A Deep Learning Approximation of Non-Stationary Solutions to Wave Kinetic Equations

Abstract

We present a deep learning approximation, stochastic optimization based, method for wave kinetic equations. To build confidence in our approach, we apply the method to a Smoluchowski coagulation equation with multiplicative kernel for which an analytic solution exists. Our deep learning approach is then used to approximate the non-stationary solution to a 3-wave kinetic equation corresponding to acoustic wave systems. To validate the neural network approximation, we compare the decay rate of the total energy with previously obtained theoretical results. A finite volume solution is presented and compared with the present method.