This paper presents a novel approach to improve the Wilcox k-ω turbulence model by using Physics Informed Neural Networks (PINN) and Neural Networks (NN) to correct the underprediction of turbulent kinetic energy common in two-equation models. The key idea is to modify the turbulent diffusion term in the k-equation by introducing a spatially varying turbulent Prandtl number σk, obtained by solving an ordinary differential equation for the turbulent viscosity using PINN with DNS data from channel flow at Reτ=5200. The resulting σk is then used to compute new damping functions Ck and Cω2 for the dissipation and destruction terms, respectively, ensuring that the model retains the correct turbulent viscosity and mean velocity predictions. To make the model applicable to arbitrary flows, three NN models are trained to predict σk, Ck, and Cω2 as functions of two local, non-dimensional parameters: the total stress normalized by friction velocity squared and the turbulent viscosity normalized by wall distance and friction velocity.

The new k-ω-PINN-NN model is validated on fully-developed channel flows at Reynolds numbers from 550 to 10,000, showing excellent agreement with DNS for mean velocity, skin friction, and turbulent kinetic energy. It also performs well on a flat-plate boundary layer at Reθ=4500, though with a slight overprediction of k. The model's generalization is further tested on the complex separated flow over a periodic hill at Re=5600, where it yields very good predictions of mean velocity and turbulent kinetic energy compared to DNS. The paper also demonstrates that the NN models can be replaced with symbolic regression (pySR) to obtain explicit algebraic expressions, facilitating implementation in commercial CFD codes. All Python scripts are made available, promoting reproducibility and further development.