Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based methods have been proposed. As these methods need a great number of forward and backward computations, they become costly in computation, in particular, when the forward or generative models are complex and the evaluation of the likelihood becomes very costly. Using Deep Neural Network surrogate models and approximate computation can become very helpful. However, accounting for the uncertainties, we need first understand the Bayesian Deep Learning and then, we can see how we can use them for inverse problems. In this work, we focus on NN, DL and more specifically the Bayesian DL particularly adapted for inverse problems. We first give details of Bayesian DL approximate computations with exponential families, then we will see how we can use them for inverse problems. We consider two cases: First the case where the forward operator is known and used as physics constraint, the second more general data driven DL methods. keyword: Neural Network, Variational Bayesian inference, Bayesian Deep Learning (DL), Inverse problems, Physics based DL.

翻译：反问题出现在任何存在间接测量的场景中。由于反问题通常是不适定的，为获得令人满意的解需要先验知识。经典方法中，已有多种正则化方法和基于贝叶斯推理的方法被提出。然而，这些方法需要大量的正向和反向计算，当正向模型或生成模型复杂且似然评估成本极高时，其计算代价十分高昂。采用深度神经网络替代模型和近似计算能极大提升效率。但为了处理不确定性，我们首先需要理解贝叶斯深度学习，进而探讨如何将其应用于反问题。本文聚焦于神经网络、深度学习，尤其是特别适用于反问题的贝叶斯深度学习。我们首先详述基于指数族的贝叶斯深度学习近似计算方法，随后探讨如何在反问题中应用这些方法。我们考虑两种情形：其一为正向算子已知且作为物理约束的情况；其二为更通用的数据驱动深度学习方法。关键词：神经网络，变分贝叶斯推理，贝叶斯深度学习，反问题，物理驱动深度学习。