Context. The interpretation of observations of atomic and molecular tracers in the galactic and extragalactic interstellar medium (ISM) requires comparisons with state-of-the-art astrophysical models to infer some physical conditions. Usually, ISM models are too timeconsuming for such inference procedures, as they call for numerous model evaluations. As a result, they are often replaced by an interpolation of a grid of precomputed models. Aims. We propose a new general method to derive faster, lighter, and more accurate approximations of the model from a grid of precomputed models for use in inference procedures. Methods. These emulators are defined with artificial neural networks (ANNs) with adapted architectures and are fitted using regression strategies instead of interpolation methods. The specificities inherent in ISM models need to be addressed to design and train adequate ANNs. Indeed, such models often predict numerous observables (e.g., line intensities) from just a few input physical parameters and can yield outliers due to numerical instabilities or physical bistabilities and multistabilities. We propose applying five strategies to address these characteristics: (1) an outlier removal procedure; (2) a clustering method that yields homogeneous subsets of lines that are simpler to predict with different ANNs; (3) a dimension reduction technique that enables us to adequately size the network architecture; (4) the physical inputs are augmented with a polynomial transform to ease the learning of nonlinearities; and (5) a dense architecture to ease the learning of simpler relations between line intensities and physical parameters. Results. We compare the proposed ANNs with four standard classes of interpolation methods, nearest-neighbor, linear, spline, and radial basis function (RBF), to emulate a representative ISM numerical model known as the Meudon PDR code. Combinations of the proposed strategies produce networks that outperform all interpolation methods in terms of accuracy by a factor of 2 in terms of the average error (reaching 4.5% on the Meudon PDR code) and a factor of 3 for the worst-case errors (33%). These networks are also 1000 times faster than accurate interpolation methods and require ten to forty times less memory. Conclusions. This work will enable efficient inferences on wide-field multiline observations of the ISM.< Réduire