Weak signal detection is a significant problem in modern detection such as mechanical fault diagnosis. The uniqueness of chaos and good learning ability of neural networks provide new ideas and framework for weak signal detection field. In this paper, Elman neural network is applied to detect and recover weak pulse signal in chaotic noise. For detection problem of weak pulse signal under chaotic noise, based on short-term predictability of chaotic observations, phase space reconstruction for observed signals is carried out. And Elman deep learning adaptive detection model (EDAD model) is established for weak pulse signal detection, and a hypothesis test is used to detect weak pulse signal from the prediction error. For the recovery of weak pulse signal under chaotic noise, a double-layer Elman deep neural network recovery model (DEDR model) is proposed, which is based on the Elman deep learning network model and single-point jump model for weak pulse signal, and it is optimized with goal of minimizing mean square prediction error of the Elman model. The profile least squares method is applied to estimate parameters of the DEDR model for difficult recovery of weak pulse signal because the DEDR model is essentially a semiparametric model with parametric and nonparametric parts. In the end, simulation experiments show that the model built in this paper can effectively detect and recover weak pulse signal in the background of chaotic noise.