Bayesian Calibration of imperfect computer models using Physics-informed priors

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. We extend this 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. To obtain the posterior distributions, we use Hamiltonian Monte Carlo sampling. We demonstrate our approach in a simulation study with hemodynamical models, which are time-dependent differential equations. Data are simulated from a more complex model than our modelling choice, and the aim is to learn physical parameters according to known mathematical connections. To demonstrate the flexibility of our approach, an example using the Heat equation, a space-time dependent differential equation where we consider a case of a biased data-acquisition process is also included. Finally, we fit the hemodynamic model using real data obtained in a medical trial.