Abstract This paper proposes a general formulation for physics-based probabilistic models that are computationally convenient for uncertainty quantification and reliability analysis of complex systems while integrating the governing physical laws. The proposed formulation starts with the prediction of the quantities of interest using differential equations that represent the governing physical laws. For computational efficiency, the solution of the governing differential equations might be approximated. The predictions from the differential equations are then improved by introducing analytical correction terms that capture those physical characteristics of the phenomenon not fully captured by the differential equations. The paper also presents nested probabilistic models for uncertain physical characteristics that are difficult to measure. Observational data are required to calibrate the nested probabilistic models and correction terms. To provide context, the paper discusses physics-based probabilistic models for a class of boundary value problems that includes as a special case, the steady advection–diffusion–reaction equation, governing a diverse range of physical, chemical, and biological phenomena. Using the Bayesian approach, the differential equations are combined with observational data and any prior information to estimate the unknown model parameters and uncertain system characteristics. The paper then formulates the reliability problem for the computation of failure probability of physical and engineering systems using the proposed physics-based probabilistic models. To illustrate, the paper considers the reliability analysis of axially loaded rock socketed drilled shafts.

中文翻译：

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基于物理的概率模型：积分微分方程和观测数据

摘要 本文提出了一种基于物理的概率模型的一般公式，该模型在计算上方便复杂系统的不确定性量化和可靠性分析，同时集成了控制物理定律。建议的公式从使用代表控制物理定律的微分方程预测感兴趣的量开始。为了计算效率，控制微分方程的解可能是近似的。然后通过引入解析校正项来改进微分方程的预测，这些项捕获微分方程未完全捕获的现象的那些物理特征。该论文还针对难以测量的不确定物理特性提出了嵌套概率模型。需要观测数据来校准嵌套概率模型和校正项。为了提供背景，本文讨论了一类边界值问题的基于物理学的概率模型，其中包括作为特例的稳定对流-扩散-反应方程，控制着各种物理、化学和生物现象。使用贝叶斯方法，微分方程与观测数据和任何先验信息相结合，以估计未知的模型参数和不确定的系统特性。然后，本文使用所提出的基于物理的概率模型，为计算物理和工程系统的失效概率制定了可靠性问题。为了显示，