Using physics-informed derivative networks to solve the forward problem of a free-convective boundary layer problem - Kaiwei Cong; Guangjin Li; Yifan Sun; Hairui Ren (Scientific Reports)
Abstract Physics-informed neural networks (PINNs) have become powerful tools for solving various nonlinear differential equations. Although several PINN-based approaches have been widely applied to some types of boundary layer problem, certain complex parameter settings or boundary conditions can still lead to training failures, regardless of whether shallow or deep networks are used. In this paper, we apply physics-informed derivative networks (PIDNs), a simple variant of PINNs to solve a classical free-convective boundary layer problem characterized by intricate parameter setups and boundary conditions. This problem is governed by a coupled system of ordinary differential equations (ODEs), derived from Kuiken’s similarity transformation. Experimental results show that PIDNs can consistently solve this ODE system with only a small, shallow network, whereas traditional PINNs cannot achieve this under the same configurations. The numerical results confirm that PIDNs produce solutions that closely match established numerical methods across all parameter settings tested and, in some cases, outperform certain analytical approaches. Notably, the model converges without relying on any known solutions beyond boundary and initial conditions during training, which is often challenging within the PINNs framework.