  Differential equations are frequently used in engineering domains, such as
modeling and control of industrial systems, where safety and performance
guarantees are of paramount importance. Traditional physics-based modeling
approaches require domain expertise and are often difficult to tune or adapt to
new systems. In this paper, we show how to model discrete ordinary differential
equations (ODE) with algebraic nonlinearities as deep neural networks with
varying degrees of prior knowledge. We derive the stability guarantees of the
network layers based on the implicit constraints imposed on the weight's
eigenvalues. Moreover, we show how to use barrier methods to generically handle
additional inequality constraints. We demonstrate the prediction accuracy of
learned neural ODEs evaluated on open-loop simulations compared to ground truth
dynamics with bi-linear terms.
