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Ordinary Differential Equations (ODEs) have recently gained a lot ofattention in machine learning. However, the theoretical aspects, e.g.,identifiability and asymptotic properties of statistical estimation are stillobscure. This paper derives a sufficient condition for the identifiability ofhomogeneous linear ODE systems from a sequence of equally-spaced error-freeobservations sampled from a single trajectory. When observations are disturbedby measurement noise, we prove that under mild conditions, the parameterestimator based on the Nonlinear Least Squares (NLS) method is consistent andasymptotic normal with $n^{-1/2}$ convergence rate. Based on the asymptoticnormality property, we construct confidence sets for the unknown systemparameters and propose a new method to infer the causal structure of the ODEsystem, i.e., inferring whether there is a causal link between systemvariables. Furthermore, we extend the results to degraded observations,including aggregated and time-scaled ones. To the best of our knowledge, ourwork is the first systematic study of the identifiability and asymptoticproperties in learning linear ODE systems. We also construct simulations withvarious system dimensions to illustrate the established theoretical results.

Identifiability and Asymptotics in Learning Homogeneous Linear ODE Systems from Discrete Observations