Source-linked AI summary
End-to-end Symmetry Preserving Inter-atomic Potential Energy Model for Finite and Extended Systems
Linfeng Zhang, Jiequn Han, Han Wang, Wissam A. Saidi, Roberto Car, Weinan E
TL;DR
Accurately and efficiently representing inter-atomic potential energy surfaces remains difficult because quantum-mechanical methods are limited by system size, while empirical potentials face accuracy and transferability limits. The paper develops DeepPot-SE, an end-to-end symmetry-preserving model, and tests it across diverse finite and extended systems with satisfactory accuracy.
Problem
Representing inter-atomic potential energy surfaces accurately and efficiently is challenging because quantum-mechanical methods are limited by system size, whereas empirical potentials are limited by accuracy and transferability.
Method
DeepPot-SE combines symmetry-preserving encoding and fitting neural networks in a single end-to-end training process using local atomic environments.
Results
DeepPot-SE describes diverse molecular and periodic systems, including metals, insulators, semiconductors, bulk crystals, surfaces, and high entropy alloys, with satisfactory accuracy.
Takeaways & Limitations
The model provides a unified potential energy framework for finite and extended systems while preserving natural symmetries and supporting scalable end-to-end modeling.
Takeaways & Limitations
The stated setup assumes inputs consist only of chemical species and atomic coordinates, with extended systems represented under periodic boundary conditions.
Abstract
from arXiv · showhide
Machine learning models are changing the paradigm of molecular modeling, which is a fundamental tool for material science, chemistry, and computational biology. Of particular interest is the inter-atomic potential energy surface (PES). Here we develop Deep Potential - Smooth Edition (DeepPot-SE), an end-to-end machine learning-based PES model, which is able to efficiently represent the PES for a wide variety of systems with the accuracy of ab initio quantum mechanics models. By construction, DeepPot-SE is extensive and continuously differentiable, scales linearly with system size, and preserves all the natural symmetries of the system. Further, we show that DeepPot-SE describes finite and extended systems including organic molecules, metals, semiconductors, and insulators with high fidelity.
1 Introduction
Accurate and efficient PES representation remains difficult because quantum-mechanical methods limit system size while empirical potentials limit accuracy and transferability. The paper proposes DeepPot-SE to satisfy fundamental PES-model requirements across finite and extended systems.
- Quantum-mechanical PES models are limited by tractable system size, whereas empirical atomic potentials are limited by accuracy and transferability.
- The target model should approach quantum-mechanical accuracy for both isolated finite systems and periodic extended systems.Finite systems are surrounded by vacuum; extended systems use periodic boundary conditions.
- A PES model should use only chemical species and atomic coordinates, remain size extensive, preserve natural symmetries, and minimize human intervention.The listed symmetries include translational, rotational, and permutational symmetry.
- Practitioners seek first-principles-level ease of use and accuracy at significantly lower computational cost.
- Existing ML models generally satisfy only subsets of the requirements, including violations involving input information or size extensivity.The cited examples are BIM-NN and GDML; the passage also contrasts them with the symmetry-preserving Deep Potential model.
- DeepPot-SE is proposed and tested as a new PES model satisfying all listed requirements, with evaluations spanning diverse systems and challenging materials.The tests include DFT data for high entropy alloys, and the methodology is presented as applicable to other symmetry-preserving ML tasks.
2 Related Work
Related work includes symmetry-preserving architectures and several machine-learning approaches for representing PES. These methods differ in their use of kernels, descriptors, interaction passes, and symmetry-aware function classes.
- Symmetry-preserving models: Spherical CNN incorporates S2 and SO(3) cross-correlations to preserve rotational invariance.
- Symmetry-preserving models: DeepSets provides a family of functions containing any permutation-invariant objective function and has been tested on several tasks.
- ML-based PES models: SOAP represents neighboring atomic densities through a smooth-similarity kernel method.
- ML-based PES models: DTNN uses nuclear charges and an inter-atomic distance matrix, with interaction passes through which atom representations influence one another pairwise.
- ML-based PES models: The related PES approaches include BIM-NN, BPNN, DeepPot, GDML, SOAP, DTNN, and SchNet.
3 Theory
DeepPot-SE represents the PES through symmetry-preserving local atomic environments and end-to-end neural sub-networks. Smooth distance weighting, invariant feature construction, and joint training support differentiability, symmetry preservation, and efficient scaling.
- Energy decomposition: The total energy is decomposed into atomic contributions, with each atom’s contribution computed from its chemical species and local environment.Neighbors are selected within a predefined cutoff radius using relative coordinates.
- Symmetry-preserving sub-networks: The model maps each local environment through encoding and fitting networks to an atomic energy while preserving translational, rotational, and permutational symmetries.The encoding network produces symmetry-preserving embedded features, and the fitting network maps them to the atomic energy.
- Smooth local environments: Relative coordinates are augmented with a smooth distance weighting that reduces distant-neighbor contributions and removes cutoff-radius discontinuities.The weighting function transitions smoothly to zero between the smooth cutoff parameter and the cutoff radius.
- Symmetry-preserving representation: The encoded feature matrix combines coordinate and axis filters to preserve rotation and permutation symmetry, while translation symmetry is preserved by using relative coordinates.The matrix is reshaped into fitting-network input that yields the atomic energy.
- Symmetry-preserving representation: Choosing M1 ∼100 and M2 ∼4 gives good empirical performance, with M1 much larger than M2 in practice.The filters are adaptive and depend on distance and chemical species rather than being fixed functions such as Gaussians.
- Training process: The encoding and fitting parameters are trained end-to-end with a loss combining energy, force, and virial errors when those labels are available.Using forces during training significantly reduces the number of snapshots needed for a good PES model.
4 Data and Experiments
The experiments evaluate DeepPot-SE on molecular, bulk, supported-cluster, and high-entropy-alloy systems using held-out snapshots and comparisons with DeePMD and other benchmarks. The unified model fits diverse compositions and phases, including especially challenging multi-component and extended systems.
- Datasets and evaluation: The evaluation covers molecular and extended datasets, including metals, semiconductors, insulators, supported Pt clusters on MoS2, and a CoCrFeMnNi high-entropy alloy.The extended-system suite includes both moderately challenging single-phase materials and more complex multi-component systems.
- Small organic molecules: The small molecular dataset contains seven organic molecules, and DeepPot-SE is trained both as one unified model and as separate models for each molecule.The unified model uses shared sub-networks for the four element types C, H, O, and N.
- Small organic molecules: All methods achieve energy and force fits for the small organic molecules, with DeepPot-SE total-energy MAEs below chemical accuracy of 0.04 eV in every case.The unified model performs slightly worse than individual models but remains generally comparable.
- Datasets and evaluation: For all systems, 90% of randomly selected snapshots are used for training and 10% for testing; the HEA test set additionally includes 16 occupations distinct from training.Table 2 reports energy and force RMSEs, with energy errors normalized by system size.
- Bulk and complex systems: Pyridine and TiO2 provide more challenging bulk tests because they contain two and three crystalline polymorphs, respectively.The broader bulk dataset also includes Cu, Ge, Si, and Al2O3, while supported Pt clusters vary in size on MoS2 slabs.
- Bulk and complex systems: A unified DeepPot-SE model captures the multi-component, extended, and variable-size structure of supported Pt clusters on MoS2 with satisfactory accuracy.These features make the supported-cluster system challenging for an end-to-end framework.
5 Summary
The paper develops and evaluates DeepPot-SE as an end-to-end potential energy model for molecular and periodic systems. It reports coverage across diverse electronic structures and material complexities, while identifying expanded datasets and easier training-data collection as future needs.
- Summary: DeepPot-SE is an end-to-end, scalable, symmetry-preserving, and accurate potential energy model tested on molecular and periodic systems.The model is evaluated across both finite and extended settings.
- Summary: For periodic systems, DeepPot-SE describes metals, insulators, semiconductors, bulk crystals, surfaces, and high-entropy alloys.The reported scope spans diverse electronic structures and degrees of complexity.
- Summary: Future work should expand datasets for more challenging scientific and engineering studies and develop strategies to ease training-data collection.The paper also points to applications of symmetry-preserving ideas in other machine-learning tasks.