Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations

Mohan Wu, Martin Lysy
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:1018-1026, 2024.

Abstract

Estimating the parameters of ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODEs are typically approximated with deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local maxima in the likelihood function – a problem which existing probabilistic solvers have yet to resolve. Here we present a novel probabilistic ODE likelihood approximation, DALTON, which can dramatically reduce parameter sensitivity by learning from noisy ODE measurements in a data-adaptive manner. Our approximation scales linearly in both ODE variables and time discretization points, and is applicable to ODEs with both partially-unobserved components and non-Gaussian measurement models. Several examples demonstrate that DALTON produces more accurate parameter estimates via numerical optimization than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood itself.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-wu24b, title = {Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations}, author = {Wu, Mohan and Lysy, Martin}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {1018--1026}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/wu24b/wu24b.pdf}, url = {https://proceedings.mlr.press/v238/wu24b.html}, abstract = {Estimating the parameters of ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODEs are typically approximated with deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local maxima in the likelihood function – a problem which existing probabilistic solvers have yet to resolve. Here we present a novel probabilistic ODE likelihood approximation, DALTON, which can dramatically reduce parameter sensitivity by learning from noisy ODE measurements in a data-adaptive manner. Our approximation scales linearly in both ODE variables and time discretization points, and is applicable to ODEs with both partially-unobserved components and non-Gaussian measurement models. Several examples demonstrate that DALTON produces more accurate parameter estimates via numerical optimization than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood itself.} }
Endnote
%0 Conference Paper %T Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations %A Mohan Wu %A Martin Lysy %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-wu24b %I PMLR %P 1018--1026 %U https://proceedings.mlr.press/v238/wu24b.html %V 238 %X Estimating the parameters of ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODEs are typically approximated with deterministic algorithms, new research on probabilistic solvers indicates that they produce more reliable parameter estimates by better accounting for numerical errors. However, many ODE systems are highly sensitive to their parameter values. This produces deep local maxima in the likelihood function – a problem which existing probabilistic solvers have yet to resolve. Here we present a novel probabilistic ODE likelihood approximation, DALTON, which can dramatically reduce parameter sensitivity by learning from noisy ODE measurements in a data-adaptive manner. Our approximation scales linearly in both ODE variables and time discretization points, and is applicable to ODEs with both partially-unobserved components and non-Gaussian measurement models. Several examples demonstrate that DALTON produces more accurate parameter estimates via numerical optimization than existing probabilistic ODE solvers, and even in some cases than the exact ODE likelihood itself.
APA
Wu, M. & Lysy, M.. (2024). Data-Adaptive Probabilistic Likelihood Approximation for Ordinary Differential Equations. Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:1018-1026 Available from https://proceedings.mlr.press/v238/wu24b.html.

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