Flow-Based Bayesian Estimation of Nonlinear Differential Equations for Modeling Biological Networks

We consider the problem of estimating parameters and unobserved trajectories in nonlinear ordinary differential equations (ODEs) from noisy and partially observed data. We focus on a class of statespace models defined from the integration of the differential equation in the evolution equation. Within a Bayesian framework, we derive a nonsequential estimation procedure that infers the parameters and the initial condition of the ODE, taking into account that both are required to fully characterize the solution of the ODE. This point of view, new in the context of state-space models, modifies the learning problem. To evaluate the relevance of this approach, we use an Adaptive Importance Sampling in a population Monte Carlo scheme to approximate the posterior probability distribution. We compare this approach to recursive estimation via Unscented Kalman Filtering on two reverse-modeling problems in systems biology. On both problems, our method improves on classical smoothing methods used in state space models for the estimation of unobserved trajectories.