Physics-based probabilistic models: Integrating differential equations and observational data

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.