Due to their uncertainty quantification, Bayesian solutions to inverse problems are the framework of choice in applications that are risk averse. These benefits come at the cost of computations that are in general, intractable. New advances in machine learning and variational inference (VI) have lowered the computational barrier by learning from examples. Two VI paradigms have emerged that represent different tradeoffs: amortized and non-amortized. Amortized VI can produce fast results but due to generalizing to many observed datasets it produces suboptimal inference results. Non-amortized VI is slower at inference but finds better posterior approximations since it is specialized towards a single observed dataset. Current amortized VI techniques run into a sub-optimality wall that can not be improved without more expressive neural networks or extra training data. We present a solution that enables iterative improvement of amortized posteriors that uses the same networks architectures and training data. The benefits of our method requires extra computations but these remain frugal since they are based on physics-hybrid methods and summary statistics. Importantly, these computations remain mostly offline thus our method maintains cheap and reusable online evaluation while bridging the approximation gap these two paradigms. We denote our proposed method ASPIRE - Amortized posteriors with Summaries that are Physics-based and Iteratively REfined. We first validate our method on a stylized problem with a known posterior then demonstrate its practical use on a high-dimensional and nonlinear transcranial medical imaging problem with ultrasound. Compared with the baseline and previous methods from the literature our method stands out as an computationally efficient and high-fidelity method for posterior inference.

翻译：由于具有不确定性量化能力，贝叶斯逆问题方法在风险厌恶型应用中成为首选框架。然而，这些优势以通常难以处理的计算为代价。机器学习与变分推断领域的新进展通过从样本中学习，降低了计算门槛。当前存在两种权衡不同的变分推断范式：摊销式与非摊销式。摊销变分推断能快速生成结果，但由于需泛化至多个观测数据集，其推理结果次优；而非摊销变分推断尽管推理速度较慢，但因专注于单个观测数据集，能获得更优的后验近似。现有摊销变分推断技术存在次优性壁垒，若不采用更具表达力的神经网络或额外训练数据则无法突破。我们提出一种能在不改变网络架构与训练数据的前提下迭代改进摊销后验的方法。该方法虽需额外计算，但由于采用物理混合方法与汇总统计量，这些计算仍保持经济性。关键在于，这些计算主要离线完成，因此我们的方法在弥合两种范式近似差距的同时，仍能保持低成本可复用的在线评估。我们将所提方法命名为ASPIRE——基于物理的汇总统计量迭代精化摊销后验。我们首先在已知后验的简化问题上验证该方法，随后将其应用于高维非线性经颅超声医学成像实际问题。与文献中的基线方法及先前方法相比，我们的方法在后验推理中兼具计算高效与高保真度的优势。