  Given a finite and noisy dataset generated with a closed-form mathematical
model, when is it possible to learn the true generating model from the data
alone? This is the question we investigate here. We show that this
model-learning problem displays a transition from a low-noise phase in which
the true model can be learned, to a phase in which the observation noise is too
high for the true model to be learned by any method. Both in the low-noise
phase and in the high-noise phase, probabilistic model selection leads to
optimal generalization to unseen data. This is in contrast to standard machine
learning approaches, including artificial neural networks, which in this
particular problem are limited, in the low-noise phase, by their ability to
interpolate. In the transition region between the learnable and unlearnable
phases, generalization is hard for all approaches including probabilistic model
selection.
