  In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that
can identify governing partial differential equations (PDEs) directly from
noisy, limited measurements of a physical system of interest. PDE-LEARN uses a
Rational Neural Network, $U$, to approximate the system response function and a
sparse, trainable vector, $\xi$, to characterize the hidden PDE that the system
response function satisfies. Our approach couples the training of $U$ and $\xi$
using a loss function that (1) makes $U$ approximate the system response
function, (2) encapsulates the fact that $U$ satisfies a hidden PDE that $\xi$
characterizes, and (3) promotes sparsity in $\xi$ using ideas from iteratively
reweighted least-squares. Further, PDE-LEARN can simultaneously learn from
several data sets, allowing it to incorporate results from multiple
experiments. This approach yields a robust algorithm to discover PDEs directly
from realistic scientific data. We demonstrate the efficacy of PDE-LEARN by
identifying several PDEs from noisy and limited measurements.
