Picard Iteration for Parameter Estimation in Nonlinear Ordinary Differential Equations

TL;DR

This paper introduces a novel Picard iteration-based framework for robust parameter estimation in nonlinear ODEs using noisy, sparse, and irregular data, ensuring convergence and improved accuracy.

Contribution

It develops a new gradient-based estimation method leveraging the Picard operator, with convergence guarantees for challenging real-world data scenarios.

Findings

Algorithms demonstrate robustness across diverse datasets

Proven convergence under specific conditions

Outperforms existing methods in accuracy and stability

Abstract

We consider the problem of using experimental time-series data for parameter estimation in nonlinear ordinary differential equations, focusing on the case where the data is noisy, sparse, irregularly sampled, includes multiple experiments, and does not directly measure the system state or its time-derivative. To account for such low-quality data, we propose a new framework for gradient-based parameter estimation which uses the Picard operator to reformulate the problem as constrained optimization with infinite-dimensional variables and constraints. We then use the contractive properties of the Picard operator to propose a class of gradient-contractive algorithms and provide conditions under which such algorithms are guaranteed to converge to a local optima. The algorithms are then tested on a battery of models and variety of datasets in order to demonstrate robustness and improvement over alternative approaches.