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Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curves

A physics-constrained Gaussian Process regression framework is developed for predicting shocked material states along the Hugoniot curve using data from a small number of shockwave simulations. The proposed Gaussian process employs a probabilistic Taylor series expansion in conjunction with the Rankine-Hugoniot jump conditions between the various shocked material states to construct a thermodynamically consistent covariance function. This leads to the formulation of an optimization problem over a small number of interpretable hyperparameters and enables the identification of regime transitions, from a leading elastic wave to trailing plastic and phase transformation waves. This work is motivated by the need to investigate shock-driven material response for materials discovery and for offering mechanistic insights in regimes where experimental characterizations and simulations are costly. The proposed methodology relies on large-scale molecular dynamics which are an accurate but expensive computational alternative to experiments. Under these constraints, the proposed methodology establishes Hugoniot curves from a limited number of molecular dynamics simulations. We consider silicon carbide as a representative material and atomic-level simulations are performed using a reverse ballistic approach together with appropriate interatomic potentials. The framework reproduces the Hugoniot curve with satisfactory accuracy while also quantifying the uncertainty in the predictions using the Gaussian Process posterior.