  Estimating the parameters of ordinary differential equations (ODEs) is of
fundamental importance in many scientific applications. While ODEs are
typically approximated with deterministic algorithms, new research on
probabilistic solvers indicates that they produce more reliable parameter
estimates by better accounting for numerical errors. However, many ODE systems
are highly sensitive to their parameter values. This produces deep local maxima
in the likelihood function -- a problem which existing probabilistic solvers
have yet to resolve. Here we present a novel probabilistic ODE likelihood
approximation, DALTON, which can dramatically reduce parameter sensitivity by
learning from noisy ODE measurements in a data-adaptive manner. Our
approximation scales linearly in both ODE variables and time discretization
points, and is applicable to ODEs with both partially-unobserved components and
non-Gaussian measurement models. Several examples demonstrate that DALTON
produces more accurate parameter estimates via numerical optimization than
existing probabilistic ODE solvers, and even in some cases than the exact ODE
likelihood itself.
