Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations

TL;DR

DALTON is a new probabilistic likelihood approximation for ODEs that reduces parameter sensitivity and improves parameter estimation accuracy, especially in complex, noisy, and partially observed systems.

Contribution

It introduces DALTON, a scalable, data-adaptive probabilistic likelihood approximation that handles non-Gaussian noise and partial observations, outperforming existing methods.

Findings

DALTON achieves more accurate parameter estimates than existing probabilistic solvers.

DALTON can sometimes outperform the exact ODE likelihood in parameter estimation.

The method scales linearly with variables and time points.

Abstract

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.