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.

翻译：当流体流经固体表面时，会形成一个薄边界层，其中流速通过粘性受表面影响，并在足够高的速度下从层流转变为湍流。理解并预测这些条件下的风动力学是流体力学中最具挑战性的科学问题之一。因此，构建能够捕捉非线性时空速度结构并以计算高效的方式进行预测的模型具有高度重要性。传统的统计方法在及时预测复杂、非线性时空结构方面能力有限，且难以同时结合底层流动物理。在本研究中，我们提出了一种模型，用于准确预测边界层速度，该模型采用深度双储层计算网络，能够捕捉边界层的复杂非线性动力学，同时通过偏微分方程（PDE）获得的惩罚项来引入物理约束。针对一维粘性流体的模拟研究表明，所提出的模型能够在准确预测的同时考虑能量损失。应用重点在于风洞中的边界层数据，使用基于纳维-斯托克斯方程适当简化推导出的PDE惩罚项，结果显示预测更符合质量守恒。