Aims. We investigate whether neural networks (NNs) can accurately differentiate between growth-rate data of the large-scale structure (LSS) of the Universe simulated via two models: a cosmological constant and Λ cold dark matter (CDM) model and a tomographic coupled dark energy (CDE) model.Methods. We built an NN classifier and tested its accuracy in distinguishing between cosmological models. For our dataset, we generated fσ8(z) growth-rate observables that simulate a realistic Stage IV galaxy survey-like setup for both ΛCDM and a tomographic CDE model for various values of the model parameters. We then optimised and trained our NN with Optuna, aiming to avoid overfitting and to maximise the accuracy of the trained model. We conducted our analysis for both a binary classification, comparing between ΛCDM and a CDE model where only one tomographic coupling bin is activated, and a multi-class classification scenario where all the models are combined.Results. For the case of binary classification, we find that our NN can confidently (with > 86% accuracy) detect non-zero values of the tomographic coupling regardless of the redshift range at which coupling is activated and, at a 100% confidence level, detect the ΛCDM model. For the multi-class classification task, we find that the NN performs adequately well at distinguishing ΛCDM, a CDE model with low-redshift coupling, and a model with high-redshift coupling, with 99%, 79%, and 84% accuracy, respectively.Conclusions. By leveraging the power of machine learning, our pipeline can be a useful tool for analysing growth-rate data and maximising the potential of current surveys to probe for deviations from general relativity.