# Learning thermodynamically constrained equations of state with uncertainty

arXiv: https://arxiv.org/abs/2306.17004 · HTML: https://mlsanity.com/paper/2306.17004/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 38 references that are in the map.
- First posted on arXiv 29 Jun 2023.

## Builds on (most cited first, 6 of 6)

- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Non-linear process convolutions for multi-output Gaussian processes](https://mlsanity.com/paper/1810.04632/) (2018)
- [Nonnegativity-Enforced Gaussian Process Regression](https://mlsanity.com/paper/2004.04632/) (2020)
- [UQpy v4.1: Uncertainty Quantification with Python](https://mlsanity.com/paper/2305.09572/) (2023)

## Built on by (most cited first, 5 of 5)

- [Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification](https://mlsanity.com/paper/2402.15115/) (2024)
- [Physics-Informed Polynomial Chaos Expansions](https://mlsanity.com/paper/2309.01697/) (2023)
- [Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curves](https://mlsanity.com/paper/2601.06655/) (2026)
- [Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models](https://mlsanity.com/paper/2601.03367/) (2026)
- [Structure-preserving uncertainty quantification for GENERIC dynamics](https://mlsanity.com/paper/2608.12624/) (2026)

## Abstract (arXiv)

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.