Deep learning-based schemes for singularly perturbed convection-diffusion problems^*

¹ Fédération de Mathématiques, CentraleSupélec FR3487 CNRS, Paris-Sacla University, France
² Ecole des Ponts ParisTech & MATHERIALS INRIA team-project, Marne-la-Vallée, France
³ Cermics, École des Ponts, INRIA, Marne-la-Vallée, France
⁴ COMMEDIA, INRIA Paris, France & LJLL, Sorbonne Université
⁵ Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB, Eindhoven, Netherlands
⁶ LJLL, Sorbonne Université, Paris, France

Abstract

Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged as an alternative to classical numerical schemes for solving Partial Differential Equations (PDEs). They are very appealing at first sight because implementing vanilla versions of PINNs based on strong residual forms is easy, and neural networks offer very high approximation capabilities. However, when the PDE solutions are low regular, an expert insight is required to build deep learning formulations that do not incur in variational crimes. Optimization solvers are also significantly challenged, and can potentially spoil the final quality of the approximated solution due to the convergence to bad local minima, and bad generalization capabilities. In this paper, we present an exhaustive numerical study of the merits and limitations of these schemes when solutions exhibit low-regularity, and compare performance with respect to more benign cases when solutions are very smooth. As a support for our study, we consider singularly perturbed convection-diffusion problems where the regularity of solutions typically degrades as certain multiscale parameters go to zero.