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.

翻译：我们提出了一种计算高效的数据驱动框架，适用于量化由微分方程描述的计算机模型中物理参数和模型公式的不确定性。我们构建了物理信息先验，这是一种多输出高斯过程先验，通过协方差函数编码模型的结构。该框架被扩展为完整的贝叶斯框架，用于量化物理参数和模型预测的不确定性。由于物理模型通常是对真实过程的不完美描述，我们通过考虑一个偏差函数，允许模型偏离观测数据。在推断过程中，使用了哈密顿蒙特卡洛方法。此外，针对大数据场景开发了近似方法，将计算复杂度从$\mathcal{O}(N^3)$降低至$\mathcal{O}(N\cdot m^2)$，其中$m \ll N$。我们的方法在仿真和真实数据案例研究中进行了验证，这些案例中的物理过程由时间相关常微分方程（心血管模型）和时空相关偏微分方程（热方程）描述。研究表明，在以下情况下，我们的建模框架能够恢复物理模型的真实参数：1）现实情况比我们的建模选择更复杂；2）数据采集过程存在偏差。同时，该方法还能产生准确的预测。此外，我们还证明了该方法在计算速度上优于传统的贝叶斯校准方法。