Refining Machine Learning Potentials through Thermodynamic Theory of Phase Transitions

TL;DR

The paper addresses the mismatch between experimental phase behavior and predictions from foundational ML potentials by introducing DiffTTC, a top-down refinement that uses Differentiable Trajectory Reweighting to minimize phase free-energy differences at target conditions. The approach is model-agnostic and compatible with existing DiffTRe workflows, enabling direct alignment of phase diagrams with experiments while preserving out-of-target properties. Applied to titanium, DiffTTC yields phase boundaries within ~50 K of experiment across 0–5 GPa, improves the physical realism of the phase diagram, and demonstrates potential applicability to multi-component systems. This work provides a practical pathway to highly accurate, application-specific ML potentials through targeted thermodynamic correction without overhauling underlying training data.

Abstract

Foundational Machine Learning Potentials can resolve the accuracy and transferability limitations of classical force fields. They enable microscopic insights into material behavior through Molecular Dynamics simulations, which can crucially expedite material design and discovery. However, insufficiently broad and systematically biased reference data affect the predictive quality of the learned models. Often, these models exhibit significant deviations from experimentally observed phase transition temperatures, in the order of several hundred kelvins. Thus, fine-tuning is necessary to achieve adequate accuracy in many practical problems. This work proposes a fine-tuning strategy via top-down learning, directly correcting the wrongly predicted transition temperatures to match the experimental reference data. Our approach leverages the Differentiable Trajectory Reweighting algorithm to minimize the free energy differences between phases at the experimental target pressures and temperatures. We demonstrate that our approach can accurately correct the phase diagram of pure Titanium in a pressure range of up to 5 GPa, matching the experimental reference within tenths of kelvins and improving the liquid-state diffusion constant. Our approach is model-agnostic, applicable to multi-component systems with solid-solid and solid-liquid transitions, and compliant with top-down training on other experimental properties. Therefore, our approach can serve as an essential step towards highly accurate application-specific and foundational machine learning potentials.

Figures (5)

• Figure 1: Differentiable Transition Temperature Correction (DiffTTC) Method. 1. The method begins with a pre-trained potential. 2. Using the pretrained model with parameters $\theta_0$, the free energy difference $\Delta G^{\text{I}\rightarrow\text{II}}_{\theta_0}$ between phases I (blue line) and II (orange line) at the experimental transition temperature $T_\text{exp}$ is computed, e.g., by extrapolating the phase free energy change in temperature from the coexistence point (red star) at $\hat{T}$. 3. DiffTTC corrects the melting temperature by iteratively refining the potential parameter $\theta$, by matching the free energy changes $\Delta F_{\theta_0\rightarrow\theta}^{\text{I}/\text{II}}$ (orange and blue points) towards dynamically adjusted targets (orange and blue stars), eventually compensating the initial free energy difference (dashed arrow).
• Figure 2: Predicted Energies and Forces: Differences in the predicted energy and forces between the pretrained and the DiffTTC-refined model on the test split versus the DFT reference energy. The right plot shows the angle of predicted forces with respect to the DFT force. The black dashed line in the left plot corresponds to a linear regression fit to the energy differences. The color corresponds to the dominant crystal structure identified via the polyhedral template matching method larsenRobustStructuralIdentification2016. Only every 70th force component is shown.
• Figure 3: Phase-diagram: Predictions of the pretrained and through DiffTTC refined MACE model are shown in comparison to the DP model wenSpecialisingNeuralNetwork2021, the foundational MACE-MP-0b3 model batatia2025foundationmodelatomisticmaterials (only BCC-liquid phase transition), and experimental measurements. The values for the DP model and the experimental data (black dashed lines) are extracted from wenSpecialisingNeuralNetwork2021. The error bars denote the uncertainty in the mean temperature in the coexistence simulations for the BCC-liquid transition and approximate the error in temperature of the free energy intersection point for the HCP-BCC transition based on error estimates reported in freitas2016 for the free energy integration methods.
• Figure 4: Lattice Parameters and Volumes: Predictions of the pretrained and refined MACE model are shown in comparison to the DP model wenSpecialisingNeuralNetwork2021 and experimental measurements. The values for the DP model and the experimental data corresponding to the black dashed lines in figures (a-b) are extracted from wenSpecialisingNeuralNetwork2021. The black crosses are taken from jspreadboroughMeasurementLatticeExpansions1959. The regression line (dotted black) and uncertainty estimates (solid black) in plot (f) are converted from density measurements from ozawaPreciseDensityMeasurement2017.
• Figure 5: Out-of-Target Properties for Liquid Titanium.a Radial Distribution Function (RDF) and Angular Distribution Function (ADF) predicted for the pretrained and DiffTTC potential at $0\ \mathrm{GPa}$ and $1960\ \mathrm{K}$ in comparison to experimental measurements for RDF holland-moritzShortrangeOrderStable2007 and ADF kimStructuralStudySupercooled2007. b Diffusion constant at $0\ \mathrm{GPa}$ estimated for the pretrained and DiffTTC potential through velocity autocorrelation function (VACF, crosses) and temporal evolution of mean-squared displacement (points) in comparison to the experimental reference data horbachImprovementComputerSimulation2009 (black points). The solid lines are a linear regression fit to the VACF measurements. The dashed lines denote two times the standard error of the regression line offset.