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 PDEs, and how the concepts of numerical analysis could be adapted to DNNs, how to develop hybrid models incorporating both NN modules and ODE/PDE solvers? On the application side, we will focus on PDEs 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.
Keywords : Neural Networks, Differential Equations
This PhD research project has been submitted for a funding request to “Sorbonne Center for Artificial Intelligence (SCAI)”. The PhD candidate selected by the project leader will therefore participate in the project selection process (including a file and an interview) to obtain funding.