Solving Stokes Flow with Hybrid ML-Simulation Methods

Abstract

Technical systems are becoming more and more complicated, making simulations of those even more complex and expensive. To reduce complexity and to make evaluation faster, nowadays neural networks are often used to build reduced order models that substitute these simulations. The neural networks are typically trained with data from simulations or measurements. But with this data-driven approach some natural laws like the conservation of energy, mass and momentum are not, or only poorly considered. In this talk two different hybrid approaches which both combine physical knowledge with neural networks are examined. First, we consider physics-informed neural networks which embed the differential equations into the loss function of a neural network. Second, we present our novel hybrid approach which incorporates the residual of the finite element formulation on a discretization into the loss function of a neural network. Both methods are trained without data from simulations or measurements, but rely on the partial differential equation itself. Finally, the methods are applied to a Stokes flow and evaluated with regard to the consideration of the incompressibility condition and computational complexity.