Datasets in engineering applications are often limited and contaminated, mainly due to unavoidable measurement noise and signal distortion. Thus, using conventional data-driven approaches to build a reliable discriminative model, and further applying this identified surrogate to uncertainty analysis remains to be very challenging. A deep learning approach is presented to provide predictions based on limited and noisy data. To address noise perturbation, the Bayesian learning method that naturally facilitates an automatic updating mechanism is considered to quantify and propagate model uncertainties into predictive quantities. Specifically, hierarchical Bayesian modeling (HBM) is first adopted to describe model uncertainties, which allows the prior assumption to be less subjective, while also makes the proposed surrogate more robust. Next, the Bayesian inference is seamlessly integrated into the DL framework, which in turn supports probabilistic programming by yielding a probability distribution of the quantities of interest rather than their point estimates. Variational inference (VI) is implemented for the posterior distribution analysis where the intractable marginalization of the likelihood function over parameter space is framed in an optimization format, and stochastic gradient descent method is applied to solve this optimization problem. Finally, Monte Carlo simulation is used to obtain an unbiased estimator in the predictive phase of Bayesian inference, where the proposed Bayesian deep learning (BDL) scheme is able to offer confidence bounds for the output estimation by analyzing propagated uncertainties. The effectiveness of Bayesian shrinkage is demonstrated in improving predictive performance using contaminated data, and various examples are provided to illustrate concepts, methodologies, and algorithms of this proposed BDL modeling technique.

中文翻译：

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具有分层先验的贝叶斯深度学习：来自有限和嘈杂数据的预测

工程应用中的数据集通常受到限制和污染，主要是由于不可避免的测量噪声和信号失真。因此，使用传统的数据驱动方法来构建可靠的判别模型，并将此确定的替代物进一步应用于不确定性分析仍然非常具有挑战性。提出了一种深度学习方法，以提供基于有限和嘈杂数据的预测。为了解决噪声扰动问题，可以自然地促进自动更新机制的贝叶斯学习方法被认为是将模型不确定性量化和传播为预测量。具体来说，首先采用分层贝叶斯建模（HBM）来描述模型的不确定性，这使得先验假设不那么主观，同时也使所提出的替代方案更加稳健。下一个，贝叶斯推理无缝集成到 DL 框架中，这反过来又通过产生感兴趣数量的概率分布而不是它们的点估计来支持概率编程。变分推理 (VI) 用于后验分布分析，其中参数空间上的似然函数难以处理的边缘化以优化格式构建，并应用随机梯度下降方法来解决此优化问题。最后，蒙特卡罗模拟用于在贝叶斯推理的预测阶段获得无偏估计量，其中所提出的贝叶斯深度学习 (BDL) 方案能够通过分析传播的不确定性为输出估计提供置信界限。