Atrial fibrillation (AF) is the most common arrhythmia and its incidence is predicted to increase. Personalized treatments for AF, such as ablation therapy, rely on sparse and noisy measurements to define the treatment strategy. The reconstruction of activation patterns from electroanatomical data remains challenging during AF due to its chaotic behavior. These activation maps can provide valuable information to optimally locate ablation lines. In cases where there is primary wave that is periodic, such as a rotor o a spiral, the problem can be recasted to estimate phase maps instead of activation maps, which will be discontinuous in this case. In this work, we aim to reconstruct phase maps of periodic electrophysiological activity from sparse measurements using physics-informed neural networks. Within this framework, we can seamlessly combine data and any physical knowledge about the system in question. In our case, the data will come from electrode recordings in contact with cardiac tissue at sparse locations, from which we will extract the phase. As physics, we recast the Eikonal equation with the phase as primary variable leading to a partial differential equation with complex numbers. We use a multi-layer perceptron that takes as input the position in the cardiac tissue and outputs the real and imaginary parts of a complex variable. The phase can be directly computed from this complex variable. We train this network to approximate the phase data and also satisfy the complex Eikonal equation for periodic activations. Our results show that we can recover the full phase map with high accuracy from sparse measurements. With the reconstructed map, we can also identify the location of phase singularities, which may be of clinical importance. This work presents the first step towards an accurate reconstruction of phase maps with physics-informed neural networks.