Traditional methods for uncertainty quantification (UQ) struggle with the curse of dimensionality when dealing with high-dimensional problems. One approach to address this challenge is to leverage the potent approximation capabilities of deep neural networks (DNNs). However, conventional DNNs often demand a substantial amount of high-fidelity (HF) training data to ensure precise predictions. Unfortunately, the availability of such data is limited due to computational or experimental constraints, primarily driven by associated costs. To mitigate these training expenses, this research introduces multi-fidelity deep neural networks (MF-DNNs), wherein a sub-network is constructed to simultaneously capture both linear and non-linear correlations between HF- and low-fidelity (LF) data. The efficacy of MF-DNNs is initially demonstrated by accurately approximating diverse benchmark functions. Subsequently, the developed MF-DNNs are employed for the first time to simulate the aleatory uncertainty propagation in 1-, 32-, and 100-dimensional contexts, considering either uniform or Gaussian distributions of input uncertainties. The UQ results affirm that MF-DNNs adeptly predict probability density distributions of quantities of interest (QoI) and their statistical moments without significant compromise of accuracy. Furthermore, MF-DNNs are applied to model the physical flow inside an aircraft propulsion system while accounting for aleatory uncertainties originating from experimental measurement errors. The distributions of isentropic Mach number are accurately predicted by MF-DNNs based on the 2D Euler flow field and few experimental data points. In conclusion, the proposed MF-DNN framework exhibits significant promise in addressing UQ and robust optimization challenges in practical engineering applications, particularly when dealing with multi-fidelity data sources.