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 data-driven method to infer networked nonlinear dynamical systems by using sparsity-promoting optimization to select a subset of nonlinear interactions representing dynamics on a network. In contrast to standard model selection methods-based upon information content for a finite number of heuristic models (order 10 or less), our model selection procedure discovers a parsimonious model from a combinatorially large set of models, without an exhaustive search. Our particular innovation is appropriate for many biological networks, where the governing dynamical systems have rational function nonlinearities with cross terms, thus requiring an implicit formulation and the equations to be identified in the null-space of a library of mixed nonlinearities, including the state and derivative terms. This method, implicit-SINDy, succeeds in inferring three canonical biological models: 1) Michaelis-Menten enzyme kinetics; 2) the regulatory network for competence in bacteria; and 3) the metabolic network for yeast glycolysis.
