Quantifying Uncertainty in Equation-of-State Models with Thermodynamically Constrained Machine Learning *

Equation-of-state (EOS) models provide an interesting challenge for uncertainty quantification and machine learning since they must be trained on very sparse data and then extrapolated over huge regions of the input parameter space. They are also subject to strong physical constraints; even small deviations from so-called thermodynamic consistency results in the failure of downstream tasks, for example radiation-hydrodynamics simulations. On the other hand, the constraints are known analytically and so can be enforced by choosing a suitable form for the EOS.

Current approaches to EOS model building and uncertainty quantification (UQ) do not capture the uncertainty in the model form, potentially underestimating the uncertainty in extrapolation regions. The usual approach is to choose a reliable functional form for the EOS and parametrically fit to the available data; any uncertainty analysis is then limited to investigating the uncertainty in the values of the parameters. This approach guarantees a useable EOS model but does not address the uncertainty in the choice of underlying functional form.

Gaussian Processes (GPs) provide a potential alternative to the current approach that can constrain the missing model uncertainty. GPs are well known to explore spaces of functions and have a strong statistical interpretation leading to meaningful uncertainties. In this work we will formulate a constrained GP that explores the space of thermodynamically consistent functions and provides an upper limit on model uncertainty in EOS tables. We will demonstrate the approach using simulation data for Boron Carbide and constrain uncertainty in the resulting EOS due to data sparsity, simulation noise and extrapolation away from available data. Finally, we will discuss further constraints arising from known limiting behavior like ideality at high temperature and the melt curve at low temperature.