Understanding the impact of numerical solvers on inference for differential equation models

Creswell, Richard; Shepherd, Katherine M.; Lambert, Ben; Mirams, Gary R.; Lei, Chon Lok; Tavener, Simon; Robinson, Martin; Gavaghan, David J.

Abstract

Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers that seem sufficiently accurate for the forward problem, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the inverse problem, i.e. for inferring the model parameters from data. We show that for both fixed step and adaptive step ODE solvers, solving the forward problem with insufficient accuracy can distort likelihood surfaces, which might become jagged, causing inference algorithms to get stuck in local ‘phantom’ optima. We demonstrate that biases in inference arising from numerical approximation of ODEs are potentially most severe in systems involving low noise and rapid nonlinear dynamics. We reanalyse an ODE change point model previously fit to the COVID-19 outbreak in Germany and show the effect of the step size on simulation and inference results. We then fit a more complicated rainfall run-off model to hydrological data and illustrate the importance of tuning solver tolerances to avoid distorted likelihood surfaces. Our results indicate that, when performing inference for ODE model parameters, adaptive step size solver tolerances must be set cautiously and likelihood surfaces should be inspected for characteristic signs of numerical issues.

Citation

Creswell, R., Shepherd, K. M., Lambert, B., Mirams, G. R., Lei, C. L., Tavener, S., Robinson, M., & Gavaghan, D. J. (2024). Understanding the impact of numerical solvers on inference for differential equation models. Journal of the Royal Society, Interface, 21(212), Article 20230369. https://doi.org/10.1098/rsif.2023.0369

• Journal Article Type: Article
• Acceptance Date: Jan 30, 2024
• Online Publication Date: Mar 6, 2024
• Publication Date: 2024-03
• Deposit Date: Apr 13, 2024
• Publicly Available Date: Apr 16, 2024
• Journal: Journal of the Royal Society Interface
• Print ISSN: 1742-5689
• Electronic ISSN: 1742-5662
• Publisher: The Royal Society
• Peer Reviewed: Peer Reviewed
• Volume: 21
• Issue: 212
• Article Number: 20230369
• DOI: https://doi.org/10.1098/rsif.2023.0369
• Keywords: truncation error, compartmental models, ordinary differential equations, Bayesian statistics, hydrological modelling, inference
• Public URL: https://nottingham-repository.worktribe.com/output/32450528
• Publisher URL: https://royalsocietypublishing.org/doi/10.1098/rsif.2023.0369
• Additional Information: Received: 2023-07-03; Accepted: 2024-01-30; Published: 2024-03-06