使用深层神经网络代理加速地下水流模拟的不确定性量化
Accelerating Uncertainty Quantification of Groundwater Flow Modelling Using a Deep Neural Network Proxy
摘要：
量化模型参数和输出的不确定性是模型驱动的地下水管理决策支持系统的关键组成部分。本文提出了一种新的算法方法，融合马尔可夫链蒙特卡罗（MCMC）和机器学习方法，以加速地下水流模型的不确定性量化。我们制定了一个贝叶斯逆问题的数学模型，考虑模型参数作为一个随机过程的基础概率分布。MCMC允许我们从这种分布中进行采样，但它也有一些局限性：在处理代价高昂的似然函数时，它可能过于昂贵，随后的样本通常高度相关，标准的梅特罗波利斯-黑斯廷斯算法受到维数灾难的影响。本文设计了一个Metropolis-Hastings的建议，利用深层神经网络（DNN）近似的地下水流模型，显着加快MCMC采样。我们修改了延迟接受（DA）模型层次结构，通过使用廉价的DNN近似运行短子链来生成提案，从而导致后续精细模型提案的去相关。使用一个简单的自适应误差模型，我们估计和校正DNN近似相对于后验分布的偏差。两个合成的例子，各向同性的二维问题，和各向异性的三维问题的方法进行测试。结果表明，与单级MCMC相比，不确定性量化的成本可以降低高达50%，这取决于所采用DNN的预计算成本和准确性。
英文摘要：
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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