  The discovery of governing equations from data has been an active field of
research for decades. One widely used methodology for this purpose is sparse
regression for nonlinear dynamics, known as SINDy. Despite several attempts,
noisy and scarce data still pose a severe challenge to the success of the SINDy
approach. In this work, we discuss a robust method to discover nonlinear
governing equations from noisy and scarce data. To do this, we make use of
neural networks to learn an implicit representation based on measurement data
so that not only it produces the output in the vicinity of the measurements but
also the time-evolution of output can be described by a dynamical system.
Additionally, we learn such a dynamic system in the spirit of the SINDy
framework. Leveraging the implicit representation using neural networks, we
obtain the derivative information -- required for SINDy -- using an automatic
differentiation tool. To enhance the robustness of our methodology, we further
incorporate an integral condition on the output of the implicit networks.
Furthermore, we extend our methodology to handle data collected from multiple
initial conditions. We demonstrate the efficiency of the proposed methodology
to discover governing equations under noisy and scarce data regimes by means of
several examples and compare its performance with existing methods.
