Scaling Up Bayesian Uncertainty Quantification for Inverse Problems Using Deep Neural Networks
Scaling Up Bayesian Uncertainty Quantification for Inverse Problems Using Deep Neural Networks
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DOI：
10.1137/21m1439456
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发表时间：
2021-01
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影响因子：
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通讯作者：
Shiwei Lan;Shuyi Li;B. Shahbaba
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作者：
Shiwei Lan;Shuyi Li;B. Shahbaba
由于不确定性量化 (UQ) 的重要性，反问题的贝叶斯方法最近在应用数学、物理和工程领域受到欢迎。然而，基于马尔可夫链蒙特卡罗（MCMC）的传统贝叶斯推理方法对于此类高维问题往往计算量大且效率低。为了解决这个问题，人们提出了几种基于代理模型的方法来加速推理过程。更具体地说，校准仿真采样 (CES) 方案已被证明在大维 UQ 问题中是成功的。在这项工作中，我们提出了一种基于仿真阶段深度神经网络模型的贝叶斯推理新 CES 方法。由此产生的算法在计算上更加高效，并且对于训练集中的变化更加鲁棒。此外，通过使用自动编码器 (AE) 进行降维，我们已经能够将贝叶斯推理方法加速三个数量级。总体而言，我们的方法（此后称为 \emph{降维仿真自动编码器蒙特卡罗 (DREAMC)} 算法）能够将贝叶斯 UQ 扩展到数千维以解决逆问题。使用两个低维（线性和非线性）反问题，我们说明了这种方法的有效性。接下来，我们将我们的方法应用于两个高维数值示例（椭圆和平流扩散），以证明其相对于现有算法的计算优势。
Due to the importance of uncertainty quantification (UQ), Bayesian approach to inverse problems has recently gained popularity in applied mathematics, physics, and engineering. However, traditional Bayesian inference methods based on Markov Chain Monte Carlo (MCMC) tend to be computationally intensive and inefficient for such high dimensional problems. To address this issue, several methods based on surrogate models have been proposed to speed up the inference process. More specifically, the calibration-emulation-sampling (CES) scheme has been proven to be successful in large dimensional UQ problems. In this work, we propose a novel CES approach for Bayesian inference based on deep neural network models for the emulation phase. The resulting algorithm is computationally more efficient and more robust against variations in the training set. Further, by using an autoencoder (AE) for dimension reduction, we have been able to speed up our Bayesian inference method up to three orders of magnitude. Overall, our method, henceforth called \emph{Dimension-Reduced Emulative Autoencoder Monte Carlo (DREAMC)} algorithm, is able to scale Bayesian UQ up to thousands of dimensions for inverse problems. Using two low-dimensional (linear and nonlinear) inverse problems we illustrate the validity of this approach. Next, we apply our method to two high-dimensional numerical examples (elliptic and advection-diffussion) to demonstrate its computational advantages over existing algorithms.