  Ordinary differential equations (ODEs) are widely used to model complex
dynamics that arises in biology, chemistry, engineering, finance, physics, etc.
Calibration of a complicated ODE system using noisy data is generally very
difficult. In this work, we propose a two-stage nonparametric approach to
address this problem. We first extract the de-noised data and their higher
order derivatives using boundary kernel method, and then feed them into a
sparsely connected deep neural network with ReLU activation function. Our
method is able to recover the ODE system without being subject to the curse of
dimensionality and complicated ODE structure. When the ODE possesses a general
modular structure, with each modular component involving only a few input
variables, and the network architecture is properly chosen, our method is
proven to be consistent. Theoretical properties are corroborated by an
extensive simulation study that demonstrates the validity and effectiveness of
the proposed method. Finally, we use our method to simultaneously characterize
the growth rate of Covid-19 infection cases from 50 states of the USA.
