Differential equations form one of the bedrock of scientific computing, while neural networks have emerged as the preferred tool of modern machine learning. These two methods are not only closely related to each other but also offer complementary strengths: the modelling power and interpretability of differential equations, and the approximation and generalization power of deep neural networks. The objective of the thesis project is to develop links between Deep Neural Networks (DNN) and Differential Equations (DE) in order to start answering central questions like: How could DNNs be used to solve DEs? How the concepts of numerical analysis could be adapted to DNNs? How to develop hybrid models incorporating both DNN modules and partial/ordinary DE solvers? On the application side, we will focus on partial DEs arising from environmental applications. They often model transport phenomena, difficult to solve and to analyze due to their sequential nature and to the high dimension of the solution space. 

PhD student: Léon Migus 

PhD supervisor: Julien Salomon 

Research laboratory: LJLL - Laboratoire Jacques-Louis Lions