Numerical simulations of high energy-density experiments require equation of state (EOS) models that relate a material's thermodynamic state variables -- specifically pressure, volume/density, energy, and temperature. EOS models are typically constructed using a semi-empirical parametric methodology, which assumes a physics-informed functional form with many tunable parameters calibrated using experimental/simulation data. Since there are inherent uncertainties in the calibration data (parametric uncertainty) and the assumed functional EOS form (model uncertainty), it is essential to perform uncertainty quantification (UQ) to improve confidence in the EOS predictions. Model uncertainty is challenging for UQ studies since it requires exploring the space of all possible physically consistent functional forms. Thus, it is often neglected in favor of parametric uncertainty, which is easier to quantify without violating thermodynamic laws. This work presents a data-driven machine learning approach to constructing EOS models that naturally captures model uncertainty while satisfying the necessary thermodynamic consistency and stability constraints. We propose a novel framework based on physics-informed Gaussian process regression (GPR) that automatically captures total uncertainty in the EOS and can be jointly trained on both simulation and experimental data sources. A GPR model for the shock Hugoniot is derived and its uncertainties are quantified using the proposed framework. We apply the proposed model to learn the EOS for the diamond solid state of carbon, using both density functional theory data and experimental shock Hugoniot data to train the model and show that the prediction uncertainty reduces by considering the thermodynamic constraints.

翻译：高能量密度实验的数值模拟需要状态方程（EOS）模型来描述材料的热力学状态变量——特别是压力、体积/密度、能量和温度。EOS模型通常采用半经验参数化方法构建，该方法假设具有物理启发的函数形式，并通过实验/模拟数据校准大量可调参数。由于校准数据存在固有不确定性（参数不确定性），且假定的EOS函数形式也存在不确定性（模型不确定性），因此必须进行不确定性量化（UQ）以提高EOS预测的置信度。模型不确定性对UQ研究具有挑战性，因为它需要探索所有可能的物理一致性函数形式空间。因此，人们常更倾向于处理参数不确定性，而忽略模型不确定性，因为参数不确定性更容易在违反热力学定律的前提下量化。本文提出了一种数据驱动的机器学习方法构建EOS模型，该方法能够自然捕获模型不确定性，同时满足必要的热力学一致性和稳定性约束。我们提出了一个基于物理信息高斯过程回归（GPR）的新框架，该框架可自动捕获EOS的总不确定性，并能同时基于模拟和实验数据源进行联合训练。推导了冲击雨贡纽线的GPR模型，并利用所提出的框架量化了其不确定性。我们应用所提模型学习碳金刚石固态的状态方程，使用密度泛函理论数据和实验冲击雨贡纽线数据训练模型，并表明考虑热力学约束可降低预测不确定性。