Solving forward problems of boundary-layer problems in ordinary differential equations via physics-informed derivative networks

In this paper, we employ a deep learning approach based on the physics-informed derivative network (PIDN) to solve forward problems of boundary layer ordinary differential equations, including the classical Blasius and Falkner–Skan equations as well as other parameter-sensitive cases. Despite using a relatively small network with only a few neurons, the data-driven solutions generated by PIDN closely match those obtained through traditional numerical methods. Compared to conventional physics-informed neural networks (PINNs), which often rely on large-scale networks, our approach significantly improves both training stability and computational efficiency with a minimal network. Moreover, evaluations across a wide range of parameter settings and interval lengths demonstrate that PIDN remains highly robust, even on extensive dimensionless intervals. Overall, these findings confirm the remarkable effectiveness and stability of the PIDN method for solving a variety of complex boundary layer problems.