Summary Lattice Boltzmann methods (LBM) are well suited to highly parallel computational fluid dynamics simulations due to their separability into a perfectly parallel collision step and a propagation step that only communicates within a local neighborhood. The implementation of the propagation step provides constraints for the maximum possible bandwidth‐limited performance, memory layout and usage of vector instructions. This article revisits and extends the work on implicit propagation on directly addressed grids started by A‐A and its shift‐swap‐streaming (SSS) formulation by reconsidering them as transformations of the underlying space filling curve. In this work, a new periodic shift (PS) pattern is proposed that imposes minimal restrictions on the implementation of collision operators and utilizes virtual memory mapping to provide consistent performance across a range of targets. Various implementation approaches as well as time dependency and performance anisotropy are discussed. Benchmark results for SSS and PS on SIMD CPUs including Intel Xeon Phi as well as Nvidia GPUs are provided. Finally, the application of PS as the propagation pattern of the open source LBM framework OpenLB is summarized.

In this paper, we explore applications of deep learning in statistical physics. We choose the Boltzmann equation as a typical example, where neural networks serve as a closure to its moment system. We present two types of neural networks to embed the convexity of entropy and to preserve the minimum entropy principle and intrinsic mathematical structures of the moment system of the Boltzmann equation. We derive an error bound for the generalization gap of convex neural networks which are trained in Sobolev norm and use the results to construct data sampling methods for neural network training. Numerical experiments demonstrate that the neural entropy closure is significantly faster than classical optimizers while maintaining sufficient accuracy.