Machine Learning-Based Troubled-Cell Indicators for RKDG Methods

Abstract

Solving partial differential equations (PDEs) numerically is an ongoing challenge, especially given the complicated PDEs that arise from scientific applications. Hyperbolic conservation laws are a specific type of PDEs arising from physical situations where a quantity such as mass, momentum, or energy is conserved in a fixed volume. However, the solutions to this class of PDEs often develop discontinuities as time evolves. These discontinuities often cause spurious oscillations in the numerical solution, reducing the solver's accuracy. To eliminate spurious oscillations, shock-capturing methods identify the location of discontinuities, labeling them as troubled cells, and smooth the solution in those cells. For this thesis, troubled-cell indicators are examined in the context of the Runge-Kutta Discontinuous Galerkin method for hyperbolic conservation laws. Unfortunately, many existing troubled-cell indicators rely on problem-dependent parameters that do not generalize across different initial conditions, conservation laws, or degrees of the solution. Therefore, the goal of this thesis is to compare the performance of machine-learning based methods, which are free of problem-dependent parameters, to a selection of existing troubled-cell indicators in a variety of one-dimensional cases. This thesis will discuss the use of support vector machines (SVMs) and decision trees as alternatives to traditional troubled-cell indicators and neural networks (created by Ray and Hesthaven, for example). While neural networks have been successful, their complicated nature inhibits interpretation of the troubled cell decision function. We show that SVMs are competitive with other troubled cell indicators in a variety of conservation law examples and analyze the SVM in comparison to the neural networks of Ray and Hesthaven.