Gradient-free physics-informed neural networks (GF-PINNs) for vortex shedding prediction in flow past square cylinders

Physics-informed neural networks (PINNs) face significant challenges to predict the vortex shedding in the flow past a two-dimensional cylinder, mainly due to complex loss landscapes, spectral bias, and a lack of inductive bias towards periodic functions. To overcome these challenges, a novel gradient-free PINN (GF-PINN) coupled with a U-Net+ + architecture is proposed. For optimizing the complex loss landscape, the skip pathways in U-Net+ + are redesigned to reduce the semantic gap between encoder and decoder feature maps. Then, the stream function instead of velocity, is used as the input and output for the neural network, ensuring flow incompressibility and reducing output dimensionality. This approach aims to overcome the inherent problems of spectral bias and the lack of inductive bias towards periodic functions in PINNs. Furthermore, gradient-free convolutional filters are employed to approximate the derivative terms in the loss function to further optimize the complex loss landscape. A series of numerical experiments and dynamic mode analyses are conducted and the results show that the vortex shedding in the wake of a square cylinder is successfully captured by the proposed model and the estimated drag coefficients and Strouhal numbers are in a good agreement with those predicted by traditional methods. In addition, numerical experiments also show that the model exhibits great capabilities of generalization and extrapolation. This work demonstrates the potential of PINN-based models to effectively solve complex fluid dynamics problems.