Print Email Facebook Twitter Solving Partial Differential Equations with Neural Networks Title Solving Partial Differential Equations with Neural Networks Author van der Meer, Remco (TU Delft Electrical Engineering, Mathematics and Computer Science) Contributor Oosterlee, Kees (mentor) Borovykh, Anastasia (mentor) Degree granting institution Delft University of Technology Programme Applied Mathematics Date 2019-07-01 Abstract Recent works have shown that neural networks can be employed to solve partial differential equations, bringing rise to the framework of physics informed neural networks.The aim of this project is to gain a deeper understanding of these novel methods, and to use these insights to further improve them. We show that solving a partial differential equation can be formulated as a multi-objective optimization problem, and use this formulation to propose several modifications to existing methods. These modifications manifest as a scaling parameter, which can improve the accuracy by orders of magnitude for certain problems when it is chosen properly. We also propose heuristic methods to approximate the optimal scaling parameter, which can be used to eliminate the need to optimize this parameter. Our proposed methods are tested on a variety of partial differential equations and compared to existing methods. These partial differential equations include the Laplace equation, which we solve in up to four dimensions, the convection-diffucsion equation and the Helmholtz equation, all of which show that our proposed modifications lead to enhanced accuracy. Subject Partial Differential EquationsNeural NetworksDeep Learningnumerical methods To reference this document use: http://resolver.tudelft.nl/uuid:c77e1bcc-7212-4234-af34-6586b628ab1c Part of collection Student theses Document type master thesis Rights © 2019 Remco van der Meer Files PDF Thesis.pdf 3.81 MB Close viewer /islandora/object/uuid:c77e1bcc-7212-4234-af34-6586b628ab1c/datastream/OBJ/view