  Measurement noise is an integral part while collecting data of a physical
process. Thus, noise removal is necessary to draw conclusions from these data,
and it often becomes essential to construct dynamical models using these data.
We discuss a methodology to learn differential equation(s) using noisy and
irregular sampled measurements. In our methodology, the main innovation can be
seen in the integration of deep neural networks with the neural ordinary
differential equations (ODEs) approach. Precisely, we aim at learning a neural
network that provides (approximately) an implicit representation of the data
and an additional neural network that models the vector fields of the dependent
variables. We combine these two networks by constraining using neural ODEs. The
proposed framework to learn a model describing 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. Moreover, a
particular structure, e.g., second-order with respect to time, can easily be
incorporated. We demonstrate the effectiveness of the proposed method for
learning models using data obtained from various differential equations and
present a comparison with the neural ODE method that does not make any special
treatment to noise.
