Inverse problems, i.e., estimating parameters of physical models from experimental data, are ubiquitous in science and engineering. The Bayesian formulation is the gold standard because it alleviates ill-posedness issues and quantifies epistemic uncertainty. Since analytical posteriors are not typically available, one resorts to Markov chain Monte Carlo sampling or approximate variational inference. However, inference needs to be rerun from scratch for each new set of data. This drawback limits the applicability of the Bayesian formulation to real-time settings, e.g., health monitoring of engineered systems, and medical diagnosis. The objective of this paper is to develop a methodology that enables real-time inference by learning the Bayesian inverse map, i.e., the map from data to posteriors. Our approach is as follows. We represent the posterior distribution using a parameterization based on deep neural networks. Next, we learn the network parameters by amortized variational inference method which involves maximizing the expectation of evidence lower bound over all possible datasets compatible with the model. We demonstrate our approach by solving examples a set of benchmark problems from science and engineering. Our results show that the posterior estimates of our approach are in agreement with the corresponding ground truth obtained by Markov chain Monte Carlo. Once trained, our approach provides the posterior parameters of observation just at the cost of a forward pass of the neural network.

翻译：逆问题，即从实验数据中估计物理模型参数，在科学与工程领域普遍存在。贝叶斯方法是黄金标准，因为它能缓解不适定问题并量化认知不确定性。由于通常无法获得解析的后验分布，因此需要借助马尔可夫链蒙特卡洛采样或近似变分推断。然而，对于每组新数据，都需要从头重新进行推断。这一缺陷限制了贝叶斯方法在实时场景（例如工程系统健康监测和医学诊断）中的适用性。本文旨在开发一种通过学习贝叶斯逆映射（即从数据到后验分布的映射）来实现实时推断的方法。我们的方法如下：首先，使用基于深度神经网络的参数化方法表示后验分布。接着，通过摊销变分推断方法学习网络参数，该方法涉及在所有与模型兼容的可能数据集上最大化证据下界的期望值。我们通过求解科学与工程领域的一组基准问题来展示所提方法。结果表明，我们方法的后验估计与马尔可夫链蒙特卡洛方法得到的相应真实值一致。一旦训练完成，我们的方法仅需一次神经网络前向传播的成本即可提供观测数据的后验参数。