Abstract: Ordinary differential equation (ODE) models are widely used to describe processes in biology, chemistry, and many other scientific fields. In this talk, we consider the estimation and assessment of such models on the basis of experimental data. In practice, the available experimental data are often noisy and sparse; furthermore, some components of the system may not be observed. To address these challenges, we developed a method of Bayesian inference based on manifold-constrained Gaussian processes (MAGI), such that derivatives of the Gaussian process must satisfy the dynamics of the differential equations. MAGI completely bypasses the need for numerical integration and is thus fast to compute. First, we will describe the MAGI framework and show how it works for estimating ODE parameters and system trajectories, including examples with unobserved system components. Second, we will illustrate how MAGI can be used to assess and select different ODE models from experimental data. Third, we will discuss further extensions and applications of MAGI to systems with time-delay parameters. This talk includes joint work with Shihao Yang (Georgia Tech), Samuel Kou (Harvard), and Yuxuan Zhao (UWaterloo).