Turbulence modeling using physics-informed neural networks: The importance of training points distribution and hyper parameters

Abstract

Physics-informed neural networks (PINNs) are employed to solve the Navier–Stokes equations with a standard k–ω turbulence model for flow over a period, focusing on how the spatial placement of training points and hyperparameters affect accuracy. Wall-bounded turbulence modeling includes challenges because variables like the specific dissipation rate (ω) exhibit high near-wall gradients. To address this, the research was conducted in three phases. First, learning rate (LR) and optimizer iterations were studied, and analysis revealed that increasing iterations does not necessarily improve prediction accuracy, and specific LR scheduling is required for improved accuracy. Second, various network capacities and sampling methodologies of random, shear-aligned, and convective were analyzed simultaneously, and revealed that random and convective sampling fail to capture near-wall turbulence characteristics, causing considerable predictive errors in ω. While shear-aligned point selection achieves lower ω error, it requires prior knowledge of the flow field. Third, a novel multi-objective Monte Carlo active learning (MOMCAL) algorithm was developed to address this issue. It evaluates spatial gradients, partial differential equation residuals, and current errors to semi-stochastically select new training points. By applying MOMCAL, the neural network easily learned the highly sensitive ω field. It reduced the ω prediction error considerably. The generalizability of MOMCAL is confirmed on two additional geometries, achieving similar reductions in ω error, and comparisons with existing adaptive refinement methods demonstrate its superior performance. Ultimately, this work demonstrates that intelligent LR scheduling combined with physics-aware point selection is essential for building reliable, high-precision PINNs turbulence models.