Richard Creswell, Katherine M Shepherd, Ben Lambert, Gary R Mirams, Chon Lok Lei, Simon Tavener, Martin Robinson, David J Gavaghan
{"title":"了解数值求解器对微分方程模型推理的影响。","authors":"Richard Creswell, Katherine M Shepherd, Ben Lambert, Gary R Mirams, Chon Lok Lei, Simon Tavener, Martin Robinson, David J Gavaghan","doi":"10.1098/rsif.2023.0369","DOIUrl":null,"url":null,"abstract":"<p><p>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 <i>forward problem</i>, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the <i>inverse problem</i>, 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.</p>","PeriodicalId":17488,"journal":{"name":"Journal of The Royal Society Interface","volume":null,"pages":null},"PeriodicalIF":3.7000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10914510/pdf/","citationCount":"0","resultStr":"{\"title\":\"Understanding the impact of numerical solvers on inference for differential equation models.\",\"authors\":\"Richard Creswell, Katherine M Shepherd, Ben Lambert, Gary R Mirams, Chon Lok Lei, Simon Tavener, Martin Robinson, David J Gavaghan\",\"doi\":\"10.1098/rsif.2023.0369\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p><p>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 <i>forward problem</i>, i.e. for obtaining an accurate simulation, might not be sufficiently accurate for the <i>inverse problem</i>, 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.</p>\",\"PeriodicalId\":17488,\"journal\":{\"name\":\"Journal of The Royal Society Interface\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.7000,\"publicationDate\":\"2024-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10914510/pdf/\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of The Royal Society Interface\",\"FirstCategoryId\":\"103\",\"ListUrlMain\":\"https://doi.org/10.1098/rsif.2023.0369\",\"RegionNum\":2,\"RegionCategory\":\"综合性期刊\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"2024/3/6 0:00:00\",\"PubModel\":\"Epub\",\"JCR\":\"Q1\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of The Royal Society Interface","FirstCategoryId":"103","ListUrlMain":"https://doi.org/10.1098/rsif.2023.0369","RegionNum":2,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/3/6 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
Understanding the impact of numerical solvers on inference for differential equation models.
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.
期刊介绍:
J. R. Soc. Interface welcomes articles of high quality research at the interface of the physical and life sciences. It provides a high-quality forum to publish rapidly and interact across this boundary in two main ways: J. R. Soc. Interface publishes research applying chemistry, engineering, materials science, mathematics and physics to the biological and medical sciences; it also highlights discoveries in the life sciences of relevance to the physical sciences. Both sides of the interface are considered equally and it is one of the only journals to cover this exciting new territory. J. R. Soc. Interface welcomes contributions on a diverse range of topics, including but not limited to; biocomplexity, bioengineering, bioinformatics, biomaterials, biomechanics, bionanoscience, biophysics, chemical biology, computer science (as applied to the life sciences), medical physics, synthetic biology, systems biology, theoretical biology and tissue engineering.