cond-mat.mtrl-sciMar 10, 2026

Flexible Cutoff Learning: Optimizing Machine Learning Potentials After Training

Authors: Rick Oerder, Jan Hamaekers

Organizations: Institute for Numerical Simulation, University of Bonn · Fraunhofer Institute for Algorithms and Scientific Computing SCAI · Institute for Numerical Simulation, University of Bonn, Friedrich-Hirzebruch-Allee 7, 53115 Bonn, Germany · Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven 1, 53757 Sankt Augustin, Germany

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Abstract

We introduce Flexible Cutoff Learning (FCL), a method for training machine learning interatomic potentials (MLIPs) whose cutoff radii can be adjusted after training. Unlike conventional MLIPs that fix the cutoff radius during training, FCL models are trained by randomly sampling cutoff radii independently for each atom. The resulting model can then be deployed with different per-atom cutoff radii depending on the application, enabling application-specific optimization of the accuracy-cost tradeoff. Using a differentiable cost model, these per-atom cutoffs can be optimized for specific target systems after training. We demonstrate FCL with a modified MACE architecture trained on the MAD dataset. For a subset featuring molecular crystals, optimized per-atom cutoffs reduce computational cost by more than 60% while increasing force errors by less than 1%. These results show that FCL enables training of a single general-purpose MLIP that can be adapted to diverse applications through post-training cutoff optimization, eliminating the need for retraining.

Figures & tables

Modelrcutr_{\text{cut}} / \SIUnitSymbolAngstrom\mathrm{\SIUnitSymbolAngstrom}kkReference
MACE-MP-06.02[ 8 ]
MACE-OFF23-S4.52[ 9 ]
MACE-OFF23-M/L5.02[ 9 ]
MACE-OFF24-M6.02[ 9 ]
SevenNet-Omni6.02[ 14 ]
UMA6.02[ 10 ]
Table 1: Cutoff radii and the highest explicit interaction order kk for selected MLIPs. Values for the cutoff radius in brackets for M3GNet and CHGNet correspond to a separate cutoff radius for three-body interactions.
Figure 1: Simplified representation of a model that depends explicitly on atom-wise cutoff radii. For a fixed geometry, the model can be evaluated for different realizations of rcut(1),...,rcut(n)r_{\text{cut}}^{(1)},...,r_{\text{cut}}^{(n)} . In this schematic figure, the model is assumed to be decomposable into on-site contributions that are computed from atom-centered atomic environments. In particular, each contribution is obtained by evaluating a function (denoted as VV ) on atomic environments of different sizes that depend on the choice of the atom-wise cutoff radii.
Figure 2: Schematic overview of the Flexible Cutoff Learning and the optimization of cutoff radii after training. (a) Per-atom cutoff radii rcut(i)r_{\text{cut}}^{(i)} are sampled during the training process. (b) Optimization of error and cost with respect to cutoff radii is performed after training. Smaller receptive fields lead to faster evaluation but may reduce the model accuracy.
Figure 3: Test errors as a function of a global cutoff radius. Results for ”Flexible Cutoff Learning” are obtained from evaluating a single model with different global cutoff radii. ”Static Cutoff Learning” correspond to a separate model for each cutoff radius. All results are averaged over three training runs with different seeds. Error bars correspond to the empirical standard deviation.
Figure 4: Energy as a function of distance in diatomic systems for different inference cutoff radii. All curves are obtained by evaluating a single FCL model.
Figure 5: Force RMSE versus average number of pairs per atom for different MAD test subsets. Each curve shows results for varying λ\lambda (from left to right: 10−2,10−3,10−4,10−5,10−610^{-2},10^{-3},10^{-4},10^{-5},10^{-6} ), demonstrating the accuracy-cost tradeoff achievable through cutoff optimization.
Figure 6: Optimized element-wise cutoff radii for λ=10−4\lambda=10^{-4} across MAD subsets. Only elements present in all four subsets are shown.
Figure 7: Optimized cutoff radii (orange) versus initial values (blue) for SHIFTML-molcrys at different λ\lambda values. Higher λ\lambda emphasizes computational cost, producing smaller cutoffs across elements.
Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

ParameterValue
Cutoff radius ( rmaxr_{\text{max}} )variable
Bessel functions10
Polynomial cutoff order5
Correlation order1
Maximum angular momentum ( ℓmax\ell_{\text{max}} )1
Interaction layers2
Appendix
Table 2: MACE model hyperparameters.
r_maxavg_num_neighbors
4.019.3
5.038.2
6.065.4
7.0104.1
Appendix
Table 3: Hyperparameters for the static MACE models with different cutoff radii. The value for avg_num_neighbors is computed for the training set of the MAD dataset.
ElementFraction
H4.20⋅10−14.20\cdot 10^{-1}
C3.27⋅10−13.27\cdot 10^{-1}
N1.05⋅10−11.05\cdot 10^{-1}
O9.49⋅10−29.49\cdot 10^{-2}
F1.77⋅10−21.77\cdot 10^{-2}
Na4.95⋅10−44.95\cdot 10^{-4}
Appendix
Table 4: Element frequencies in SHIFTML-molcrys training data.
Figure 8: RMSE of energy per atom as a function of a global cutoff radius
Figure 9: Optimizing the per-element cutoff radii on different subsets of the MAD dataset for different values of from left to right) improves the error-cost ratio of a model. A single FCL-model is evaluated on different subsets from the MAD dataset of the test data using the element-wise cutoff radii RE\mathcal{R}_{E} from the optimization process with different values for λ∈{10−2,10−3,10−4,10−5,10−6}\lambda\in\{10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}\} (from left to right). Shows the RMSE of the energy per atom.
Figure 10: For the full range on the x-axis: Energy as a function of distance in diatomic systems for different values of the cutoff radius. All curves are obtained by evaluating a single model.
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