Machine-learning-enhanced collision operator for the lattice Boltzmann method based on invariant networks

Integrating machine learning (ML) techniques in established numerical solvers represents a modern approach to enhance computational fluid dynamics simulations. Within the lattice Boltzmann method (LBM), the collision operator serves as an ideal entry point to incorporate machine learning (ML) techniques to enhance its accuracy and stability. In this work, an invariant neural network is constructed, acting on an equivariant collision operator, optimizing the relaxation rates of nonphysical moments. This optimization enhances robustness to symmetry transformations and ensures consistent behavior across geometric operations. The proposed neural collision operator (NCO) is trained using forced isotropic turbulence simulations driven by spectral forcing, ensuring stable turbulence statistics. The desired performance is achieved by minimizing the energy spectrum discrepancy between direct numerical simulations and underresolved simulations over a specified wave number range. The loss function is further extended to tailor numerical dissipation at high wave numbers, ensuring robustness without compromising accuracy at low and intermediate wave numbers. The NCO's performance is demonstrated using three-dimensional Taylor-Green vortex (TGV) flows, where it accurately predicts the dynamics even in highly underresolved simulations. Compared to other LBM models, such as the BGK and KBC operators, the NCO exhibits improved accuracy while maintaining stability. In addition, the operator shows robust performance in alternative configurations, including turbulent three-dimensional cylinder flow. Finally, an alternative training procedure using time-dependent quantities is introduced. It is based on a reduced TGV model along with newly proposed symmetry boundary conditions. The reduction in memory consumption enables training at significantly higher Reynolds numbers, successfully leading to stable yet accurate simulations.