  Measurement noise is an integral part while collecting data of a physical
process. Thus, noise removal is a necessary step to draw conclusions from these
data, and it often becomes quite essential to construct dynamical models using
these data. We discuss a methodology to learn differential equation(s) using
noisy and sparsely sampled measurements. In our methodology, the main
innovation can be seen in of integration of deep neural networks with a
classical numerical integration method. Precisely, we aim at learning a neural
network that implicitly represents the data and an additional neural network
that models the vector fields of the dependent variables. We combine these two
networks by enforcing the constraint that the data at the next time-steps can
be given by following a numerical integration scheme such as the fourth-order
Runge-Kutta scheme. The proposed framework to learn a model predicting the
vector field is highly effective under noisy measurements. The approach can
handle scenarios where dependent variables are not available at the same
temporal grid. We demonstrate the effectiveness of the proposed method to
learning models using data obtained from various differential equations. The
proposed approach provides a promising methodology to learn dynamic models,
where the first-principle understanding remains opaque.
