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

TL;DR

This paper introduces Lattice-Equivariant Neural Networks (LENNs), which incorporate lattice symmetries into neural network models for fluid dynamics, improving accuracy, stability, and computational efficiency in simulating complex flows.

Contribution

The paper develops a novel class of neural networks that are equivariant to lattice symmetries, enabling scalable, stable, and efficient surrogate models for lattice Boltzmann fluid simulations.

Findings

LENNs outperform non-symmetric networks in stability and accuracy.

LENNs are approximately ten times faster than group-averaged networks in 3D.

The approach is validated on 2D and 3D fluid flow regimes, including turbulent flows.

Abstract

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.