Machine Learning for Partial Differential Equations: Leveraging Neural Networks and Supercomputing

Partial Differential Equations are a fundamental way of modeling phenomena in fluid dynamics, with nonlinear Burger’s equation serving as a key modeling tool for studying nonlinear wave propagation, shock formations, and turbulence. In this research we investigate Deep Neural Networks (DNNs) and Deep Operator Networks (DeepONets) to train Burger’s Equation. One of the main challenges in using NN to predict the numerical solution is to obtain a domain independent neural solver. The objective of this study is to investigate the effect of increasing the depth of the DNN, training data size, and range of initial conditions required to achieve optimal performance. The training data consists of CFD simulation of the Burger’s equation for different values of viscosity, different domain sizes and different initial conditions, for given boundary conditions.