标题： 基于离散损失优化的PDE反问题贝叶斯推断 显示英文标题

标题： Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss

摘要： 反问题在科学、工程和医学中的许多应用中至关重要，这些应用涉及数据同化、设计和成像。 它们的求解是从噪声数据和部分可观测过程中推断复杂系统的参数或潜在状态。 当测量值是对系统的不完整或间接观察时，需要额外的知识来准确解决反问题。 采用偏微分方程（PDEs）形式的系统物理模型是一种有效的方法来填补这一空白。 特别是，优化离散损失（ODIL）方法在鲁棒性和计算成本方面表现出巨大的潜力。 在本工作中，我们引入了B-ODIL，这是ODIL的一种贝叶斯扩展，它将ODIL的PDE损失作为先验知识，并将其与描述数据的似然结合。 B-ODIL采用基于PDE的反问题的贝叶斯公式，以推断具有量化不确定性的解决方案。 我们在一系列涉及一维、二维和三维PDE的合成基准测试中展示了B-ODIL的能力。 我们展示了B-ODIL在从MRI扫描中估计患者大脑中的肿瘤浓度及其不确定性方面的应用，使用的是三维肿瘤生长模型。 显示英文摘要

摘要： 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.