This paper is dedicated to control theoretically explainable application of autoencoders to optimal fault detection in nonlinear dynamic systems. Autoencoder-based learning is a standard machine learning method and widely applied for fault (anomaly) detection and classification. In the context of representation learning, the so-called latent (hidden) variable plays an important role towards an optimal fault detection. In ideal case, the latent variable should be a minimal sufficient statistic. The existing autoencoder-based fault detection schemes are mainly application-oriented, and few efforts have been devoted to optimal autoencoder-based fault detection and explainable applications. The main objective of our work is to establish a framework for learning autoencoder-based optimal fault detection in nonlinear dynamic systems. To this aim, a process model form for dynamic systems is firstly introduced with the aid of control theory, which also leads to a clear system interpretation of the latent variable. The major efforts are made on the development of a control theoretic solution to the optimal fault detection problem, in which an analog concept to minimal sufficient statistic, the so-called lossless information compression, is introduced and proven for dynamic systems and fault detection specifications. In particular, the existence conditions for such a latent variable are derived, based on which a loss function and further a learning algorithm are developed. This learning algorithm enables optimally training of autoencoders to achieve an optimal fault detection in nonlinear dynamic systems. A case study on three-tank system is given at the end of this paper to illustrate the capability of the proposed autoencoder-based fault detection and to explain the essential role of the latent variable in the proposed fault detection system.