  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.
