Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors
Michail Spitieris, Ingelin Steinsland; 24(108):1−39, 2023.
Abstract
In this paper, we present a computationally efficient data-driven framework for quantifying uncertainty in physical parameters and model formulation of computer models represented by differential equations. Our approach involves constructing physics-informed priors, which are multi-output Gaussian Process (GP) priors that encode the model’s structure in the covariance function. We extend this framework into a fully Bayesian approach that not only quantifies the uncertainty of physical parameters but also provides uncertainty estimates for model predictions. We acknowledge that physical models are often imperfect descriptions of real processes, so we allow the model to deviate from observed data by introducing a discrepancy function. To perform inference, we utilize Hamiltonian Monte Carlo. Additionally, we develop approximations for handling big data, significantly reducing the computational complexity from $\\mathcal{O}(N^3)$ to $\\mathcal{O}(N\\cdot m^2),$ where $m \\ll N.$ We demonstrate the effectiveness of our approach through simulation and real data case studies involving time-dependent Ordinary Differential Equations (ODEs) (specifically cardiovascular models) and space-time dependent Partial Differential Equations (PDEs) (specifically the heat equation). Our studies show that our modeling framework can accurately recover the true parameters of physical models even in cases where the reality is more complex than our modeling choice and the data acquisition process is biased. Furthermore, we demonstrate that our approach is computationally faster than traditional Bayesian calibration methods.
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