: PyTorch-based Lattice Boltzmann Framework](https://mlsanity.com/paper/2106.12929/) (2021)

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

- [Fast-Forward Lattice Boltzmann: Learning Kinetic Behaviour with Physics-Informed Neural Operators](https://mlsanity.com/paper/2509.22411/) (2025)

## Abstract (arXiv)

We present a new class of equivariant neural networks, hereby dubbed Lattice-Equivariant Neural Networks (LENNs), designed to satisfy local symmetries of a lattice structure. Our approach develops within a recently introduced framework aimed at learning neural network-based surrogate models Lattice Boltzmann collision operators. Whenever neural networks are employed to model physical systems, respecting symmetries and equivariance properties has been shown to be key for accuracy, numerical stability, and performance. Here, hinging on ideas from group representation theory, we define trainable layers whose algebraic structure is equivariant with respect to the symmetries of the lattice cell. Our method naturally allows for efficient implementations, both in terms of memory usage and computational costs, supporting scalable training/testing for lattices in two spatial dimensions and higher, as the size of symmetry group grows. We validate and test our approach considering 2D and 3D flowing dynamics, both in laminar and turbulent regimes. We compare with group averaged-based symmetric networks and with plain, non-symmetric, networks, showing how our approach unlocks the (a-posteriori) accuracy and training stability of the former models, and the train/inference speed of the latter networks (LENNs are about one order of magnitude faster than group-averaged networks in 3D). Our work opens towards practical utilization of machine learning-augmented Lattice Boltzmann CFD in real-world simulations.