Physics-Informed Neural Networks for Viscous Flow Past a NACA 0012 Airfoil at Zero Angle of Attack: A Comparative Study of Standard and Curriculum Learning Training Strategies

Abstract Physics-informed neural networks (PINNs) can solve the Navier–Stokes equations without a mesh, yet at moderate Reynolds numbers they often converge to fields with a small training loss and almost no pressure. We study this problem for steady, two-dimensional, incompressible flow past a NACA0012 airfoil at 𝛼 = 0◦ and Re ∈ {100, 500, 1000, 2000, 5000, 10,000}, comparing two ways of training the same network (eight hidden layers of twenty tanh neurons, velocity–pressure formulation). The standard PINN is trained from a new Glorot initialization at each Reynolds number. The curriculum PINN is trained once along Re = 500 → 1000 → 2000 → 5000 → 10,000, with every stage starting from the weights of the previous one; numerically, this is a natural-parameter continuation in Re. At matched Reynolds numbers the curriculum network finishes with a loss 3.2 to 28.5 times lower and gives a physical pressure coefficient at all five stages, with stagnation peaks between +0.87 and +1.15. The standard network is physical only at Re = 100 and Re = 1000, and above Re = 2000 its 𝐶𝑝 is of order 10−3. The curriculum 𝐶𝑝 agrees closely with ANSYS Fluent at Re = 500, 1000, 5000 and 10,000. Inside the network, the curriculum PINN keeps more of its capacity in use (54.47% sparsity against 83.28% at Re = 1000), its deepest layer becomes more active at every stage, and its 𝑥-momentum residual is lower at every matched Reynolds number. Its neurons also become less correlated with pressure as Re increases: the mean correlation magnitude drops by 83%, from 0.407 to 0.070, while the velocity correlation stays near 0.10. We read this as the network following the thinning of the boundary layer. Taken together, the measurements explain why warm-started training works where independent training does not, rather than only showing that it does.