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，即从实验数据中估计物理模型参数，在科学与工程领域普遍存在。贝叶斯公式是黄金标准，因其能缓解不适定性问题并量化认知不确定性。由于解析后验分布通常无法获得，研究者通常采用马尔可夫链蒙特卡洛采样或近似变分推断。然而，每遇到新的数据集都需要从头重新进行推断。这一缺陷限制了贝叶斯公式在实时场景中的应用，例如工程系统的健康监测和医疗诊断。本文的目标是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：我们将后验分布参数化为数据的函数。本文概述了实现这一目标的两种不同途径。第一种方法通过神经网络实现，使用摊销全秩高斯引导体对后验分布进行参数化。第二种方法采用条件归一化流引导体，利用条件可逆神经网络处理目标后验分布高度复杂的情况。在两种方法中，我们均通过摊销变分推断学习网络参数，这涉及在模型兼容的所有可能数据集上最大化证据下界的期望。我们通过求解一系列科学与工程领域的基准问题来验证所提方法。结果表明，我们的方法得到的后验估计与通过马尔可夫链蒙特卡洛获得的对应真实结果一致。训练完成后，我们的方法仅需神经网络一次前向传播的计算代价，即可为给定观测提供后验分布。