用贝叶斯方法融合物理模型，实现带不确定性的微分方程逆问题求解。
Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss
- 将PDE损失作为先验，结合数据似然进行贝叶斯推断
- 在1-3维合成数据上验证了对参数和不确定性估计的准确性
- 适用于医学影像中肿瘤浓度等关键参数的高可信度反演
逆问题是科学、工程和医学中数据融合、设计与成像的关键。其目标是从含噪观测数据和部分可观测过程中推断复杂系统的参数或隐状态。当测量结果仅为系统不完整或间接的视图时，需引入额外知识以准确求解逆问题。采用偏微分方程（PDE）形式的物理模型是填补这一空白的有效手段。特别是离散损失优化方法（ODIL）在鲁棒性和计算成本方面表现优异。本文提出B-ODIL，即ODIL的贝叶斯扩展，将ODIL的PDE损失作为先验，并与描述数据的似然函数结合，构建基于贝叶斯框架的PDE逆问题求解方法，实现解的不确定性量化。我们在一维、二维和三维合成基准上验证了B-ODIL的性能。进一步展示了该方法在三维肿瘤生长模型下，从脑部MRI扫描中估计肿瘤浓度及其不确定性的应用能力。
原文摘要 · Abstract (English)
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.
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