Solving thin boundary layer problems with physics-informed machine learning inspired by perturbation theory*

Thin boundary layers commonly arise in high Reynolds number flows and high Peclet number convective transport. An accurate numerical solution to these problems requires a high-resolution mesh near the wall. In the case of high Schmidt number mass transport that commonly arises in biotransport problems, the extremely thin nature of these boundary layers makes traditional numerical solution a tedious task. Modern scientific computing approaches such as physics-informed neural networks (PINN) provide an alternative mesh-free strategy. However, PINN cannot resolve the sharp gradients in thin boundary layers. In this talk, we present a boundary-layer PINN (BL-PINN) approach inspired by the classical perturbation theory. We demonstrate how PINN architecture could be designed in a theory-guided fashion to replicate the singular perturbation theory using asymptotic expansions. In benchmark convective transport problems, we demonstrate that BL-PINN can solve thin boundary layer problems with good accuracy. We also discuss parametric re-evaluation of the solution in BL-PINN without the need for retraining. Finally, we leverage the hybrid data-driven and physics-based framework offered by PINN to demonstrate the utility of BL-PINN for solving inverse problems in boundary layers.