Studies on the use of neural networks as surrogate models for partial differential equations

• Grant number: 19/04018-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Effective date (Start): May 01, 2019
• Effective date (End): December 31, 2019
• Field of knowledge: Interdisciplinary Subjects
• Principal Investigator: Renato Fernandes Cantão
• Grantee: Bruno Augusto Veloso Nunes Silva
• Host Institution: Centro de Ciências e Tecnologias para a Sustentabilidade (CCTS). Universidade Federal de São Carlos (UFSCAR). Sorocaba , SP, Brazil

Abstract

In this work we propose to assess the application of deep neural networks as surrogate models for traditional differential models. Neural networks learning process will be guided not only by a training set -- as usual -- but also by adding constraints imposing characteristics from the phenomenon being modelled. The training set will be formed from samples coming from solutions of model problems, analytical when available or high fidelity numerical ones obtained with the aid of classical methods. This methodology will be tested with linear advection and Burgers' equations. The approximating capabilities of the neural networks will be assessed in terms of their deepness, number of neurons, activation function choice and size of the training sets. Variations on the initial and boundary conditions and in the spatiotemporal domain will also be evaluated. Comparisons will be made with high fidelity solutions obtained through standard numerical methods.