  Inferring the structure and dynamics of network models is critical to
understanding the functionality and control of complex systems, such as
metabolic and regulatory biological networks. The increasing quality and
quantity of experimental data enable statistical approaches based on
information theory for model selection and goodness-of-fit metrics. We propose
an alternative method to infer networked nonlinear dynamical systems by using
sparsity-promoting $\ell_1$ optimization to select a subset of nonlinear
interactions representing dynamics on a fully connected network. Our method
generalizes the sparse identification of nonlinear dynamics (SINDy) algorithm
to dynamical systems with rational function nonlinearities, such as biological
networks. We show that dynamical systems with rational nonlinearities may be
cast in an implicit form, where the equations may be identified in the
null-space of a library of mixed nonlinearities including the state and
derivative terms; this approach applies more generally to implicit dynamical
systems beyond those containing rational nonlinearities. This method,
implicit-SINDy, succeeds in inferring three canonical biological models:
Michaelis-Menten enzyme kinetics, the regulatory network for competence in
bacteria, and the metabolic network for yeast glycolysis.
