  We develop a framework for estimating unknown partial differential equations
from noisy data, using a deep learning approach. Given noisy samples of a
solution to an unknown PDE, our method interpolates the samples using a neural
network, and extracts the PDE by equating derivatives of the neural network
approximation. Our method applies to PDEs which are linear combinations of
user-defined dictionary functions, and generalizes previous methods that only
consider parabolic PDEs. We introduce a regularization scheme that prevents the
function approximation from overfitting the data and forces it to be a solution
of the underlying PDE. We validate the model on simulated data generated by the
known PDEs and added Gaussian noise, and we study our method under different
levels of noise. We also compare the error of our method with a Cramer-Rao
lower bound for an ordinary differential equation. Our results indicate that
our method outperforms other methods in estimating PDEs, especially in the low
signal-to-noise regime.
