Classical reservoir simulators are built upon different underlying models: geological models integrating all the knowledge of the subsurface properties, fluid flow models integrating reservoir fluid physical properties, wells, and surface installations. The construction of such models however is very time and resources consuming. In the case of mature fields, where historical production data are available, data driven models can represent a suitable alternative or can be complementary to classical reservoir modelling as they require much less computation time and allocated resources. Among such models are Capacitance Resistive Models (CRMs), based on set of coupled ordinary differential equations representing material balance. These aim to predict flow rate in a reservoir using only dynamic data of production rates, water injections and Bottom Hole Pressure (BHP). In addition, CRMs can explain the underlying connectivity between several injectors and producers that could be a valuable information for dynamic synthesis and for better understanding of fluid flows in the reservoir. Most of the current work on CRMs optimizes a nonlinear multivariate regression of the CRM's parameters. Such optimization needs a closed form solution of the CRM ODEs. which is only possible under conditions: constant or linear variation in injection or in BHP. Once we have optimized the CRM's parameters, we can use the closed form solution to perform forecasting. The aim of this work is to propose a complete approach to optimize the CRM's parameters and forecast future production. This approach is not based on any assumptions on injections or on producers' BHP. To this end, we introduce a new approach based on a deep learning strategy: Physics-Informed Neural Networks (PINNs) for CRMs. This paper is organized as follows: first we introduce the related work on CRMs. Second, we detail the theory of CRMs and PINNs. Our approach, called Deep-CRMs, is presented in the third section. We focus on the mathematical description of Deep-CRMs and show experiments in order to compare our approach to the nonlinear multivariate regression of the closed form solution. These experiments are based on two datasets: the first is a synthetic dataset generated using ECLIPSE® and SISMAGE®, and the second is a real field dataset provided by one of our affiliates.