Multiphase MCMC sampling for parameter inference in nonlinear ordinary differential equations

Traditionally, ODE parameter inference relies on solving the system of ODEs and assessing fit of the estimated signal with the observations. However, nonlinear ODEs often do not permit closed form solutions. Using numerical methods to solve the equations results in prohibitive computational costs, particularly when one adopts a Bayesian approach in sampling parameters from a posterior distribution. With the introduction of gradient matching, we can abandon the need to numerically solve the system of equations. Inherent in these efficient procedures is an introduction of bias to the learning problem as we no longer sample based on the exact likelihood function. This paper presents a multiphase MCMC approach that attempts to close the gap between efficiency and accuracy. By sampling using a surrogate likelihood, we accelerate convergence to the stationary distribution before sampling using the exact likelihood. We demonstrate that this method combines the efficiency of gradient matching and the accuracy of the exact likelihood scheme

Inference of dynamic systems from noisy and sparse data via manifold-constrained Gaussian processes

Parameter estimation for nonlinear dynamic system models, represented by ordinary differential equations (ODEs), using noisy and sparse data is a vital task in many fields. We propose a fast and accurate method, MAGI (MAnifold-constrained Gaussian process Inference), for this task. MAGI uses a Gaussian process model over time-series data, explicitly conditioned on the manifold constraint that derivatives of the Gaussian process must satisfy the ODE system. By doing so, we completely bypass the need for numerical integration and achieve substantial savings in computational time. MAGI is also suitable for inference with unobserved system components, which often occur in real experiments. MAGI is distinct from existing approaches as we provide a principled statistical construction under a Bayesian framework, which incorporates the ODE system through the manifold constraint. We demonstrate the accuracy and speed of MAGI using realistic examples based on physical experiments