  In the recent years, deep learning approaches have shown much promise in
modeling complex systems in the physical sciences. A major challenge in deep
learning of PDEs is enforcing physical constraints and boundary conditions. In
this work, we propose a general framework to directly embed the notion of an
incompressible fluid into Convolutional Neural Networks, and apply this to
coarse-graining of turbulent flow. These physics-embedded neural networks
leverage interpretable strategies from numerical methods and computational
fluid dynamics to enforce physical laws and boundary conditions by taking
advantage the mathematical properties of the underlying equations. We
demonstrate results on three-dimensional fully-developed turbulence, showing
that this technique drastically improves local conservation of mass, without
sacrificing performance according to several other metrics characterizing the
fluid flow.
