PhaseForgePlus: Gradient-Optimized CALPHAD Model

• PhaseForgePlus is a computational pipeline that integrates machine-learned interatomic potentials, ATAT-based thermodynamics, and gradient-informed optimization to generate physically informed CALPHAD models.
• It streamlines the complete modeling process from initial Gibbs energy construction to parameter tuning, ensuring phase diagram predictions align closely with experimental equilibria.
• The workflow achieves significant speedups and accuracy improvements by leveraging analytic gradients and a reproducible, open-stack design, preserving critical physical priors.

PhaseForgePlus is a computationally efficient, fully open-source workflow for physically-informed CALPHAD model generation and parameter fitting. It was introduced through a Pt–W case study as an end-to-end pipeline that combines machine-learned interatomic potentials, ATAT-based alloy thermodynamics, and gradient-informed optimization using experimental data, with the stated goal of producing reliable initial Gibbs energy models and then tuning a compact set of thermodynamic parameters so that the final phase diagram closely matches experimental phase equilibria without deviating far from the physically grounded baseline (Kunselman et al., 1 Aug 2025).

1. Definition, scope, and positioning

PhaseForgePlus was created to bridge three communities and toolchains: machine-learned interatomic potentials (MLPs), ATAT-based alloy thermodynamics for physically grounded Gibbs energy models, and modern gradient-based parameter optimization using experimental data through PyCalphad and ESPEI with the Jansson derivative method (Kunselman et al., 1 Aug 2025). In this formulation, the workflow is not limited to parameter regression in isolation; it unifies data generation, model construction, and parameter tuning under a reproducible, open stack that avoids proprietary databases and licenses.

Its novelty is framed against prior CALPHAD assessment practice in two ways. First, it uses MLPs to generate initial Gibbs energy descriptions within ATAT/PhaseForge, providing physics-based starting points instead of ad hoc polynomials or purely database-driven guesses. Second, it replaces black-box or MCMC-only parameter assessment with analytically differentiable, gradient-informed optimization grounded in the Jansson derivative method. The stated consequence is orders-of-magnitude speedups while preserving physical priors inherited from the MLP-derived model (Kunselman et al., 1 Aug 2025).

The demonstrated scope in Pt–W includes data preparation for BCC and FCC solution phases, liquid-phase energy estimation, unary end members, handling mechanically unstable phases such as FCC W and BCC Pt, Gibbs energy model construction, TDB export, gradient-informed maximum-likelihood fitting, and validation against experimental phase equilibria and thermochemical consistency checks (Kunselman et al., 1 Aug 2025). This places PhaseForgePlus at the interface between ML-informed thermodynamic model generation and experimental CALPHAD reassessment, rather than as a replacement for CALPHAD formalism itself.

2. Workflow architecture and software stack

The operational stack centers on PhaseForge, ATAT, Grace, PyCalphad, ESPEI, and SciPy. In the Pt–W study, Special Quasirandom Structures for FCC and BCC up to level 2 were taken from the ATAT database; relaxations and total energies were computed with Grace, specifically the GRACE-2L-OMAT foundation model; unary end-member thermodynamic properties were taken from a PhaseForge database based on VASP DFT; and the resulting TDB file generated in PhaseForge was imported directly into PyCalphad for calculations (Kunselman et al., 1 Aug 2025).

Component	Implementation in PhaseForgePlus	Role
ATAT interface	SQS for FCC and BCC; inflection detection	Structure generation and unstable unary handling
MLP layer	Grace (GRACE-2L-OMAT)	Relaxations and total energies
Thermodynamic construction	PhaseForge	TDB construction and liquid energies
Assessment layer	PyCalphad and ESPEI	Likelihood evaluation and parameter fitting
Optimizer	SciPy conjugate gradient	Gradient-informed optimization

The data pipeline is explicitly bifurcated. On the model-construction side, MLP/DFT data are used by PhaseForge to construct a TDB containing solution phases and liquid energies, with liquid energies obtained using a ternary-search method with a 50 K offset. On the assessment side, experimental phase equilibria from ASM Alloy Phase Diagram Database #101202 are converted into a log-likelihood composed of ZPF residuals and non-equilibrium thermochemical constraints, specifically mixing enthalpies at 298 K (Kunselman et al., 1 Aug 2025).

The optimization module uses ESPEI gradients via the Jansson derivative method, with SciPy conjugate gradient as the algorithm and simple preconditioning by scaling large temperature-independent parameters by 1000. The repository is identified as https://github.com/dogusariturk/PhaseForgePlus, and the named dependencies include ATAT, PhaseForge, Grace MLP, PyCalphad, ESPEI with PR #268 for gradient support, and SciPy (Kunselman et al., 1 Aug 2025).

3. Thermodynamic formulation and derivative-based optimization

For the Pt–W binary substitutional solution phases, the Gibbs energy is written as

$$G^{\phi}(T,x) = x\,G^{\circ,\phi}_{\mathrm{Pt}}(T) + (1-x)\,G^{\circ,\phi}_{\mathrm{W}}(T) + RT\,[x\ln x + (1-x)\ln(1-x)] + G^{\text{ex},\phi}(T,x),$$

where $\phi \in \{\mathrm{FCC}, \mathrm{BCC}\}$ and the excess term is modeled by Redlich–Kister polynomials,

$$G^{\text{ex},\phi}(T,x) = x(1-x)\sum_{k=0}^{m} L^{\phi}_{k}(T)\,(2x-1)^k.$$

In the Pt–W assessment, binary interactions up to level 2 were included, and linear temperature dependence was introduced for the 0th- and 1st-order interaction parameters,

$$L^{\phi}_{k}(T)=a^{\phi}_{k}+b^{\phi}_{k}T \qquad \text{for } k\in\{0,1\},$$

with remaining terms temperature-independent unless specified otherwise (Kunselman et al., 1 Aug 2025).

The fitting problem is cast as maximum likelihood, equivalently weighted least squares under normal residuals, with log-likelihood

$$l(\boldsymbol{\theta}) = \sum_{i=1}^{n} \left[ \ln\!\left(\frac{w_i}{\sigma_i\sqrt{2\pi}}\right) - \frac{1}{2} \left(\frac{w_i\,X_i(\boldsymbol{\theta})}{\sigma_i}\right)^2 \right].$$

The gradient is

$$\frac{\partial l}{\partial \boldsymbol{\theta}} = -\sum_{i=1}^{n} \left( \frac{w_i^2\,X_i}{\sigma_i^2} \right) \frac{\partial X_i}{\partial \boldsymbol{\theta}},$$

with $\partial X_i/\partial \boldsymbol{\theta}$ obtained analytically using the Jansson derivative method (Kunselman et al., 1 Aug 2025). The role of that method is to provide analytic gradients of residuals involving equilibria without re-running nested numerical differentiation; parameters are treated as external conditions, and envelope-theorem-like results are applied to minimized Gibbs energies.

For phase-equilibria data, the ZPF residual driving force for phase $\alpha$ in datum $i$ is

$$X^{\mathrm{zpf}}_{i,\alpha} = \sum_{A}\bar{\mu}_{A}\,x^{\alpha}_{A} - G^{\alpha}(T,\{x^{\alpha}_{A}\};\boldsymbol{\theta}),$$

where the target hyperplane chemical potentials $\phi \in \{\mathrm{FCC}, \mathrm{BCC}\}$0 are constructed per the derivative-method cases used for reported or unreported vertices. For non-equilibrium thermochemical constraints such as mixing enthalpy at fixed composition, the residual is

$\phi \in \{\mathrm{FCC}, \mathrm{BCC}\}$1

This formulation makes gradient-informed optimization central rather than auxiliary. The workflow therefore differs from black-box calibration in that exact parameter derivatives of equilibrium observables are part of the main algorithmic design (Kunselman et al., 1 Aug 2025).

4. Pt–W case study

The Pt–W demonstration includes FCC on the Pt-rich side, BCC on the W-rich side, and liquid (Kunselman et al., 1 Aug 2025). The initial MLP/ATAT/PhaseForge model already reproduced the correct qualitative stability ordering, but it exhibited two important quantitative deficiencies: underestimated Pt solubility in BCC W at high temperatures, where the experimental solubility is greater than 5%, and an equilibrium temperature for the solid/liquid reactions more than 1000 K lower than experiments, yielding a eutectic versus peritectic misclassification (Kunselman et al., 1 Aug 2025).

The parameterization began with 16 tunable parameters, excluding SGTE end-member parameters and short-range order terms from tuning, and then added 4 additional parameters to introduce linear temperature dependence in $\phi \in \{\mathrm{FCC}, \mathrm{BCC}\}$2 and $\phi \in \{\mathrm{FCC}, \mathrm{BCC}\}$3 for FCC and BCC (Kunselman et al., 1 Aug 2025). The paper does not report explicit numerical parameter values; instead, it reports the total number of tuned coefficients and the resulting phase-diagram behavior.

After gradient-based optimization, the tuned model showed excellent agreement with ASM-extracted phase equilibria across the diagram, with ZPF points matched and two-phase fields consistent (Kunselman et al., 1 Aug 2025). The reported physical changes were limited: the liquid description required minimal adjustment, while small increases in enthalpy and entropy for the mechanically unstable unary end members, FCC W and BCC Pt, together with modest modifications to interaction parameters, were sufficient to correct phase boundaries and invariant temperatures. The paper characterizes these as “slight adjustments,” and notes that despite very small energy and entropy changes, with tens of J/mol scale implied, the phase diagram shifted by large amounts, including correction of deviations greater than 1000 K in key invariant temperatures (Kunselman et al., 1 Aug 2025).

Computationally, the entire assessment from the MLP-informed model to tuned CALPHAD parameters can be completed in less than a day on local single-core resources. The optimization required approximately 150 function evaluations in total, and near-optimal solutions within 0.1% of the final log-likelihood were reached after roughly 80 evaluations (Kunselman et al., 1 Aug 2025). Compared with ESPEI’s default MCMC-based optimization, the gradient-informed approach is reported as achieving one to three orders of magnitude speedups in related studies and similarly fast convergence in this case.

5. Relation to PhaseForge and MLIP-based CALPHAD workflows

PhaseForgePlus is most naturally understood as operating downstream of the PhaseForge workflow introduced for MLIP-enabled phase-diagram construction (Zhu et al., 20 Jun 2025). PhaseForge embeds MLIP calculators, via MaterialsFramework, into the ATAT pipeline. It automates SQS generation, relaxation and energy evaluation, robust relaxation with inflection detection for mechanically unstable phases, optional vibrational entropy treatment, liquid free energies via MLIP-driven molecular dynamics, and CALPHAD interaction fitting and TDB assembly (Zhu et al., 20 Jun 2025).

This predecessor workflow provides the physically grounded Gibbs energy descriptions that PhaseForgePlus later tunes. In PhaseForge, minimal Gibbs free-energy models are anchored to SGTE unary data, with ideal configurational entropy and optional phonon contributions; phase diagrams are then obtained by minimizing Gibbs energy over compositions and phases in external CALPHAD packages (Zhu et al., 20 Jun 2025). PhaseForge therefore establishes the thermodynamic baseline, while PhaseForgePlus adds a derivative-based experimental calibration layer.

The distinction matters because the PhaseForge paper explicitly states that it does not mention “PhaseForgePlus” (Zhu et al., 20 Jun 2025). Its primary emphasis is efficient MLIP-based thermodynamic model construction and MLIP benchmarking, not gradient-based reassessment against ZPF and thermochemical data. PhaseForgePlus, by contrast, reuses the ATAT/PhaseForge-derived TDB representation and augments it with analytic-gradient optimization through ESPEI and PyCalphad (Kunselman et al., 1 Aug 2025).

A common misunderstanding is to equate an MLP-informed baseline with a complete phase-diagram assessment. The Pt–W study shows that a physically grounded initial model can still misclassify invariants or underpredict solubilities, and that small parameter corrections, especially for mechanically unstable unary end members and low-order interaction terms, can have disproportionately large effects on equilibria (Kunselman et al., 1 Aug 2025). The workflow is thus best viewed as a hybrid methodology: MLPs and ATAT generate the prior structure of the thermodynamic description, while experimental data enforce quantitative consistency.

6. Reproducibility, limitations, and prospective extensions

The reproducibility protocol is explicit. It consists of obtaining the repository, generating SQS structures, running Grace, building the TDB with PhaseForge, preparing the ASM APD ZPF dataset together with non-equilibrium mixing enthalpy constraints from the PhaseForge initial model, enabling gradients in ESPEI/PyCalphad, applying the conjugate-gradient optimizer, and reproducing the optimized phase diagram together with Gibbs, enthalpy, and entropy plots (Kunselman et al., 1 Aug 2025).

The main limitations are also stated directly. Mechanically unstable unary end members require estimation through inflection detection, and the resulting accuracy can strongly influence phase boundaries. The Pt–W study included only FCC, BCC, and liquid; intermetallic phases in other systems would require sublattice models and possibly more complex MLP or DFT sampling. Parameter extrapolation beyond the training or data range remains risky, and the regularization supplied by mixing enthalpy constraints helps but does not fully remove extrapolation uncertainties (Kunselman et al., 1 Aug 2025).

The stated future directions are extension to multicomponent systems such as Au–Pt–W, uncertainty quantification atop gradient-based fitting, and active learning loops in which MLP or DFT sampling is expanded in regions of high sensitivity or high uncertainty (Kunselman et al., 1 Aug 2025). These directions preserve the workflow’s central design principle: compact CALPHAD parameterizations constrained simultaneously by physically grounded energetics and experimental observations.

A separate, prospective line of development is suggested by the modular free-energy and automatic symbolic differentiation capability described for MOOSE/MARMOT (Schwen et al., 2017). In that framework, phase-field solvers such as Cahn–Hilliard and Allen–Cahn are separated from model-specific constitutive information; free-energy contributions are encapsulated as self-contained objects; derivatives are generated by automatic symbolic differentiation; and just-in-time compilation reduces the cost of evaluating differentiated expressions while yielding performance on par with carefully hand-coded derivatives (Schwen et al., 2017). This suggests a plausible route by which CALPHAD models produced in PhaseForgePlus could be coupled to multiphase chemo-mechanical simulations through modular free-energy composition, KKS or WBM formulations, and exact Jacobian assembly. The demonstrated Pt–W PhaseForgePlus workflow, however, is a CALPHAD model-construction and parameter-tuning study, not a reported multiphase-field implementation (Kunselman et al., 1 Aug 2025).