A Unified Approach for Sparse Dynamical System Inference from Temporal Measurements

Abstract: 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.