FIG. 2.

Workflow for uncovering coarse-scale PDEs. First, we compute macroscopic variables $u$ and $v$ from the lattice Boltzmann simulation data [see Eq. (18) and Fig. 1] and estimate their spatial derivatives (e.g., by finite difference schemes on the lattice). After that, we employ machine-learning algorithms (here, Gaussian process regression or artificial neural networks) to identify “proper” time derivatives $u_t$ and $v_t$ from an original input data domain directly (no feature selection among several spatial derivatives) or from a reduced input data domain (feature selection among several spatial derivatives) using ARD in Gaussian processes or diffusion maps. We then simulate the identified coarse-scale PDE for given coarse initial conditions (${\mathbf{u}}_{\mathbf{0}},{\mathbf{v}}_{\mathbf{0}}$).