Finite-volume methods with dense neural networks

Abstract

Hyperbolic conservation laws are an important part in classical physics to be able to mathematically describe the actions of nature. To obtain approximate solutions of such problems, several numerical methods have been developed, most of which with both advantages and disadvantages in terms of accuracy, efficiency and implementation simplicity. Inspired by modern computer science we will in this thesis propose numerical methods based on flux approximations obtained by using dense neural networks (DNNs). We will investigate the accuracy and efficiency by performing experiments with Burgers' equation. The main result of this thesis is a proposed numerical method for approximating solutions of two-dimensional nonlinear conservation laws. As there does not exist exact solution formulas of such two-dimensional problems, a possible approach is to use fine-resolution solvers in order to properly approximate the solutions. These solvers are extremely time consuming, and the hope is that the use of pre-trained DNN models will lead to a precise and efficient numerical method. We will also explore the possibility of using a physics-informed loss-function for approximating solutions of one-dimensional conservation laws, and further discuss how this may be applied to the two-dimensional methods. The DNN based numerical methods tested in this thesis yielded promising results with respect to both accuracy and efficiency. Due to time limitations of this study we have restricted ourselves to only studying Burgers' equation with a narrow sample of parameters. Thus, some uncertainty follows with the results, and thereby uncertainty in the conclusions. However, there are strong indications that the proposed models are valuable, given the right set of parameters.