Flow-control techniques are extensively studied in fluid mechanics, as a means to reduce energy losses related to friction, both in fully-developed and spatially-developing flows. These techniques typically rely on closed-loop control systems that require an accurate representation of the state of the flow to compute the actuation. Such representation is generally difficult to obtain without perturbing the flow. For this reason, in this work we propose a fully-convolutional neural-network (FCN) model trained on direct-numerical-simulation (DNS) data to predict the instantaneous state of the flow at different wall-normal locations using quantities measured at the wall. Our model can take as input the heat-flux field at the wall from a passive scalar with Prandtl number Pr = ν/α = 6 (where ν is the kinematic viscosity and α is the thermal diffusivity of the scalar quantity). The heat flux can be accurately measured also in experimental settings, paving the way for the implementation of a non-intrusive sensing of the flow in practical applications.