Aims: We introduce a new deep-learning approach for the reconstruction of 3D dust density and temperature distributions from multi-wavelength dust emission observations on the scale of individual star-forming cloud cores (<0.2pc). Methods: We construct a training data set by processing cloud cores from the Cloud Factory simulations with the POLARIS radiative transfer code to produce synthetic dust emission observations at 23 wavelengths between 12 and 1300 \(\mu\)m. We simplify the task by reconstructing the cloud structure along individual lines of sight and train a conditional invertible neural network (cINN) for this purpose. The cINN belongs to the group of normalising flow methods and is able to predict full posterior distributions for the target dust properties. We test different cINN setups, ranging from a scenario that includes all 23 wavelengths down to a more realistically limited case with observations at only seven wavelengths. We evaluate the predictive performance of these models on synthetic test data. Results: We report an excellent reconstruction performance for the 23-wavelengths cINN model, achieving median absolute relative errors of about 1.8% in \(\log(n/m^{-3})\) and 1% in \(\log(T_{dust}/K)\), respectively. We identify trends towards overestimation at the low end of the density range and towards underestimation at the high end of both the density and temperature values, which may be related to a bias in the training data. Limiting our coverage to a combination of only seven wavelengths, we still find a satisfactory performance with average absolute relative errors of about 3.3% and 2.5% in \(\log(n/m^{-3})\) and \(\log(T_{dust}/K)\). Conclusions: This proof-of-concept study shows that the cINN-based approach for 3D reconstruction of dust density and temperature is very promising and even compatible with a more realistically constrained wavelength coverage.