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.
翻译:在这项工作中,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的不确定性,以差异方程为代表。我们为差异方程为不同的方程构建了物理知情前程,这是多输出Gaussian进程(GP)的前程,将模型的结构编码在共变函数中。我们将这一框架扩展到一个完全的巴伊西亚框架,使物理参数和模型预测的不确定性量化。由于物理模型通常对真实过程的描述不完善,我们允许该模型通过考虑差异函数而偏离观察到的数据。使用了推断汉密尔顿·蒙特卡洛(HMC)取样。这项工作的动机是需要为个人治疗高血压的心脏肝动力学提供可解释的参数。我们使用的模型是动脉型Windkesels模型,它代表心脏的心动动力,通过具有实际解释医学利益参数的不同方程。作为大多数物理模型,Windkesel模型是对真实过程的不完美的描述。我们模拟了我们从一个比较复杂的物理模型中提取数据的方法。我们模拟了从一个比较复杂的物理变量模型和已知的数学联系,而随后的精确的精确的精确的精确的精确的计算参数,我们又选择了模型的精确的精确的精确的精确的计算。我们为了模型的精确的精确的精确的数值。我们做了选择。