Benchmarking PDE problems via machine learning for enhanced computational modeling

The rapid advancement of mathematical modeling and scientific computing has become instrumental in addressing complex challenges across diverse fields. This proposed project seeks to leverage the interdisciplinary platform provided by the CEMRACS (Centre d’Été Mathématique de Recherche Avancée en Calcul Scientifique) summer school to contribute to the cutting-edge landscape of mathematical research and its practical applications. In the context of the 2023 CEMRACS summer school, this project aims to delve into scientific machine learning. By synergizing insights from partial differential equations (PDE) and state of the art machine learning techniques, our project seeks to provide a benchmark for some PDE problems using multiple learning techniques. We begin by considering the conventional Poisson equation and subsequently adapt it to cater to geosciences scenarios like reservoir simulation. We would like to extend our sincere gratitude to IFP Energies Nouvelles for their sponsorship, and to Thibault Faney for his supervising which has played a pivotal role in enabling the realization of this project.