  Discovering the partial differential equations underlying spatio-temporal
datasets from very limited and highly noisy observations is of paramount
interest in many scientific fields. However, it remains an open question to
know when model discovery algorithms based on sparse regression can actually
recover the underlying physical processes. In this work, we show the design
matrices used to infer the equations by sparse regression can violate the
irrepresentability condition (IRC) of the Lasso, even when derived from
analytical PDE solutions (i.e. without additional noise). Sparse regression
techniques which can recover the true underlying model under violated IRC
conditions are therefore required, leading to the introduction of the
randomised adaptive Lasso. We show once the latter is integrated within the
deep learning model discovery framework DeepMod, a wide variety of nonlinear
and chaotic canonical PDEs can be recovered: (1) up to $\mathcal{O}(2)$ higher
noise-to-sample ratios than state-of-the-art algorithms, (2) with a single set
of hyperparameters, which paves the road towards truly automated model
discovery.
