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. We extend this 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. To obtain the posterior distributions, we use Hamiltonian Monte Carlo sampling. We demonstrate our approach in a simulation study with hemodynamical models, which are time-dependent differential equations. Data are simulated from a more complex model than our modelling choice, and the aim is to learn physical parameters according to known mathematical connections. To demonstrate the flexibility of our approach, an example using the Heat equation, a space-time dependent differential equation where we consider a case of a biased data-acquisition process is also included. Finally, we fit the hemodynamic model using real data obtained in a medical trial.
翻译:我们引入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编制中的不确定性,以差异方程式为代表; 我们构建了物理知情的前题,即多输出GP前题,将模型的结构编码为共差函数; 我们将此扩展为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全地描述真实过程, 我们允许该模型通过考虑差异功能而偏离观察到的数据; 为了获得后方分布, 我们使用汉密尔顿·蒙特卡洛取样。 我们在模拟研究中展示了我们的方法, 模拟模型是取决于时间的偏差方程式。 数据是模拟模型比我们所选择的更复杂的模型模拟的, 目的是根据已知的数学联系来学习物理参数。 为了展示我们的方法的灵活性, 举例来说, 使用Heat方程式, 一种根据空间时间的差方程式, 我们认为存在数据获取偏差的情况。 最后, 我们用在医学试验中获得的真实数据来匹配 hemovical模型。