Machine-Learning-Compatible CALPHAD-type Optimization from Phase Equilibria by Auto-differentiation
Abstract
To accurately determine phase boundaries and phase transitions, thermodynamic models that describe phase free energies often have to be optimized based on experimentally observed phase equilibria. While different approaches exist for thermodynamic optimization, they are often implemented in ways that are not compatible with machine learning workflows requiring differentiable calculation of the loss function. In this work, we derive a phase-equilibrium loss function based on thermodynamic potentials that can be efficiently evaluated and enables gradient-based optimization by auto-differentiation in the \texttt{PyTorch} package. By minimizing this loss function, general thermodynamic model parameters can be optimized with respect to experimental phase-equilibria data. Using thermodynamic models in the CALculation of PHAse Diagram (CALPHAD) framework, we illustrate successful and efficient optimization in different systems, including ternary systems with more than 100 parameters. As the loss function is defined independently of the details of the thermodynamic models, it can be used to optimize machine learning thermodynamic models in general. In particular, we demonstrate top-down optimization of an atomistic potential from target phase equilibria.
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BibTeX
@misc{zhang2026machinelearningcompatible,
title = {{Machine-Learning-Compatible CALPHAD-type Optimization from Phase Equilibria by Auto-differentiation}},
author = {Wenhao Zhang and Jean-Claude Crivello and Yusuke Matsuoka and Toshiyuki Koyama and Taichi Abe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.00516},
eprint = {2608.00516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00516},
}
RIS
TY - UNPB
TI - Machine-Learning-Compatible CALPHAD-type Optimization from Phase Equilibria by Auto-differentiation
AU - Wenhao Zhang
AU - Jean-Claude Crivello
AU - Yusuke Matsuoka
AU - Toshiyuki Koyama
AU - Taichi Abe
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2608.00516
UR - https://arxiv.org/abs/2608.00516
AB - To accurately determine phase boundaries and phase transitions, thermodynamic models that describe phase free energies often have to be optimized based on experimentally observed phase equilibria. While different approaches exist for thermodynamic optimization, they are often implemented in ways that are not compatible with machine learning workflows requiring differentiable calculation of the loss function. In this work, we derive a phase-equilibrium loss function based on thermodynamic potentials that can be efficiently evaluated and enables gradient-based optimization by auto-differentiation in the \texttt{PyTorch} package. By minimizing this loss function, general thermodynamic model parameters can be optimized with respect to experimental phase-equilibria data. Using thermodynamic models in the CALculation of PHAse Diagram (CALPHAD) framework, we illustrate successful and efficient optimization in different systems, including ternary systems with more than 100 parameters. As the loss function is defined independently of the details of the thermodynamic models, it can be used to optimize machine learning thermodynamic models in general. In particular, we demonstrate top-down optimization of an atomistic potential from target phase equilibria.
ER -