Calculating N-body simulations has been an extremely time and resource consuming process for researchers. There have been many schemes trying to approximate the correct positions of celestial objects of simulations. In this paper, we propose using Neural Networks in the physical and in the Fourier domain in order to correct a Particle Mesh scheme, primarily in the smaller scales i.e., the smaller details of the N-body simulations. In addition, we used a recently proposed in the literature technique to train our models i.e. through an Ordinary Differential Equations (ODE) solver. We present our promising results of the different types of Neural Networks that we experimented with.