We introduce a computational efficient data-driven framework suitable for the quantification of the uncertainty in physical parameters of computer models, represented by differential equations. We construct physics-informed priors for time-dependent differential equations, which are multi-output Gaussian process (GP) priors that encode the model's structure in the covariance function. We extend this into a fully Bayesian framework which allows quantifying the uncertainty of physical parameters and model predictions. Since physical models are usually 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 (HMC) sampling.

This work is primarily motivated by the need for interpretable parameters for the hemodynamics of the heart for personal treatment of hypertension. The model used is the arterial Windkessel model, which represents the hemodynamics of the heart through differential equations with physically interpretable parameters of medical interest. As most physical models, the Windkessel model is an imperfect description of the real process. To demonstrate our approach we simulate noisy data from a more complex physical model with known mathematical connections to our modeling choice. We show that without accounting for discrepancy, the posterior of the physical parameters deviates from the true value while accounting for discrepancy gives reasonable quantification of physical parameters uncertainty and reduces the uncertainty in subsequent model predictions.