  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.
