  We introduce DeepMoD, a Deep learning based Model Discovery algorithm.
DeepMoD discovers the partial differential equation underlying a
spatio-temporal data set using sparse regression on a library of possible
functions and their derivatives. A neural network approximates the data and
constructs the function library, but it also performs the sparse regression.
This construction makes it extremely robust to noise, applicable to small data
sets, and, contrary to other deep learning methods, does not require a training
set. We benchmark our approach on several physical problems such as the
Burgers', Korteweg-de Vries and Keller-Segel equations, and find that it
requires as few as $\mathcal{O}(10^2)$ samples and works at noise levels up to
$75\%$. Motivated by these results, we apply DeepMoD directly on noisy
experimental time-series data from a gel electrophoresis experiment and find
that it discovers the advection-diffusion equation describing this system.
