The "time-varying loudness" (TVL) model of Glasberg and Moore calculates "instantaneous loudness" every 1 ms, and this is used to generate predictions of short-term loudness, the loudness of a short segment of sound, such as a word in a sentence, and of long-term loudness, the loudness of a longer segment of sound, such as a whole sentence. The calculation of instantaneous loudness is computationally intensive and real-time implementation of the TVL model is difficult. To speed up the computation, a deep neural network (DNN) was trained to predict instantaneous loudness using a large database of speech sounds and artificial sounds (tones alone and tones in white or pink noise), with the predictions of the TVL model as a reference (providing the "correct" answer, specifically the loudness level in phons). A multilayer perceptron with three hidden layers was found to be sufficient, with more complex DNN architecture not yielding higher accuracy. After training, the deviations between the predictions of the TVL model and the predictions of the DNN were typically less than 0.5 phons, even for types of sounds that were not used for training (music, rain, animal sounds, and washing machine). The DNN calculates instantaneous loudness over 100 times more quickly than the TVL model. Possible applications of the DNN are discussed.