We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder-decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Bayesian inference on millions of uncertain network parameters. This approach achieves state of the art performance in terms of predictive accuracy and uncertainty quantification in comparison to other approaches in Bayesian neural networks as well as techniques that include Gaussian processes and ensemble methods even when the training data size is relatively small. To evaluate the performance of this approach, we consider standard uncertainty quantification benchmark problems including flow in heterogeneous media defined in terms of limited data-driven permeability realizations. The performance of the surrogate model developed is very good even though there is no underlying structure shared between the input (permeability) and output (flow/pressure) fields as is often the case in the image-to-image regression models used in computer vision problems. Studies are performed with an underlying stochastic input dimensionality up to $4,225$ where most other uncertainty quantification methods fail. Uncertainty propagation tasks are considered and the predictive output Bayesian statistics are compared to those obtained with Monte Carlo estimates.

本文提出了一种使用卷积神经网络实现的关于具有随机偏微分方程的不确定性建模的代理模型，该方法使用变分梯度下降算法对参数进行“近似贝叶斯推断”，可以在处理不确定性时实现和其他方法相比具有最先进的预测精度和不确定性量化，即使训练数据量相对较小，也能获得非常好的性能。

基于贝叶斯深度卷积编码-解码网络的代理建模和不确定性量化