Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curves

TL;DR

This work addresses predicting shock-response Hugoniot states from limited molecular-dynamics data by embedding Rankine-Hugoniot constraints and thermodynamic laws directly into a Gaussian Process surrogate. The method constructs a Taylor-series-based physics-informed covariance and uses a multi-wave Rankine-Hugoniot formulation to model the leading elastic wave and trailing plastic and phase-transformation waves, with uncertainty quantified via the GP posterior. Applied to 3C-SiC along the [001] direction, the approach yields thermodynamically consistent Hugoniot curves for $u_s$, $\rho$, $P$, and $T$ across regimes, and identifies regime transitions with quantified predictive uncertainty. This framework offers data-efficient, uncertainty-aware surrogates for shock physics and lays groundwork for active-learning–driven autonomous materials discovery under extreme conditions.

Abstract

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.

Figures (16)

• Figure 1: Schematic representation of shock wave propagation in a material element.
• Figure 2: Shock velocity ($u_s$) vs. particle velocity ($u_p$) for single crystal SiC in the [0, 0, 1] orientation. Each 'x' denotes a mean value +/- one standard deviation from MD simulations. Four regions are denoted. In the 'Elastic' regime, a single elastic wave forms as shown by the blue x's. In the 'Elastic-Plastic' regime, a leading elastic wave forms (blue x's) and is trailed by a plastic wave (red x's) whose shock velocity increases with increasing particle velocity until the two waves merge. In the 'Plastic-Phase Transformation' regime, a leading plastic wave forms (blue x's) with a trailing phase transformation wave (green x's). In the 'Overdriven' regime, a single phase transformation leading wave forms whose shock velocity increases with increasing particle velocity.
• Figure 3: Schematic of shock wave propagation resulting in a two-wave structure.
• Figure 4: Shock wave simulations in MD: (a) Schematic setup; (b) Elastic wave propagation from $u_p=0.75$ km/s. (c) Dual elastic-plastic wave formation from $u_p=1.5$ km/s. (d) Dual plastic-phase transformation wave formation from $u_p=3.5$ km/s. (e) Overdriven shock propagation. In all figures, colors show atomic shear strains.
• Figure 5: Profiles of (a) temperature, (b) mass density, (c) total stress $\sigma_{zz}$ along shock direction (z: [001]), and (d) particle velocity from MD shock simulations at (1) $u_p=0.75$ km/s, (2) $u_p=1.5$ km/s, (3) $u_p=3.5$ km/s, and (4) $u_p=4.5$ km/s.