A Deep Neural Network-Based Shock Capturing Method

Deep neural networks (DNN’s) have the ability to approximate extremely complicated nonlinear functions. Due to their flexibility, DNN’s present a promising solution to developing highly accurate shock-capturing methods. Because shock capturing methods, such as WENO5, often make use of complicated nonlinear functions that are hand designed to accurately predict weak solutions to hyperbolic conservation laws, we teach a DNN to approximate the best possible nonlinear function to avoid relying on human intuition in the design of the numerical scheme. We structure the model as a directed acyclic graph network to embed arbitrarily high orders of accuracy into the solution that the DNN outputs. Additionally, a convolutional layer is used to reduce the dimensionality of the network, since the derivative can be computed using only local information. We also explore training the DNN using a recurrent architecture to optimize the model weights such that long-term error is minimized and stable solutions are encouraged.