论文标题
使用物理知识的贝叶斯对不完美的计算机模型进行校准
Bayesian Calibration of Imperfect Computer Models using Physics-Informed Priors
论文作者
论文摘要
我们引入了一个计算有效的数据驱动框架，适合量化物理参数中的不确定性和计算机模型的模型公式，以微分方程表示。我们构建了物理知识的先验，它们是多输出的GP先验，它们在协方差函数中编码模型的结构。这将扩展到一个完全贝叶斯的框架，该框架量化了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，因此我们允许该模型通过考虑差异函数来偏离观察到的数据。为了推断，使用了汉密尔顿蒙特卡洛。 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).在研究中，结果表明，在1）现实比我们的建模选择更复杂的情况下，我们的建模框架可以恢复物理模型的真实参数，而2）数据采集过程有偏见，同时也产生准确的预测。此外，证明我们的方法比传统的贝叶斯校准方法更快。
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.
