Development of an uncertainty-aware equation of state for gold

This study introduces a framework that employs Gaussian Processes (GPs) to develop high-fidelity equation of state (EOS) tables, essential for modeling material properties across varying temperatures and pressures. GPs offer a robust predictive modeling approach and are especially adept at handling uncertainties systematically. By integrating Error-in-Variables (EIV) into the GP model, we adeptly navigate uncertainties in both input parameters (like temperature and density) and output variables (including pressure and other thermodynamic properties). Our methodology is demonstrated using first-principles density functional theory (DFT) data for gold, observing its properties over maximum density compression (up to 100 g/cc) and extreme temperatures within the warm dense matter region (reaching 300 eV). Furthermore, we assess the resilience of our uncertainty propagation within the resultant EOS tables under various conditions, including data scarcity and the intrinsic noise of experiments and simulations.

💡 Research Summary

The paper presents a novel framework for constructing an uncertainty‑aware equation of state (EOS) for gold (Au) that simultaneously provides high‑fidelity thermodynamic values and rigorous uncertainty quantification. Traditional EOS generation methods treat input variables such as density and temperature as exact, propagating only implicit model errors. This can lead to under‑estimation of uncertainty, especially when data are sparse or extrapolation is required. To overcome these limitations, the authors adopt a Bayesian regression approach based on Gaussian Processes (GPs) and extend it with an Error‑in‑Variables (EIV) formulation that explicitly incorporates uncertainties in both inputs and outputs.

In the methodological core, the latent thermodynamic function f(x) maps a state vector x (e.g., density ρ and temperature T) to an observable such as free energy or pressure. Each training datum (x_i, y_i) is modeled as y_i = f(˜x_i) + ε_y,i with ε_y,i ∼ N(0, σ_y,i²) and ˜x_i = x_i – ε_x,i with ε_x,i ∼ N(0, Σ_x,i). When Σ_x,i = 0 the model reduces to a standard noisy‑output GP; otherwise the input noise is folded into an effective output variance σ_eff,i² ≈ σ_y,i² + ∇f(˜x_i)ᵀ Σ_x,i ∇f(˜x_i). This coupling makes the training iterative: the GP hyper‑parameters are optimized while the effective variances are updated until convergence.

The authors employ stationary kernels with automatic relevance determination (ARD). The baseline is the squared‑exponential (SE) kernel, but Matérn 3/2 and 5/2 kernels are also examined to test sensitivity to smoothness assumptions. Because the free energy is decomposed into three physically distinct contributions—cold curve, electron‑thermal, and ion‑thermal—each component is modeled by a separate GP. The total Helmholtz free energy is reconstructed as F(ρ,T) = F_cold(ρ) + F_vib(ρ,T) + F_elec(ρ,T). Analytic differentiation of the GP surrogate yields pressure and other thermodynamic derivatives together with propagated uncertainties, enabling a fully Bayesian EOS table.

The training data consist of extensive density‑functional theory (DFT) calculations. Multiple exchange‑correlation functionals (LDA, PBE, PBEsol) and spin‑orbit coupling are explored to capture systematic model spread. An optimized norm‑conserving ONCV pseudopotential with 33 valence states is used to generate a dense grid covering densities from 0.5 ρ₀ to 5 ρ₀ and temperatures up to 300 eV (≈3.5 × 10⁶ K). Electron‑thermal contributions are obtained by solving the Kohn‑Sham equations with Fermi‑Dirac occupations; ion‑thermal terms are derived from self‑consistent phonon calculations for solids and Born‑Oppenheimer molecular dynamics for liquids. Output uncertainties σ_y,i are assigned based on convergence criteria (energy <1 meV/atom, pressure <0.1 kbar) and on known DFT systematic errors; input uncertainties Σ_x,i are set to zero for pure DFT data but become non‑zero when experimental Hugoniot points are incorporated (density error bars treated as input noise, pressure errors as output noise).

The resulting uncertainty‑aware EOS, labeled U790, is benchmarked against diamond‑anvil‑cell (DAC) measurements, toroidal DAC data, and shock‑wave Hugoniot experiments. Compared with the established LLNL tables L790 and Y790, U790 shows agreement within 2–3 % over most of the pressure range up to ~600 GPa. At the highest pressures (≈27 g/cc, >30 GPa) discrepancies rise to 5–8 %, which the authors attribute to DFT functional choices and electron‑thermal modeling. Importantly, the GP provides credible intervals that widen in sparsely sampled regions, offering a quantitative measure of confidence that traditional tables lack.

Computationally, the authors acknowledge the O(N³) scaling of exact GP inference but demonstrate that sparse approximations and inducing‑point methods (as implemented in the GPz library) enable training on several thousand DFT points with manageable resources. They also discuss extensions to multi‑fidelity GP (co‑kriging) where low‑fidelity simulations guide the global trend and high‑fidelity experiments anchor local accuracy, without any single fidelity dominating the fit.

In conclusion, the paper delivers a practical, scalable pipeline for generating EOS tables that embed uncertainty directly into the model. This approach facilitates rigorous error propagation in high‑energy‑density physics simulations, improves model validation against heterogeneous data sources, and opens pathways for extending uncertainty‑aware EOS generation to other materials and multi‑phase systems. Future work will address broader material sets, phase‑transition modeling, and integration of the U‑EOS into real‑time simulation workflows.