🔬 materials science

A latent-space extrapolation grade built into graph atomic cluster expansion foundation potentials

This paper introduces the calibrated Mahalanobis (CALM) extrapolation grade, a computationally efficient, per-atom uncertainty metric integrated into GRACE foundation potentials that reliably detects structural and chemical extrapolation by correlating latent-space distances with force errors to guide targeted data collection.

Original authors: Yury Lysogorskiy, Anton Bochkarev, Ralf Drautz

Published 2026-10-01

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CC BY 4.0

Original authors: Yury Lysogorskiy, Anton Bochkarev, Ralf Drautz

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

In the world of materials science, researchers use powerful computer programs to predict how atoms behave. These programs, known as machine-learning potentials, act as a bridge between the slow, exact calculations of quantum physics and the fast, large-scale simulations needed to design new batteries, stronger metals, or more efficient solar cells. By learning from vast libraries of known atomic arrangements, these models can instantly guess the energy and forces acting on atoms in a new situation. This speed has unlocked the ability to simulate entire crystals and complex chemical reactions that were previously too expensive to compute. However, a critical blind spot remains: when these models encounter a situation they have never seen before, they often fail silently. They might produce a result that looks perfectly reasonable but is wildly wrong, offering no warning that the prediction has drifted far outside the realm of what the model actually knows.

To solve this problem, a team of researchers at Ruhr-University Bochum has developed a new way to measure confidence in real-time. They introduced a system called the calibrated Mahalanobis extrapolation grade, or simply the "calm" grade, which acts as an internal compass for these atomic models. Imagine a map where the model knows every street in a specific city. If a driver stays on those streets, the map is reliable. But if the driver suddenly finds themselves in a desert or a different country, the map is useless. The new system gives every single atom a score that tells the computer exactly how far it has strayed from the known streets. If the score is low, the atom is in familiar territory. If the score is high, the atom is in unknown territory, and the computer knows to treat the result with extreme caution.

The researchers built this system into a foundation model called GRACE, which is designed to work across a wide range of elements and chemical environments. Instead of running multiple copies of the model to check for errors—a method that is accurate but computationally expensive and slow—they engineered a single, lightweight check that runs alongside the main calculation. This check looks at the mathematical "fingerprint" of an atom's neighborhood. It compares this fingerprint to a library of fingerprints gathered from the training data. If the new fingerprint is too different from anything the model has seen before, the system flags it. Crucially, this check is calibrated so that a specific score, set to one, marks the boundary between what the model knows and what it is guessing.

The team tested this approach rigorously using two massive datasets containing millions of atomic configurations. They found that the score correlates strongly with the actual error in the model's predictions. When the score was low, the model's guesses about how atoms push and pull on each other were accurate. When the score rose above the threshold, the errors grew significantly, sometimes by orders of magnitude. The system proved effective at spotting two distinct types of danger. First, it detected structural surprises, such as atoms squeezed too tightly together or pulled too far apart, or atoms sitting on the surface of a material where their neighbors are missing. Second, it caught chemical surprises, such as a material made of a mix of elements in proportions the model had never seen during its training. In one test involving a mix of molybdenum and tungsten, the system correctly identified that the model was struggling when the concentration of tungsten exceeded the limits of the training data, even though the atoms themselves were chemically similar to what the model knew.

Beyond simply flagging errors, the researchers showed that this score could be used to actively improve the model. By using the score to guide a computer simulation, they could steer the simulation toward the most uncertain regions of the atomic landscape. Instead of letting the simulation wander randomly, they used the score as a bias to push the atoms into configurations where the model was least confident. This allowed them to collect new, high-value data points exactly where they were needed most. When they used these new data points to retrain the model, the resulting potential was significantly better at handling high temperatures and liquid states than a model trained on the same amount of data collected without this guidance. This suggests that the system does not just identify problems; it provides a roadmap for fixing them efficiently.

The method is also remarkably efficient. Adding this confidence check to a simulation increases the computing time by only a tiny fraction, often less than one percent for larger, more complex models. This means it can be used in every single step of a long simulation without slowing down the research. The researchers demonstrated that this small cost buys a massive gain in reliability, allowing scientists to run simulations with the confidence that they are not unknowingly drifting into error. By integrating this "calm" grade directly into the foundation models, the team has provided a tool that makes these powerful simulations safer and more trustworthy, ensuring that when scientists look at the atomic world, they know exactly when they are looking at a reflection of reality and when they are looking at a guess.

Technical Summary: CALM Extrapolation Grade for Graph Atomic Cluster Expansion Foundation Potentials

Problem Statement
Foundation machine-learning interatomic potentials (MLIPs), such as GRACE, MACE-MP, and NequIP, enable fast atomistic simulations across broad configurational and chemical spaces by training on large datasets (e.g., OMat24, SMAX). While these models approach Density Functional Theory (DFT) accuracy within their training domains, they lack native uncertainty quantification (UQ). Consequently, simulations cannot distinguish between interpolation (within the training distribution) and silent extrapolation (outside the distribution), where prediction errors can grow by orders of magnitude. Existing UQ methods, such as deep ensembles, are computationally prohibitive for foundation models due to the need for multiple training runs and inference passes. Other methods, like Bayesian approximations or evidential regression, often require architectural changes, additional training, or repeated stochastic forward passes. There is a need for an atom-resolved, calibrated extrapolation grade that adds minimal computational overhead and can be evaluated alongside energies and forces in a single model pass.

Methodology
The authors introduce the Calibrated Mahalanobis (CALM) extrapolation grade, denoted as $\gamma$ . This method is integrated into the GRACE foundation models and operates as follows:

Feature Representation: The method utilizes a latent feature space derived from the invariant many-body basis of the MLIP. To handle the wide range of basis magnitudes and ensure robustness under structural deformation (compression and stretching), the authors employ a specific feature construction:
- The full invariant product basis is normalized to unit $L_2$ norm.
- A fixed random projection (Johnson-Lindenstrauss transform) reduces the dimensionality (e.g., to 128 dimensions).
- Logarithmic norm channels of the full basis and individual reduction blocks are appended to the projection to preserve magnitude information lost during normalization.
Clustering and Mahalanobis Distance: For each chemical element, the training data in the latent feature space is clustered. For a given atom with latent feature vector $z$ , the method identifies its nearest cluster centroid $k^*$ . The Mahalanobis distance $\sigma$ is calculated as:
$\sigma = \sqrt{(z - \mu_{e,k^*})^T \Sigma_{e,k^*}^{-1} (z - \mu_{e,k^*})}$
where $\mu$ and $\Sigma$ are the mean and covariance of the assigned cluster.
Calibration: The distance $\sigma$ is normalized by a cluster-specific threshold $\theta_{e,k^*}$ to produce the grade $\gamma = \sigma / \theta_{e,k^*}$ . The threshold is defined as the median of the training-set distances plus three standard deviations (estimated via Median Absolute Deviation). Thus, $\gamma = 1$ marks the calibrated boundary of the training support; $\gamma \le 1$ indicates interpolation, while $\gamma > 1$ indicates extrapolation.
Differentiability and Aggregation: Within a fixed cluster assignment, $\gamma$ is differentiable with respect to atomic positions. The authors also define a weighted neighborhood-averaged grade ( $\gamma_{nbr}$ ) to improve the localization of force-error uncertainty.
Computational Efficiency: The grade is evaluated in a single model pass alongside energies and forces. It requires only the calculation of distances to stored cluster statistics, incurring a computational overhead of less than 1% to 6% depending on model depth.

Key Contributions

CALM Grade: A piecewise differentiable, per-atom extrapolation grade integrated directly into GRACE foundation models.
Feature Engineering: Demonstration that a normalized invariant basis with logarithmic norm channels is superior to raw or linearly projected bases for detecting both structural (compression/stretching) and chemical extrapolation.
Calibration Strategy: A method to set a universal threshold ( $\gamma=1$ ) across different elements and clusters based on training-distance distributions, allowing for consistent interpretation of extrapolation risk.
Uncertainty-Guided Sampling: Utilization of the spatial gradient of $\gamma$ to bias molecular dynamics (MD) simulations toward uncertain regions, facilitating active learning and dataset expansion without retraining the base model during sampling.

Results

Validation on Controlled Tests: The CALM grade successfully identifies structural extrapolation in elemental prototypes (Al, Cu, Fe, Si, Ti, W) under both compression and stretching, outperforming linear projected bases which fail to detect stretching when neighbors leave the cutoff.
Correlation with Error: On the OMat24-val (in-distribution) and SMAX (out-of-distribution) datasets, $\gamma$ correlates with atomic force errors. The Spearman correlation reaches 0.497 on OMat24-val and 0.617 on SMAX. Neighborhood averaging ( $\gamma_{nbr}$ ) improves these correlations further.
Comparison with Alternatives: CALM achieves higher force-error correlations than D-optimality (MaxVol) grades constructed on the same features and performs comparably to deep ensembles (which require 3x inference cost) while being significantly cheaper.
Materials Discovery Benchmark: Applied to the WBM formation-energy benchmark, filtering structures with $\gamma > 1$ reduces the mean absolute error (MAE) of formation energies from 17.7 to 16.3 meV/atom while retaining 80% of the structures.
Active Learning: In uncertainty-guided MD (HAL) on Aluminum and Silicon, sampling along the gradient of $\gamma$ collects configurations with larger force errors than unbiased MD. Potentials upfitted with these uncertainty-biased datasets show improved accuracy at high temperatures and in liquid phases compared to those trained on unbiased data.
Foundation Model Expansion: When applied to a foundation model trained on SMAX, the method successfully identified configurations with higher force errors than the training set, providing candidates for extending the model's training domain.

Significance and Claims
The paper claims that the CALM grade provides a practical, low-cost solution for uncertainty quantification in foundation MLIPs. By being differentiable and computable in a single pass, it enables real-time monitoring of simulation reliability and guides data acquisition strategies. The authors emphasize that while the correlation between $\gamma$ and error varies with dataset and chemistry, the method offers a calibrated, element-specific boundary that separates interpolation from extrapolation. The work demonstrates that uncertainty indicators can be intrinsic to the potential evaluation process, adding only percent-level computational cost while significantly improving the reliability of simulations in unknown regimes and facilitating the expansion of training datasets through targeted sampling.

Technical Summary: CALM Extrapolation Grade for Graph Atomic Cluster Expansion Foundation Potentials

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