Robust & Accurate Potentials (PRAPs)

• PRAPs is a comprehensive framework that integrates high-dimensional feature representations, density-based extrapolation tests, and active learning to construct robust machine-learning potentials.
• It employs statistical methods such as local sampling density estimation and committee model uncertainty to monitor and calibrate performance in diverse, data-sparse regimes.
• The framework generalizes to various domains including molecular dynamics, optical potentials, and empirical interatomic models while ensuring quantitative accuracy and reproducibility.

The Plan for Robust and Accurate Potentials (PRAPs) is a comprehensive conceptual and algorithmic framework for developing, validating, and maintaining machine-learning-based potentials and other model Hamiltonians that remain reliable across interpolation and extrapolation regimes. Emerging from contemporary challenges in both atomistic and optical potential modeling, PRAPs combines high-dimensional representation strategies, statistical domain assessment, active data augmentation, and uncertainty quantification into a unified workflow. The core objective is to ensure quantitative accuracy and adaptive robustness of potential models as they are deployed in data-sparse, chemically diverse, or extrapolative regions commonly encountered in large-scale molecular dynamics, structure prediction, and quantum simulation tasks (Zeni et al., 2021).

1. High-Dimensional Representation and Invariant Feature Construction

PRAPs begins by encoding local atomic or environmental information into high-dimensional feature vectors that offer systematic control over physical invariances and expressivity. For machine learning interatomic potentials (MLIPs), the pipeline operates on local atomic densities:

$$\rho_i(\mathbf{r}) = \sum_{j: r_{ij} \leq r_c} u(\mathbf{r}_{ji} - \mathbf{r})$$

where $u(\cdot)$ is a kernel (e.g., Gaussian for SOAP, Dirac delta for ACE), and the expansion of $\rho_i$ onto truncated radial functions and spherical harmonics yields coefficients $c_{nlm}^j$:

$$\rho_i(\mathbf{r}) \approx \sum_{j \in \mathrm{env}(i)} \sum_{n=0}^{n_{\max}} \sum_{l=0}^{l_{\max}} \sum_{m=-l}^{l} c_{nlm}^j f_n(r_{ji}) Y_{lm}(\hat{\mathbf{r}}_{ji})$$

These coefficients constitute a feature vector $x_i \in \mathbb{R}^D$, which is further processed into rotational and permutational invariants—either through contraction up to order $N+1$ (ACE) or power/bispectrum components (SOAP) (Zeni et al., 2021). This paradigm is readily generalized to descriptor learning frameworks for other systems, such as DeepPot-SE or graph-based universal representations (Matsumura et al., 26 Nov 2024, Qi et al., 2023).

2. Extrapolation, Convex Hulls, and Statistical Density Criteria

A key empirical observation underpinning PRAPs is that, in high-dimensional feature spaces ($D \gg 10$), most operational predictions fall outside the convex hull of the training set—a regime traditionally considered extrapolative. For the convex hull defined as:

$$\mathrm{CH}(\{x_i\}) = \{x \mid x = \sum_{i=1}^M \lambda_i x_i,\, \lambda_i \geq 0,\, \sum_i \lambda_i = 1\}$$

linear programming can test membership, but nearly all test points of chemical relevance may lie outside $\mathrm{CH}$. Therefore, PRAPs advocates replacing traditional interpolation tests with a local sampling density estimator in representation space (Zeni et al., 2021).

Density is quantified as:

$\rho(x^*) = \frac{(k^* - 1)}{MV^*}$

$V^* = \omega_d r_{k^*}^d$

where $V^*$ is the volume of the ball enclosing the $k^*$ nearest neighbors (with $d$ estimated as the intrinsic dimension via the TwoNN estimator), and $M$ is the training set size. An acceptability threshold $\rho_{\min}$ or $-\log \rho_{\max}$ defines the robust domain. Quantitative cross-validation—by binning force-MAE versus $-\log \rho$—permits the calibration of extrapolation warnings. This density-driven strategy is empirically validated across materials classes (ice, metals, gold clusters), showing a monotonic increase in error as density drops (Zeni et al., 2021).

3. Active Learning, Training Set Design, and Uncertainty Quantification

The PRAPs workflow features a dynamic, data-driven methodology for expansion and refinement of the training set:

1. Diverse Initial Sampling: Early training set $S_0$ is obtained from molecular dynamics (MD) or Monte Carlo (MC) explorations covering the full range of relevant thermodynamic and compositional variation.
2. Adaptive Augmentation: Sparse regions (low feature density or high model uncertainty) encountered during production or screening are flagged for reference calculation, with the new data points appended to the training set for retraining.
3. Committee Models: Ensembles of models (bootstrapped subsets of the training data) provide prediction variance as an orthogonal measure of uncertainty. This ensemble deviation strongly correlates with regions of low sampling density (Zeni et al., 2021, Matsumura et al., 26 Nov 2024).
4. Clustering for Representative Diversity: Feature-space clustering or farthest-point sampling promotes the selection of globally distinctive configurations, ensuring data efficiency and transferability.

These principles extend to robust holographic light potentials and large-scale empirical potentials, where systematic descriptor downselection (e.g., via CUR, FPS, or D-optimal design strategies) ensures that the reduced training and feature sets maximize the accuracy-to-cost ratio (imbalzano et al., 2018, Schroff et al., 2022).

4. Algorithmic Workflow for PRAPs Construction and Maintenance

A typical PRAPs-driven pipeline consists of:

1. Feature Computation and Dimensionality Analysis: Construct the feature matrix, compute intrinsic dimension $d$, and select an appropriate density estimator.
2. Model Training: Fit a regression or neural network (e.g., ridge, kernel, or deep learning approaches) to map features to atomic energies and forces.
3. Adaptive Domain Monitoring: For each new configuration in MD or screening runs, compute its feature representation, evaluate $-\log \rho$, and determine domain membership.
4. Active Query and Retraining: Out-of-domain structures (by density/uncertainty) are submitted to quantum reference computations (e.g., DFT), appended to the database, and the model is retrained periodically (Zeni et al., 2021, Matsumura et al., 26 Nov 2024).
5. Threshold Optimization: Cross-validate on held-out data, setting extrapolation thresholds at the "knee" where the MAE crosses the chosen tolerance (typ. 1 kcal/mol), and validate using ancillary distance metrics.

Extrapolation-detection and retraining cycles are iteratively interleaved with production, ensuring that the coverage of the feature space remains commensurate with the regions actually sampled.

5. Extensions: Generalized PRAPs Strategies Across Domains

While PRAPs originated in atomistic potential construction, the essential methodology generalizes to other physical and data-driven potential modeling contexts:

• Holographic Light Potentials: PRAPs provides sub-2% RMS error and 15–40% efficiency by integrating sub-pixel crosstalk modeling, feedback correction, vortex removal, and optimized cost-function minimization over phase patterns (Schroff et al., 2022).
• Empirical and ML Interatomic Potentials: In frameworks like RAMPAGE, PRAPs directs the assembly, cross-calibration, and error-bounded optimization of multi-component potentials for alloys, incorporating explicit uncertainty quantification and robust benchmarking (Weiss et al., 2022).
• Moment Tensor and Bond-Order Potentials: PRAPs-type workflows automate D-optimal candidate selection, error stratification, and Pareto optimization for transferability versus accuracy, with benchmarking against DFT error and thermodynamic consistency (Roberts et al., 13 Dec 2025, Subramanyam et al., 2023).
• Graph Neural Networks and Data-Efficient Potentials: Leveraging equivariant architectures (e.g., NequIP) and DIRECT sampling, PRAPs achieves data-efficient, high-fidelity models across molecules and complex materials systems while automating coverage and reducing extrapolation-induced loss of accuracy (Batzner et al., 2021, Qi et al., 2023).

6. Practical Guidelines and Best Practices

The implementation of PRAPs mandates attention to the following:

• Systematically monitor both sampling density ($\rho(x^*)$) and prediction variance (committee uncertainty or model deviation).
• Employ cross-validation for all error-vs.-density calibrations and propagate domain definitions quantitatively.
• Integrate automated workflows for feature selection, reference configuration downselection, and active retraining to reduce human bias and enhance reproducibility.
• For new material systems, prioritize chemically informed initial sampling, use data-driven feature embedding (e.g., graph-based featurizers followed by PCA/UMAP), and maintain flexibility in retraining and validation cycles (Qi et al., 2023).
• Publish full training protocols, parameter sets, performance metrics, and test scripts to support transparency and future benchmarking.

PRAPs offers a statistically rigorous and algorithmically flexible approach for constructing potentials and effective Hamiltonians that balance accuracy and robustness in both high-throughput and highly specialized modeling applications. The workflow is supported by empirical studies, algorithmic validation, and cross-domain generalizability (Zeni et al., 2021, Matsumura et al., 26 Nov 2024, Qi et al., 2023, Roberts et al., 13 Dec 2025).