Adaptive semiparametric Bayesian differential equations via sequential Monte Carlo

TL;DR

This paper introduces a Bayesian approach using sequential Monte Carlo methods for estimating parameters in nonlinear differential equations, avoiding costly numerical solvers and handling complex likelihood surfaces.

Contribution

It presents a novel adaptive semiparametric Bayesian framework with a sequential Monte Carlo algorithm for efficient inference in nonlinear DEs.

Findings

Effective parameter estimation demonstrated on ODEs and delay differential equations.

Avoids expensive numerical solvers through nonparametric function representation.

Provides an R package for practical implementation.

Abstract

Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated from noisy measurements of the dynamic system. Generally, there is no closed-form solution for nonlinear DEs, and the likelihood surface for the parameter of interest is multi-modal and very sensitive to different parameter values. We propose a Bayesian framework for nonlinear DE systems. A flexible nonparametric function is used to represent the dynamic process such that expensive numerical solvers can be avoided. A sequential Monte Carlo algorithm in the annealing framework is proposed to conduct Bayesian inference for parameters in DEs. In our numerical experiments, we use examples of ordinary differential equations and delay differential equations to demonstrate the effectiveness of the proposed algorithm. We developed an R package that is available at \url{https://github.com/shijiaw/smcDE}.