Accelerating uncertainty quantification of groundwater flow modelling using deep neural networks

M. B. Lykkegaard, T. Dodwell, D. Moxey

Comput. Meth. Appl. Mech. Eng., vol. 383, pp. 113895 (2021)

@article{lykkegaard-2021,
  title = {Accelerating uncertainty quantification of groundwater flow modelling using deep neural networks},
  author = {Lykkegaard, M. B. and Dodwell, T. and Moxey, D.},
  year = {2021},
  journal = cmame,
  volume = {383},
  pages = {113895},
  url = {https://www.sciencedirect.com/science/article/pii/S0045782521002322},
  doi = {10.1016/j.cma.2021.113895},
  abstract = {This paper presents a novel algorithmic approach which fuses Markov Chain Monte Carlo (MCMC) and Machine Learning methods to accelerate the uncertainty quantification of fluid flow in a heterogeneous porous medium, such as groundwater flow. We formulate the governing mathematical model as a Bayesian inverse problem, permitting us to consider the model parameters as a random process with an underlying probability distribution. MCMC allows us to sample from this distribution given some real observations of the system, 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 the model, trained on samples from the prior parameter distribution, to significantly accelerate the Bayesian computations. The approach is developed by modifying a delayed acceptance (DA) model hierarchy, whereby, instead of merely screening proposals with a coarse model before passing them to the fine, proposals are generated by running short subchains using an inexpensive DNN approximation in conjunction with the preconditioned Crank-Nicolson (pCN) transition kernel. As a result, the proposal distribution inherits its dimension-independence from the pCN kernel and subsequent fine model proposals are less correlated. Using a simple adaptive error model, we estimate and correct for the bias of the DNN approximation with respect to the posterior distribution on-the-fly. The approach is tested on a synthetic example, using different DNNs trained on a varying number of prior samples. The results show that the cost of uncertainty quantification using our novel approach can be reduced by up to 75\% compared to single-level pCN MCMC, depending on the precomputation cost and accuracy of the employed DNN.}
}

Quantifying uncertainty in groundwater flow means sampling a distribution over the model parameters, and Markov chain Monte Carlo becomes prohibitively expensive when every sample needs a full flow simulation. This paper trains a deep neural network approximation of the model on samples from the prior and uses it to generate proposals by running short subchains, correcting for the network's bias as it goes with an adaptive error model. On a synthetic problem the cost fell by up to 75% compared with single-level preconditioned Crank-Nicolson MCMC.