Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomplete or indirect view of the system, additional knowledge is required to accurately solve the inverse problem. Adopting a physical model of the system in the form of partial differential equations (PDEs) is a potent method to close this gap. In particular, the method of optimizing a discrete loss (ODIL) has shown great potential in terms of robustness and computational cost. In this work, we introduce B-ODIL, a Bayesian extension of ODIL, that integrates the PDE loss of ODIL as prior knowledge and combines it with a likelihood describing the data. B-ODIL employs a Bayesian formulation of PDE-based inverse problems to infer solutions with quantified uncertainties. We demonstrate the capabilities of B-ODIL in a series of synthetic benchmarks involving PDEs in one, two, and three dimensions. We showcase the application of B-ODIL in estimating tumor concentration and its uncertainty in a patient's brain from MRI scans using a three-dimensional tumor growth model.

翻译：反问题对于涉及数据同化、设计与成像的科学、工程和医学众多应用至关重要。其求解旨在从含噪声数据与部分可观测过程中推断复杂系统的参数或隐状态。当测量数据仅提供系统的不完整或间接视图时，需要额外知识才能准确求解反问题。采用偏微分方程（PDE）形式的系统物理模型是弥合此差距的有效方法。其中，离散损失优化（ODIL）方法已在鲁棒性与计算成本方面展现出巨大潜力。本文提出B-ODIL——ODIL的贝叶斯扩展框架，将ODIL的PDE损失作为先验知识，并与描述数据的似然函数相结合。B-ODIL采用基于PDE的反问题的贝叶斯表述，以推断具有量化不确定性的解。我们通过一系列涉及一维、二维及三维PDE的合成基准测试验证了B-ODIL的性能。最后，我们利用三维肿瘤生长模型，展示了B-ODIL在根据MRI扫描估计患者脑部肿瘤浓度及其不确定性的实际应用。