Physics-informed neural networks with trainable sinusoidal activation functions for approximating the solutions of the Navier-Stokes equations

We present TSA-PINN, a novel Physics-Informed Neural Network (PINN) that leverages a Trainable Sinusoidal Activation (TSA) mechanism to approximate solutions to the Navier-Stokes equations. By incorporating neuron-wise sinusoidal activation functions with trainable frequencies and a dynamic slope recovery mechanism, TSA-PINN achieves superior accuracy and convergence. Its ability to dynamically adjust activation frequencies enables efficient modeling of complex fluid behaviors, reducing training time and computational cost. Our testing goes beyond canonical problems, to study less-explored and more challenging scenarios, which have typically posed difficulties for prior models. Various numerical tests underscore the efficacy of the TSA-PINN model across five different scenarios. These include steady-state two-dimensional flows in a lid-driven cavity at two different Reynolds numbers; a cylinder wake problem characterized by oscillatory fluid behavior; and two time-dependent three-dimensional turbulent flow cases. In the turbulent cases, the focus is on detailed near-wall phenomena—including the viscous sub-layer, buffer layer, and log-law region—as well as the complex interactions among eddies of various scales. Both numerical and quantitative analyses demonstrate that TSA-PINN offers substantial improvements over conventional PINN models. This research advances physics-informed machine learning, setting a new benchmark for modeling dynamic systems in scientific computing and engineering.