The digital twin of optical systems can imitate its response to outer environments through connecting outputs from data-driven optical element models with numerical simulation methods, which could be used for system design, test and troubleshooting. Data-driven optical element models are essential blocks in digital twins. It can not only transform data obtained from sensors in real optical systems to states of optical elements in digital twins, but also simulate behaviors of optical elements with real measurements as prior conditions. For ground based optical telescopes, the digital twin of atmospheric turbulence phase screens is an important block to be developed. The digital twin of atmospheric turbulence phase screens should be able to generate phase screens with infinite length and high similarities to real measurements. In this paper, we propose a novel method to build the digital twin of atmospheric turbulence phase screens. Our method uses two deep neural networks to learn mapping functions between the space of parameters and the space of phase screens and vice versa. Meanwhile, a forecasting deep neural network is proposed to generate parameters for the next phase screen according to parameters extracted from a previous phase screen. The method proposed in this paper could be used to directly produce phase screens with infinite length and of any temporal or spatial power spectral density that follows statistical distributions of real measurements, which makes it an appropriate block in digital twins of ground based optical systems.