Towards Identifiability of Interventional Stochastic Differential Equations

We study identifiability of stochastic differential equations (SDE) under multiple interventions. Our results give the first provable bounds for unique recovery of SDE parameters given samples from their stationary distributions. We give tight bounds on the number of necessary interventions for linear SDEs, and upper bounds for nonlinear SDEs in the small noise regime. We experimentally validate the recovery of true parameters in synthetic data, and motivated by our theoretical results, demonstrate the advantage of parameterizations with learnable activation functions in application to gene regulatory dynamics. Stochastic dynamical systems are ubiquitous as models for natural data. They are perfectly suited for application to time-series data, and therefore also a good candidate to characterize systems that reach a steady state in the limit. If a system is governed by some stochastic differential equation (SDE) and the same system is observed under different interventions, ideally one would learn the underlying parameters governing the dynamics, and guarantee accurate prediction under new interventions. However, in many natural settings, data is modeled as following an SDE even if one does not have access to explicit trajectories. Studies of ecological systems focus on the long-term survival of multiple species modeled by the quasi-stationary state of SDEs with environmental factors as perturbations (Hening & Li, 2021). The application of flow cytometry to protein signaling networks under perturbation (Sachs et al., 2005) is destructive and yields protein quantification at one time point, modeled using the stationary distributions of linear SDEs in V arando & Hansen (2020).