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:我们引入了一个计算高效的数据驱动框架,适合于量化计算机模型物理参数和模型拟订的不确定性,以差异方程式为代表。我们构建了一个计算高效的数据驱动框架,以量化计算机模型物理参数和模型拟订的不确定性,我们以差异方程式为代表。我们构建了一个物理知情的前置程序,即多输出GP前端,将模型的结构编码为共差函数。这个框架将扩展至完全的巴伊西亚框架,对物理参数和模型预测的不确定性进行量化。由于物理模型往往不完善地描述真实过程,我们允许模型通过考虑一个差异功能偏离观察到的数据。为了推断,我们使用了汉密尔顿·蒙特·卡洛。此外,我们开发了大数据的近似点,将计算复杂性从$\mathcal{O}(N3/3)减为$\mathcal{O}(N\cdot m%2)到$\\\\\cd m2,在模拟和实时模型的精确度选择中,我们模拟框架可以追溯到更精确的物理模型。