Accelerating uncertainty quantification of groundwater flow modelling using a deep neural network proxy

Mikkel B. Lykkegaard; Tim J. Dodwell; David Moxey

doi:10.1016/j.cma.2021.113895

Accelerating uncertainty quantification of groundwater flow modelling using a deep neural network proxy

Mikkel B. Lykkegaard^*, Tim J. Dodwell, David Moxey

^*Corresponding author for this work

Engineering

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Abstract

Quantifying the uncertainty in model parameters and output is a critical component in model-driven decision support systems for groundwater management. This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate uncertainty quantification for groundwater flow models. We formulate the governing mathematical model as a Bayesian inverse problem, considering model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution, but it comes with some limitations: it can be prohibitively expensive when dealing with costly likelihood functions, subsequent samples are often highly correlated, and the standard Metropolis–Hastings algorithm suffers from the curse of dimensionality. This paper designs a Metropolis–Hastings proposal which exploits a deep neural network (DNN) approximation of a groundwater flow model, to significantly accelerate MCMC sampling. We modify a delayed acceptance (DA) model hierarchy, whereby proposals are generated by running short subchains using an inexpensive DNN approximation, resulting in a decorrelation of subsequent fine model proposals. Using a simple adaptive error model, we estimate and correct the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on two synthetic examples; a isotropic two-dimensional problem, and an anisotropic three-dimensional problem. The results show that the cost of uncertainty quantification can be reduced by up to 50% compared to single-level MCMC, depending on the precomputation cost and accuracy of the employed DNN.

Original language	English
Article number	113895
Journal	COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Volume	383
DOIs	https://doi.org/10.1016/j.cma.2021.113895
Publication status	Published - 1 Sept 2021

Keywords

Deep neural networks
Groundwater flow
Markov chain Monte Carlo
Surrogate models
Uncertainty quantification

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10.1016/j.cma.2021.113895

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title = "Accelerating uncertainty quantification of groundwater flow modelling using a deep neural network proxy",

abstract = "Quantifying the uncertainty in model parameters and output is a critical component in model-driven decision support systems for groundwater management. This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate uncertainty quantification for groundwater flow models. We formulate the governing mathematical model as a Bayesian inverse problem, considering model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution, but it comes with some limitations: it can be prohibitively expensive when dealing with costly likelihood functions, subsequent samples are often highly correlated, and the standard Metropolis–Hastings algorithm suffers from the curse of dimensionality. This paper designs a Metropolis–Hastings proposal which exploits a deep neural network (DNN) approximation of a groundwater flow model, to significantly accelerate MCMC sampling. We modify a delayed acceptance (DA) model hierarchy, whereby proposals are generated by running short subchains using an inexpensive DNN approximation, resulting in a decorrelation of subsequent fine model proposals. Using a simple adaptive error model, we estimate and correct the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on two synthetic examples; a isotropic two-dimensional problem, and an anisotropic three-dimensional problem. The results show that the cost of uncertainty quantification can be reduced by up to 50% compared to single-level MCMC, depending on the precomputation cost and accuracy of the employed DNN.",

keywords = "Deep neural networks, Groundwater flow, Markov chain Monte Carlo, Surrogate models, Uncertainty quantification",

author = "Lykkegaard, {Mikkel B.} and Dodwell, {Tim J.} and David Moxey",

note = "Funding Information: This work was funded as part of the Water Informatics Science and Engineering Centre for Doctoral Training (WISE CDT) under a grant from the Engineering and Physical Sciences Research Council (EPSRC), UK , grant number EP/L016214/1 . TD was funded by a Turing AI Fellowship, UK ( 2TAFFP\100007 ). DM acknowledges support from the EPSRC Platform Grant PRISM, UK ( EP/R029423/1 ). The authors have no competing interests. Data supporting the findings in this study are available in the Open Research Exeter (ORE, https://ore.exeter.ac.uk/repository/ ) data repository. Publisher Copyright: {\textcopyright} 2021 The Author(s) Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

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doi = "10.1016/j.cma.2021.113895",

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N1 - Funding Information: This work was funded as part of the Water Informatics Science and Engineering Centre for Doctoral Training (WISE CDT) under a grant from the Engineering and Physical Sciences Research Council (EPSRC), UK , grant number EP/L016214/1 . TD was funded by a Turing AI Fellowship, UK ( 2TAFFP\100007 ). DM acknowledges support from the EPSRC Platform Grant PRISM, UK ( EP/R029423/1 ). The authors have no competing interests. Data supporting the findings in this study are available in the Open Research Exeter (ORE, https://ore.exeter.ac.uk/repository/ ) data repository. Publisher Copyright: © 2021 The Author(s) Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/9/1

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N2 - Quantifying the uncertainty in model parameters and output is a critical component in model-driven decision support systems for groundwater management. This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate uncertainty quantification for groundwater flow models. We formulate the governing mathematical model as a Bayesian inverse problem, considering model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution, but it comes with some limitations: it can be prohibitively expensive when dealing with costly likelihood functions, subsequent samples are often highly correlated, and the standard Metropolis–Hastings algorithm suffers from the curse of dimensionality. This paper designs a Metropolis–Hastings proposal which exploits a deep neural network (DNN) approximation of a groundwater flow model, to significantly accelerate MCMC sampling. We modify a delayed acceptance (DA) model hierarchy, whereby proposals are generated by running short subchains using an inexpensive DNN approximation, resulting in a decorrelation of subsequent fine model proposals. Using a simple adaptive error model, we estimate and correct the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on two synthetic examples; a isotropic two-dimensional problem, and an anisotropic three-dimensional problem. The results show that the cost of uncertainty quantification can be reduced by up to 50% compared to single-level MCMC, depending on the precomputation cost and accuracy of the employed DNN.

AB - Quantifying the uncertainty in model parameters and output is a critical component in model-driven decision support systems for groundwater management. This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate uncertainty quantification for groundwater flow models. We formulate the governing mathematical model as a Bayesian inverse problem, considering model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution, but it comes with some limitations: it can be prohibitively expensive when dealing with costly likelihood functions, subsequent samples are often highly correlated, and the standard Metropolis–Hastings algorithm suffers from the curse of dimensionality. This paper designs a Metropolis–Hastings proposal which exploits a deep neural network (DNN) approximation of a groundwater flow model, to significantly accelerate MCMC sampling. We modify a delayed acceptance (DA) model hierarchy, whereby proposals are generated by running short subchains using an inexpensive DNN approximation, resulting in a decorrelation of subsequent fine model proposals. Using a simple adaptive error model, we estimate and correct the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on two synthetic examples; a isotropic two-dimensional problem, and an anisotropic three-dimensional problem. The results show that the cost of uncertainty quantification can be reduced by up to 50% compared to single-level MCMC, depending on the precomputation cost and accuracy of the employed DNN.

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KW - Markov chain Monte Carlo

KW - Surrogate models

KW - Uncertainty quantification

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M1 - 113895

ER -

Accelerating uncertainty quantification of groundwater flow modelling using a deep neural network proxy

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