Learning Physics between Digital Twins with Low-Fidelity Models and Physics-Informed Gaussian Processes

A digital twin is a computer model that represents an individual, for example, a component, a patient or a process. In many situations, we want to gain knowledge about an individual from its data while incorporating imperfect physical knowledge and also learn from data from other individuals. In this paper, we introduce and demonstrate a fully Bayesian methodology for learning between digital twins in a setting where the physical parameters of each individual are of interest. For each individual, the methodology is based on Bayesian calibration with model discrepancy. Through the discrepancy, modelled as a Gaussian process, the imperfect low-fidelity physical model is accounted for. Using ideas from Bayesian hierarchical models, a joint probabilistic model of digital twins is constructed by connecting them through a new level in the hierarchy. For the physical parameters, the methodology can be seen as using a prior distribution in the individual model that is the posterior of the corresponding hyperparameter in the joint model. For learning the imperfect physics between individuals two approaches are introduced, one that assumes the same discrepancy for all individuals and one that can be seen as using a prior learned from all individuals for the parameters of the Gaussian processes representing the discrepancies. Based on recent advances related to physics-informed priors, Hamiltonian Monte Carlo methods and using these for inverse problems we set up an inference methodology that allows our approach to be computational feasible also for physical models based on partial differential equations and individual data that are not aligned. The methodology is demonstrated in two synthetic case studies, a toy example previously used in the literature extended to more individuals and an example based on a cardiovascular differential equation model relevant for the treatment of hypertension.

Bayesian Calibration of imperfect computer models using Physics-informed priors

In this work we introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters of computer models, represented by differential equations. We construct physics-informed priors for 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. For inference Hamiltonian Monte Carlo (HMC) sampling is used. This work is 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 when accounting for discrepancy gives reasonable quantification of physical parameters uncertainty and reduces the uncertainty in subsequent model predictions.