  Temporal variations in biological systems and more generally in natural
sciences are typically modelled as a set of Ordinary, Partial, or Stochastic
Differential or Difference Equations. Algorithms for learning the structure and
the parameters of a dynamical system are distinguished based on whether time is
discrete or continuous, observations are time-series or time-course, and
whether the system is deterministic or stochastic, however, there is no
approach able to handle the various types of dynamical systems simultaneously.
In this paper, we present a unified approach to infer both the structure and
the parameters of nonlinear dynamical systems of any type under the restriction
of being linear with respect to the unknown parameters. Our approach, which is
named Unified Sparse Dynamics Learning (USDL), constitutes of two steps. First,
an atemporal system of equations is derived through the application of the weak
formulation. Then, assuming a sparse representation for the dynamical system,
we show that the inference problem can be expressed as a sparse signal recovery
problem, allowing the application of an extensive body of algorithms and
theoretical results. Results on simulated data demonstrate the efficacy and
superiority of the USDL algorithm as a function of the experimental setup
(sample size, number of time measurements, number of interventions, noise
level). Additionally, USDL's accuracy significantly correlates with theoretical
metrics such as the exact recovery coefficient. On real single-cell data, the
proposed approach is able to induce high-confidence subgraphs of the signaling
pathway. USDL algorithm has been integrated in SCENERY
(\url{http://scenery.csd.uoc.gr/}); an online tool for single-cell mass
cytometry analytics.
