Reconstruction of Audio Signal from its Absolute Spectrogram
One thing commonly done (for example in the source separation community) is to use the phase data of the original signal (before transformation where applied to it) - the result is much better than null or random phase, and not so far from algorithms aiming at reconstructing the phase information from scratch.

A classic reconstruction algorithm is Griﬃn&amp;Lim's, described in the paper "Signal estimation from modiﬁed short-time Fourier transform". This is an iterative algorithm, each iteration requires a full STFT / inverse STFT, which makes it quite costly.

This problem is indeed an active area of research, a search for STFT + reconstruction + magnitude will yield plenty of papers aiming at improving on Griffin&amp;Lim in terms of signal quality and/or computational efficiency.