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模型，该模型描述材料热力学状态变量——即压力、体积/密度、能量和温度之间的关联。传统上，状态方程模型采用半经验参数化方法构建，该方法基于具备物理知识的函数形式，并通过实验或模拟数据校准大量可调参数。由于标定数据存在固有不确定性（参数不确定性），且假定的函数形式存在模型不确定性，量化不确定性对于提升状态方程预测的可靠性至关重要。模型不确定性给不确定性量化研究带来挑战，因其需探索所有物理一致函数形式的空间。因此，实践中常忽略模型不确定性而仅关注参数不确定性——后者更易在不违反热力学定律的前提下量化。本文提出一种数据驱动的机器学习方法构建状态方程模型，该方法能在满足必要热力学一致性和稳定性约束的同时，自然捕捉模型不确定性。我们提出基于物理约束高斯过程回归的新型框架，可自动捕获状态方程中的总不确定性，并支持在模拟与实验数据源上联合训练。基于该框架推导了冲击雨贡纽状态的高斯过程回归模型，并量化其不确定性。将所提模型应用于碳金刚石固态状态方程的学习，利用密度泛函理论数据与实验冲击雨贡纽数据训练模型，结果表明考虑热力学约束可降低预测不确定性。