Employing Machine Learning in the Study of Differential Equations Related to Nuclear Engineering
DescriptionDifferential equations model a variety of phenomena in sciences and engineering. The main focus of this project is on time-dependent partial differential equations known as conservation laws that are used to model multi-phase flows. In particular, these equations are of interest to engineers who design nuclear reactor pipes to ensure the optimal flow of gasses and liquids. Most of differential equations that describe the real-world phenomena cannot be solved exactly and one has to utilize an approximation method to obtain an approximate solution.
Our project's emphasis is on relatively recent methods for solving differential equations that utilize machine learning and, in particular, neural networks, motivated by the universal approximation theorem. We review two such approaches. In the first approach a trial solution is proposed consisting of two parts – the first part, satisfying the initial/boundary conditions, with no adjustable parameters, and the second part consists of a neural network that is trained to satisfy the differential equation by minimizing the squared error loss. In the second approach, the solution is approximated by a deep neural network, derived using automatic differentiation. This results in a physics informed neural network. Within the two approaches, the common neural network parameters are learned by minimizing the mean squared error loss consisting of two terms – the first term corresponding to the initial/boundary conditions and the second term corresponding to the governing differential equation. The goal of this project is to utilize machine/deep learning methods in study of conservation laws that model multi-phase flows.