Bayesian Solution Uncertainty Quantification for Differential Equations

TL;DR

This paper introduces a Bayesian framework for quantifying discretization uncertainty in differential equations, improving inference accuracy and providing a flexible, scalable approach applicable to complex, large-scale systems.

Contribution

It proposes a novel probabilistic formalism for modeling discretization uncertainty and develops algorithms with proven convergence and computational efficiency.

Findings

The framework effectively captures discretization uncertainty in differential equations.

The sampling scheme is consistent and converges at first order.

Application to protein dynamics demonstrates practical utility.

Abstract

We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are constructed from a coarsely discretized approximation to system equations. A formalism is proposed for inferring a fixed but a priori unknown model trajectory through Bayesian updating of a prior process conditional on model information. A one-step-ahead sampling scheme for interrogating the model is described, its consistency and first order convergence properties are proved, and its computational complexity is shown to be proportional to that of numerical explicit one-step solvers. Examples illustrate the flexibility of this framework to deal with a wide variety of complex and large-scale systems. Within the calibration problem, discretization uncertainty defines a layer in the Bayesian hierarchy, and a Markov chain Monte Carlo algorithm that targets this posterior distribution is presented. This formalism is used for inference on the JAK-STAT delay differential equation model of protein dynamics from indirectly observed measurements. The discussion outlines implications for the new field of probabilistic numerics.