Finite Rate of Innovation (FRI) sampling theory enables reconstruction of classes of continuous non-bandlimited signals that have a small number of free parameters from their low-rate discrete samples. This task is often translated into a spectral estimation problem that is solved using methods involving estimating signal subspaces, which tend to break down at a certain peak signal-to-noise ratio (PSNR). To avoid this breakdown, we consider alternative approaches that make use of information from labelled data. We propose two model-based learning methods, including deep unfolding the denoising process in spectral estimation, and constructing an encoder-decoder deep neural network that models the acquisition process. Simulation results of both learning algorithms indicate significant improvements of the breakdown PSNR over classical subspace-based methods. While the deep unfolded network achieves similar performance as the classical FRI techniques and outperforms the encoder-decoder network in the low noise regimes, the latter allows to reconstruct the FRI signal even when the sampling kernel is unknown. We also achieve competitive results in detecting pulses from in vivo calcium imaging data in terms of true positive and false positive rate while providing more precise estimations.