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

TL;DR

This paper introduces a deep learning-based stochastic optimization method to approximate non-stationary solutions of wave kinetic equations, validated through comparison with analytical and finite volume solutions.

Contribution

The paper presents a novel deep learning approach for solving non-stationary wave kinetic equations, demonstrating its effectiveness on both analytical and numerical benchmarks.

Findings

Accurately approximates solutions to wave kinetic equations.

Matches decay rates of total energy with theoretical predictions.

Outperforms finite volume methods in certain scenarios.

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.