Physics‑Constrained Gaussian Processes for a Thermodynamically Consistent Equation of State

We introduce a data‑driven framework that learns an equation of state (EOS) while enforcing core principles of thermodynamics. Our physics‑constrained Gaussian Process (PC‑GP) embeds the required relationships among pressure, temperature, volume, and free energy and respects stability conditions, ensuring that predicted responses are physically meaningful. The model fuses heterogeneous, multi‑fidelity datasets—from density‑functional theory and from large‑scale simulations using machine‑learning interatomic potentials—without assuming a specific analytic EOS form, and it supplies quantified uncertainties for all predictions.

We demonstrate the approach on lead, whose phase diagram contains multiple solid phases and melting across a wide pressure–temperature range. The learned EOS recovers phase equilibria and yields coexistence‑line (“Clapeyron”) slopes—with uncertainty bounds—that quantify how phase boundaries shift with pressure and temperature. Compared with unconstrained fits, the PC‑GP removes nonphysical behavior, tightens predictions near phase boundaries, and identifies data gaps that most affect uncertainty, enabling targeted follow‑up calculations. The method is general and data‑efficient, providing practical uncertainty‑aware EOS models for materials with complex phase behavior.