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

Deep learning-based numerical schemes such as Physically Informed Neural Networks (PINNs) have recently emerged an alternative to classical 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 high approximation capabilities. However, when the PDE solutions low regular, expert insight required build deep learning formulations that do not incur in variational crimes. Optimization solvers also significantly challenged, can potentially spoil final quality approximated solution due convergence bad local minima, generalization In this paper, we present exhaustive study merits limitations these exhibit low-regularity, compare performance with respect more benign cases smooth. As a support our study, consider singularly perturbed convection-diffusion problems where regularity typically degrades certain multiscale parameters go zero.