Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors
This work addresses uncertainty quantification for researchers and practitioners using differential equation-based models in fields like cardiovascular modeling and heat transfer, offering an incremental improvement in computational efficiency.
The paper tackles the problem of quantifying uncertainty in physical parameters and model predictions for imperfect computer models, using physics-informed priors and a Bayesian framework with discrepancy functions, and demonstrates accurate parameter recovery and faster computation compared to traditional methods.
We introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters and model formulation of computer models, represented by differential equations. We construct physics-informed priors, which are multi-output GP priors that encode the model's structure in the covariance function. This is extended into a fully Bayesian framework that quantifies the uncertainty of physical parameters and model predictions. Since physical models often are imperfect descriptions of the real process, we allow the model to deviate from the observed data by considering a discrepancy function. For inference, Hamiltonian Monte Carlo is used. Further, approximations for big data are developed that reduce the computational complexity from $\mathcal{O}(N^3)$ to $\mathcal{O}(N\cdot m^2),$ where $m \ll N.$ Our approach is demonstrated in simulation and real data case studies where the physics are described by time-dependent ODEs describe (cardiovascular models) and space-time dependent PDEs (heat equation). In the studies, it is shown that our modelling framework can recover the true parameters of the physical models in cases where 1) the reality is more complex than our modelling choice and 2) the data acquisition process is biased while also producing accurate predictions. Furthermore, it is demonstrated that our approach is computationally faster than traditional Bayesian calibration methods.