A Physics-Informed, Deep Double Reservoir Network for Forecasting Boundary Layer Velocity

When a fluid flows over a solid surface, it creates a thin boundary layer where the flow velocity is influenced by the surface through viscosity, and can transition from laminar to turbulent at sufficiently high speeds. Understanding and forecasting the wind dynamics under these conditions is one of the most challenging scientific problems in fluid dynamics. It is therefore of high interest to formulate models able to capture the nonlinear spatio-temporal velocity structure as well as produce forecasts in a computationally efficient manner. Traditional statistical approaches are limited in their ability to produce timely forecasts of complex, nonlinear spatio-temporal structures which are at the same time able to incorporate the underlying flow physics. In this work, we propose a model to accurately forecast boundary layer velocities with a deep double reservoir computing network which is capable of capturing the complex, nonlinear dynamics of the boundary layer while at the same time incorporating physical constraints via a penalty obtained by a Partial Differential Equation (PDE). Simulation studies on a one-dimensional viscous fluid demonstrate how the proposed model is able to produce accurate forecasts while simultaneously accounting for energy loss. The application focuses on boundary layer data on a wind tunnel with a PDE penalty derived from an appropriate simplification of the Navier-Stokes equations, showing forecasts more compliant with mass conservation.