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 parameterize the posterior distribution as a function of data. This work outlines two distinct approaches to do this. The first method involves parameterizing the posterior using an amortized full-rank Gaussian guide, implemented through neural networks. The second method utilizes a Conditional Normalizing Flow guide, employing conditional invertible neural networks for cases where the target posterior is arbitrarily complex. In both approaches, we learn the network parameters by amortized variational inference which involves maximizing the expectation of evidence lower bound over all possible datasets compatible with the model. We demonstrate our approach by solving 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 distribution for a given observation just at the cost of a forward pass of the neural network.

翻译：逆问题即从实验数据估计物理模型参数，在科学与工程领域普遍存在。贝叶斯框架因其能够缓解不适定性问题并量化认知不确定性，成为黄金标准方法。由于解析后验分布通常难以获得，研究者常采用马尔可夫链蒙特卡洛采样或近似变分推断。然而，每处理一组新数据都需要重新进行推断计算，这一缺陷限制了贝叶斯框架在实时场景中的应用，例如工程系统健康监测和医学诊断。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）实现实时推断的方法论。我们的方法如下：将后验分布参数化为数据的函数。本文提出了两种不同的实现方案。第一种方法通过神经网络实现摊销全秩高斯引导来参数化后验分布。第二种方法采用条件归一化流引导，针对目标后验分布任意复杂的情况使用条件可逆神经网络。两种方法均通过摊销变分推断学习网络参数，该过程涉及最大化与模型兼容的所有可能数据集上的证据下界期望。我们通过求解科学与工程领域的一组基准问题验证了该方法。结果表明，我们方法的后验估计与马尔可夫链蒙特卡洛获得的相应真值一致。训练完成后，本方法仅需一次神经网络前向传播即可提供给定观测值的后验分布。