Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations
This addresses a bottleneck in scientific applications where ODE parameter estimation is crucial, offering a novel solution for improved reliability.
The paper tackles the problem of parameter sensitivity in ordinary differential equations (ODEs), which causes deep local maxima in likelihood functions, by introducing DALTON, a data-adaptive probabilistic likelihood approximation. The result is more accurate parameter estimates, outperforming existing probabilistic solvers and sometimes even the exact likelihood.
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.