Enhancing Lattice Kinetic Schemes for Fluid Dynamics with Lattice-Equivariant Neural Networks

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Abstract

A new class of equivariant neural networks is presented, hereby dubbed lattice-equivariant neural networks (LENNs), designed to satisfy local symmetries of a lattice structure. The 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, trainable layers are defined whose algebraic structure is equivariant with respect to the symmetries of the lattice cell. The presented method naturally allows for efficient implementations, in terms of both memory usage and computational costs, supporting scalable training/testing for lattices in two spatial dimensions and higher (in which the size of symmetry group grows). The approach is validated andtested considering 2D and 3D flowing dynamics, both in laminar and turbulent regimes. It is compared with group-averaged-based symmetric networks and with plain, nonsymmetric, networks, showing how the presented 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.) The work in this paper opens toward practical use of machine learning-augmented lattice Boltzmann CFD in real-world simulations.
Original languageEnglish
Pages (from-to)716-731
Number of pages16
JournalAIAA Journal
Volume63
Issue number2
Early online date17 Dec 2024
DOIs
Publication statusPublished - Feb 2025

Funding

A.G. gratefully acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program under project number 20240740PRD1 and the Center for Non-Linear Studies for this work. A.C. Acknowledges the support of a starting grant from Eindhoven Artificial Intelligence Systems Institute. This work was partially funded by the Dutch Research Council (NWO) through the UNRAVEL project (with Project No. OCENW.GROOT.2019.044).

Keywords

  • Computational Fluid Dynamics
  • Computational Modeling
  • Conservation of Mass
  • Conservation of Momentum Equations
  • Convolutional Neural Network
  • Flow Conditions
  • Lattice Boltzmann Equation
  • Machine Learning
  • Stochastic Gradient Descent
  • Taylor Green Vortex

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