Adjoint-based Training of Embedded Neural-Network Models for Particle-laden Flow*

Machine learning methods are attractive for representing difficult to describe physics within overall simulation. We consider, in particular, particles in turbulence, and present an approach in which the closure terms added are optimized in a way that is fully coupled with the physics represented in the resolved governing equations. The adjoint of the full system---the combined neural network and governing equations---is solved to provide the sensitivity of flow predictions to the network weights. In this sense it fully includes the known physics. This formulation is then demonstrated for particles in model flows and in turbulence. The benefits of this approach, such as the extrapolative robustness, are discussed along with the challenges, the principal challenge being the added complexity in the training process needing solution of adjoint governing equations. More advanced automatic differentiation methods promise to aid this primary challenge.