力学中的不确定性量化：统一的贝叶斯视角
Uncertainty quantification in mechanics: A unified Bayesian perspective
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中文总结 AI 辅助
研究力学中不确定性量化的正向和反向问题，基于贝叶斯概率论提供统一理论框架，可无缝纳入模型选择、实验设计等子问题，尤其适用于生物力学，解决其因多种因素导致的变异性和不确定性。
中文摘要 AI 辅助
不确定性量化（UQ）在实验力学中至关重要，在计算力学中也尤为关键，表现为正向和反向问题这两种基本类型。正向问题关注输入不确定性如何传播到感兴趣的量，反向问题旨在从实验观测或模拟中推断未知参数。由于高效传播通常需要大量评估来计算边际输出分布，因此需要快速、数据驱动的替代模型。我们区分了两种反向任务：输入不确定性的识别和校准以及替代模型的构建，统称为基于替代模型的UQ。基于概率推理和部分信念概念，我们证明贝叶斯概率论为解决这两种问题类型提供了统一的理论框架。我们还表明贝叶斯推理允许无缝纳入基本子问题，包括用于识别最可能模型规格的模型选择和用于通过识别最大化关于参数的预期信息增益的实验或模拟来优化数据收集的实验设计等。虽然该理论框架是针对一般力学问题提出的，但特别强调了生物力学，因其存在固有的生物异质性、患者特异性变异性和噪声数据，变异性和不确定性尤为明显。
英文摘要
Uncertainty quantification (UQ) is essential to experimental mechanics, but has become particularly relevant in computational mechanics, manifesting in two fundamental problem types: forward and inverse problems. The former addresses how input uncertainties propagate to the quantities of interest, whereas the latter aims to infer unknown parameters from experimental observations or simulations. Since efficient propagation typically requires a prohibitive number of evaluations to compute marginal output distributions, the development of fast, data-driven surrogate models becomes necessary. Thus, we can distinguish between two inverse tasks: (i) the identification and calibration of input uncertainties, and (ii) the construction of surrogates, a methodology collectively referred to as surrogate-based UQ. Building on probabilistic reasoning and the concept of partial belief, we demonstrate that Bayesian probability theory provides a unified theoretical framework for addressing both problem types. We further show that Bayesian inference allows for the seamless incorporation of essential subproblems, including model selection for identifying the most probable model specifications and experimental design for optimizing data collection by identifying experiments or simulations that maximize expected information gain about parameters, among others such as connections to sensitivity analysis or the use of special priors like random fields. While this theoretical framework is presented for general mechanical problems, particular emphasis is placed on biomechanics, where variability and uncertainty is especially pronounced due to inherent biological heterogeneity, patient-specific variability, and noisy data.
