  While there has been a surge of recent interest in learning differential
equation models from time series, methods in this area typically cannot cope
with highly noisy data. We break this problem into two parts: (i) approximating
the unknown vector field (or right-hand side) of the differential equation, and
(ii) dealing with noise. To deal with (i), we describe a neural network
architecture consisting of tensor products of one-dimensional neural shape
functions. For (ii), we propose an alternating minimization scheme that
switches between vector field training and filtering steps, together with
multiple trajectories of training data. We find that the neural shape function
architecture retains the approximation properties of dense neural networks,
enables effective computation of vector field error, and allows for graphical
interpretability, all for data/systems in any finite dimension $d$. We also
study the combination of either our neural shape function method or existing
differential equation learning methods with alternating minimization and
multiple trajectories. We find that retrofitting any learning method in this
way boosts the method's robustness to noise. While in their raw form the
methods struggle with 1% Gaussian noise, after retrofitting, they learn
accurate vector fields from data with 10% Gaussian noise.
