Elliptic partial differential equations (PDEs) are common in many areas of physics, from the Poisson equation in plasmas and incompressible flows to the Helmholtz equation in electromagnetism. Their numerical solution requires to solve linear systems which can become a bottleneck in terms of performance. The rise of computational power and inherent speed of GPUs offers exciting opportunities to solve PDEs by recasting them in terms of optimization problems. In plasma fluid simulations, the Poisson equation is solved, coupled to the charged species transport equations. We introduce PlasmaNet (https://gitlab.com/cerfacs/plasmanet), an open-source library written to study neural networks in plasma simulations. Previous work using PlasmaNet has shown significant speedup using neural networks to solve the Poisson equation compared to classical linear system solvers on this problem. Results also showed that coupling the neural network Poisson solver to plasma transport equations is a viable option in terms of accuracy. In this work, we attempt to solve a new class of elliptic differential equations, the screened Poisson equations using neural networks. These equations are used to infer the photoionization source term from the ionization rate in streamer discharges. The same methodology as that adopted for the Poisson equation is followed. A simulation running with three neural networks, one to solve the Poisson equation and two to solve the photoionization equations, yields accurate results, extending the range of applicability of the method developed previously.