🔬 materials science

Machine learning interatomic potentials for solid-state precipitation

This paper introduces novel error metrics and data-generation schemes to streamline the parameterization of machine learning interatomic potentials, successfully applying them to model complex solid-state precipitation in Mg-Nd alloys and revealing competition between order-disorder and structural transformations during aging.

Original authors: Lorenzo Piersante, Anirudh Raju Natarajan

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Lorenzo Piersante, Anirudh Raju Natarajan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Metals are rarely used in their pure form; they are usually mixed with other elements to create alloys that are stronger, lighter, or more resistant to heat. One of the most powerful ways to strengthen these materials is through a process called precipitation hardening. Imagine a metal as a vast, orderly city of atoms. When the metal is heated and then cooled, tiny clusters of different atoms can form within this city, like small, dense islands appearing in a calm sea. These islands act as roadblocks for the defects that move through the metal when it is under stress, effectively stopping the material from bending or breaking. To design better alloys, engineers need to understand exactly how these islands form, what shapes they take, and how they grow over time. However, watching these atomic events happen in real life is nearly impossible because they occur too quickly and on a scale too small for even the most powerful microscopes to capture directly.

For decades, scientists have relied on computer simulations to peek into this invisible world. The most accurate simulations use a method that calculates the behavior of every single electron in an atom, but this approach is so computationally expensive that it can only model a few hundred atoms for a split second. To study the formation of precipitates, which involves thousands of atoms over longer periods, researchers need a shortcut. They use machine learning models that learn the rules of atomic interaction from the accurate electron calculations and then apply those rules to much larger systems. The challenge has been teaching these models to correctly predict not just the energy of a single atom, but the complex, multi-stage dance of atoms rearranging themselves into new crystal structures. If the model gets the energy landscape wrong, even by a tiny amount, it might predict that a precipitate forms when it shouldn't, or that it takes a shape that never exists in reality.

In a recent study, researchers at the École Polytechnique Fédérale de Lausanne tackled this problem by creating a new, highly accurate computer model for a specific magnesium alloy containing a small amount of neodymium. This alloy is a prototype for lightweight materials used in aerospace and automotive industries, where strength is critical. The team developed a sophisticated strategy to train their machine learning model, ensuring it could capture the subtle energy differences that dictate whether atoms stay mixed or separate into distinct phases. Instead of just feeding the computer random data, they designed a system that specifically looked for the pathways atoms take when they transform from one crystal structure to another. They focused on the specific journey atoms make when the metal's internal structure shifts from a hexagonal arrangement to a cubic one, a transition that is central to how this alloy strengthens.

The researchers discovered that their new model could successfully reproduce the complex early stages of precipitation observed in real experiments. The model showed that the formation of these strengthening islands is a competition between two different types of changes: one where atoms simply rearrange their order within the existing structure, and another where the entire structure shifts into a new shape. The simulations revealed that the transition to the new, stronger cubic structure happens without a significant energy barrier, suggesting that these new phases can form very easily and continuously as the metal ages. This finding helps explain why the alloy develops its strength so effectively, as the atoms can smoothly flow into the most stable configuration without getting stuck in intermediate, less stable states.

To ensure their model was trustworthy, the team invented new ways to check its accuracy that went beyond standard error measurements. They created a scoring system that specifically tested whether the model could correctly rank the stability of different atomic arrangements, prioritizing the low-energy states that actually occur in nature. This approach allowed them to fine-tune the model until it correctly predicted the ground state of the material, avoiding the common pitfall where computer models accidentally favor the wrong crystal structure. The results confirmed that the model could accurately predict the energy of various defects and the forces holding the atoms together, providing a reliable digital twin of the physical alloy.

The study concludes that this new approach to training machine learning models offers a powerful tool for understanding and designing advanced materials. By combining a vast and diverse set of training data with these specialized validation metrics, the researchers were able to create a potential that generalizes well to situations it had never seen before, such as the interfaces between different crystal phases. Their work suggests that the continuous transition between the hexagonal and cubic structures is a key feature of the alloy's behavior, a detail that might have been missed by less precise models. This level of detail provides a clearer picture of how precipitation hardening works at the atomic level, offering a roadmap for engineers to design stronger, lighter alloys by controlling the precise conditions under which these atomic islands form and grow.

Technical Summary: Machine Learning Interatomic Potentials for Solid-State Precipitation

Problem Statement
Machine learning interatomic potentials (MLIPs) are increasingly used to model atomistic phenomena, yet parameterizing them to accurately capture solid-state phase transformations, particularly precipitation in multi-component alloys, remains challenging. Precipitation often involves complex sequences of order-disorder transitions and structural transformations (e.g., hcp to bcc) that require potentials to reproduce not only ground-state thermodynamics but also metastable phases, energy barriers, and defect energetics. Standard error metrics (e.g., RMSE, MAE) often fail to detect critical failures in phase stability rankings or transformation pathways, leading to potentials that may yield incorrect ground states or miss barrierless transition mechanisms. Specifically, in dilute Mg-Nd alloys, the precipitation sequence involves Guinier-Preston (GP) zones, $\beta''$ , $\beta'$ , $\beta_1$ , and equilibrium phases, where the $\beta_1$ phase forms via a Burgers mechanism (hcp $\to$ bcc). Existing potentials, such as modified embedded-atom method (MEAM) models, have been shown to produce zero-Kelvin ground states inconsistent with Density Functional Theory (DFT) or experiment, limiting their utility for studying nucleation mechanisms.

Methodology
The authors present a comprehensive strategy for parameterizing an MLIP based on the Atomic Cluster Expansion (ACE) framework for the Mg-Nd binary alloy. The approach integrates systematic data generation, active learning, and novel validation metrics tailored to precipitation physics.

Data Generation and Active Learning:
- Training Set Construction: The dataset encompasses diverse structural and chemical environments, including intermetallic prototypes, volumetric distortions, planar defects (stacking faults, grain boundaries), liquid structures, and solute-defect interactions.
- Symmetry-Distinct Pathways: A custom algorithm enumerates symmetrically distinct transformation pathways connecting chemical decorations on different parent crystal structures (e.g., hcp to bcc). This addresses the complexity introduced by chemical ordering, which breaks parent symmetry and renders pathways equivalent in pure elements inequivalent in ordered alloys.
- Iterative Refinement: An active learning loop was employed. A preliminary potential relaxed $\sim$ 0.5 million symmetrically distinct orderings. Configurations within 10 meV/atom of the convex hull (or new ground states) were identified, re-evaluated with DFT, and added to the training set until convergence.
Novel Validation Metrics:
- Weighted Kendall- $\tau$ ( $\tau_w$ ): To assess phase stability, the authors introduce the weighted Kendall- $\tau$ coefficient. Unlike global error metrics, $\tau_w$ quantifies the similarity in the ordering of energy spectra between DFT and MLIP. A weighting function prioritizes low-energy polymorphs, ensuring the potential correctly predicts ground states and metastable phases near the convex hull.
- Semi-Grand Canonical Generalization: For multi-component systems, $\tau_w$ is computed as a function of the exchange chemical potential ( $\tilde{\mu}$ ). This "semi-grand canonical $\tau_w$ " evaluates the potential's ability to predict phase stability and coexistence chemical potentials across the composition range.
- Transformation Pathway Classification: Structural phase transitions are classified into six qualitative energy profile classes (e.g., barrierless, barriered, intermediate states). A confusion matrix compares MLIP predictions against DFT classifications to ensure qualitative agreement in transformation landscapes, not just quantitative energy matches.
Potential Parameterization:
- The Mg-Nd potential was trained using the pacemaker code with L1/L2 regularization. Special attention was paid to weighting structures near the convex hull and those with Nd compositions between 0 and 0.25.

Key Results

Thermodynamic Accuracy: The resulting MLIP successfully reproduces the low-temperature thermodynamics of the Mg-Nd system. The weighted Kendall- $\tau$ analysis confirms that the potential correctly identifies the dhcp ground state for Nd and the stable phases for Mg-Nd up to $x_{Nd} \approx 0.25$ . While a spurious ground state appeared at $x_{Nd} = 1/3$ (limiting applicability for high Nd concentrations), the potential remains accurate for dilute alloys relevant to precipitation.
Structural and Mechanical Properties: The potential accurately predicts lattice parameters, elastic constants, and bulk moduli for various phases ( $\beta'$ , $\beta''$ , $\beta_1$ , $\beta_e$ ) and matches DFT values for surface and stacking fault energies.
Transformation Landscapes: The MLIP faithfully reproduces the energy landscape of the Burgers transformation (hcp $\to$ bcc). The confusion matrix indicates high agreement in classifying transformation pathways, though some misclassifications occur in barriered pathways (classes 3–4) due to small energy differences ( $\sim$ 4.5 meV/atom) within the validation error margin.
Defect Energetics: The potential captures Nd-vacancy interactions and binding energies to stacking faults and grain boundaries with reasonable accuracy, though the Nd-vacancy binding in the hcp matrix showed a slight discrepancy (repulsion vs. attraction) that warrants caution for kinetic parameter calculations.
Precipitation Simulation: Large-scale atomistic simulations using the MLIP revealed:
- Nucleation Mechanisms: The formation of $\beta_1$ precipitates yields the largest energy reduction, followed by $\beta'$ .
- Competition: Early-stage aging shows competition between order-disorder transitions (forming $\beta'$ , GP zones) and structural transformations.
- Continuous Transition: The simulations suggest a continuous transition between high-symmetry hcp and bcc structures during aging, where atoms in $\beta_1$ precipitates gradually relax from ideal hcp positions toward bcc coordination as the precipitate grows.
- Barrierless Nucleation: At 0 K, the large thermodynamic driving force results in barrierless nucleation for the observed precipitates.

Significance and Claims
The paper claims to provide a general framework for parameterizing MLIPs specifically for solid-state precipitation and structural phase transitions. The primary contribution is the development of tailored error metrics (weighted Kendall- $\tau$ and semi-grand canonical extensions) and data-generation schemes (symmetry-distinct pathway enumeration) that go beyond standard statistical errors. These tools allow researchers to rigorously validate whether an MLIP captures the correct thermodynamic hierarchy and transformation mechanisms, which are critical for modeling nucleation.

The authors demonstrate that this methodology enables the creation of an MLIP for Mg-Nd that generalizes well to structures and interfaces not explicitly included in the training set. The resulting potential successfully reproduces the complex, multi-stage precipitation behavior observed experimentally, including the competition between different precipitate morphologies and the continuous structural evolution from hcp to bcc. The authors conclude that these methodologies are applicable to other engineering alloy systems, including Al, Ti, and high-entropy alloys, offering a robust path toward reliable atomistic modeling of precipitation hardening.

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