In this work a reduced order model based on adaptive finite element meshes and a correction term obtainedby using an artificial neural network (FAN-ROM) is presented. The idea is to run a high-fidelity simulationby using an adaptively refined finite element mesh, and compare the results obtained with those of a coarsemesh finite element model. From this comparison, a correction forcing term can be computed for eachtraining configuration. A model for the correction term is built by using an artificial neural network, andthe final reduced order model is obtained by putting togetherthe coarse mesh finite element model, plus theartificial neural network model for the correction forcing term.The methodology is applied to non-linear solid mechanics problems, transient quasi-incompressible flows,and a fluid-structure interaction problem. The results of the numerical examples show that the FAN-ROMis capable of improving the simulation results obtained in coarse finite element meshes at a reducedcomputational cost.