This paper presents a adaptive dynamic programming-based fault detection and isolation (FDI) scheme to detect and isolate faults in an aircraft jet engine. To this end, the weights in Actor-Critic neural networks are first tuned to learn the input-output map of the jet engine considering its multiple working modes. The convergences of the trainings in Critic-Actor neural networks are strictly proved without knowing the drift dynamics and the input dynamics in the presence of unknown nonlinearities and approximation errors. Using the residuals that are generated by measuring the difference of each network output and the measured engine output, various criteria are established for accomplishing the fault diagnosis task, that addresses the problem of fault detection and isolation of the system components. A number of simulation studies are carried out for combustion chamber of a single-spool jet engine to demonstrate and illustrate the advantages, capabilities, and performance of our proposed fault diagnosis scheme.