Probability density forecast offers the whole distributions of forecasting targets, which brings greater flexibility and practicability than the other probabilistic forecast models such as prediction interval (PI) and quantile forecast. However, existing density forecast models have introduced various constraints on forecasted distributions, which has limited their ability to approximate real distributions and may result in suboptimality. In this paper, a distribution-free density forecast model based on deep learning is proposed, in which the real cumulative density functions (CDFs) of forecasting target are approximated by a large-capacity positive-weighted deep neural network (NN). Benefiting from the universal approximation ability of NNs, the range of forecasted distributions has been proven to contain all the distributions with continuous CDFs, which is superior to existing models' considering both width and accordance with reality. Three tests from different scenarios were implemented for evaluation, i.e., very-short-term wind power, wind speed, and day-ahead electricity price forecast, in which the proposed density forecast model has shown superior performance over the state of the art.