Physically-informed artificial neural networks for atomistic modeling of materials

G. P. Purja Pun; R. Batra; R. Ramprasad; Y. Mishin

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Physically-informed artificial neural networks for atomistic modeling of materials

G. P. Purja Pun, R. Batra, R. Ramprasad, Y. Mishin

arXiv:1808.01696v4Aug 5, 2018cond-mat.mtrl-sci

TL;DR

Traditional potentials transfer better but lack DFT-level accuracy, while mathematical ML potentials interpolate accurately yet extrapolate poorly to unknown structures. The paper combines an analytical bond-order potential with neural-network regression, and the resulting aluminum PINN potential preserves interpolation accuracy while improving extrapolation and agreement with physical-property data.

Problem
Traditional potentials are less accurate than DFT, while mathematical ML potentials transfer poorly beyond the DFT configurations used for training.
Method
PINN potentials use a neural network to predict local-environment-dependent parameters for an analytical bond-order potential trained on a large DFT dataset.
Results
The aluminum PINN potential achieves comparable interpolation accuracy to a mathematical NN and more accurate force and energy predictions on tested unknown configurations.
Takeaways & Limitations
Physics-based ML potentials provide a more reliable basis for extrapolating atomistic energies beyond the configurations represented in the training database.

Abstract

from arXiv · show

Large-scale atomistic computer simulations of materials heavily rely on interatomic potentials predicting the potential energy and Newtonian forces on atoms. Traditional interatomic potentials are based on physical intuition but contain few adjustable parameters and are usually not accurate. The emerging machine-learning (ML) potentials achieve highly accurate interpolation between the energies in a large DFT database but, being purely mathematical constructions, suffer from poor transferability to unknown structures. We propose a new approach that can drastically improve the transferability of ML potentials by informing them of the physical nature of interatomic bonding. This is achieved by combining a rather general physics-based model (analytical bond-order potential) with a neural-network regression. The network adjusts the parameters of the physics-based model on the fly during the simulations according to the local environments of individual atoms. This approach, called the physically-informed neural network (PINN) potential, is demonstrated by developing a general-purpose PINN potential for Al. The potential provides a DFT-level accuracy of energy predictions and excellent agreement with experimental and DFT data for a wide range of physical properties. We suggest that the development of physics-based ML potentials is the most effective way forward in the field of atomistic simulations.

I. INTRODUCTION

Traditional potentials are fast and physically grounded but less accurate, whereas mathematical ML potentials improve interpolation yet transfer poorly to unknown atomic environments. The paper proposes combining physical bonding models with neural networks to retain accuracy while improving transferability.

Motivation: Traditional interatomic potentials enable large-scale simulations but are less accurate than DFT while often transferring reasonably to unseen configurations.Their physical basis supports predictions when the chemical bonding remains consistent with the assumed model.
Motivation: Machine-learning potentials accurately interpolate between DFT configurations but generally extrapolate poorly to unknown atomic environments.Their purely mathematical construction provides no physical basis for reliable behavior outside the training domain.
Proposed approach: The proposed PINN approach combines a physics-based interatomic model with neural-network adaptability to improve ML-potential transferability.It is trained on a large DFT dataset while incorporating basic physics and chemistry of atomic interactions.
Objective: The work targets transferability to unknown environments while preserving the high training accuracy of mathematical machine-learning potentials.The stated application focus includes defected solids and liquids in single-component systems, with future extension to multicomponent systems.

II. PHYSICALLY-INFORMED NEURAL NETWORK POTENTIALS

PINN potentials insert a physics-based interatomic model between local-environment descriptors and predicted atomic energies. A neural network predicts environment-specific potential parameters, enabling physically guided interpolation and extrapolation without sacrificing the direct network’s representational accuracy.

Architecture: The physics-based model converts the predicted parameters and atomic coordinates into local energies, which are summed to obtain the total energy.The network supplies parameters while the interatomic potential performs the subsequent energy mapping.
Architecture: PINN potentials map local atomic fingerprints to environment-specific parameters of a physics-based interatomic potential.The parameters are generated continuously rather than stored as discrete parameter sets for reference structures.
Accuracy: The intermediate physics-based step need not reduce accuracy because a feedforward neural network can represent the same real-valued mapping as direct energy prediction.The network can adjust its architecture, weights, and biases to reproduce the direct mapping.
Transferability: Physics-guided extrapolation is intended to remain physically meaningful even when neural-network parameter predictions are imperfect for unknown environments.The same physical structure may also improve interpolation accuracy.
Physics model: The analytical bond-order potential was chosen to capture bonding effects across both covalent and metallic materials.Its functional form includes pairwise repulsion and angular dependence of chemical bond strength.

III. EXAMPLE: PINN POTENTIAL FOR AL

The aluminum PINN potential matches DFT accuracy on interpolation and shows stronger performance on extrapolation and physical-property tests. Its adaptive, physics-based parameters improve behavior for configurations far from equilibrium, including forces, volumes, defects, and thermal properties.

Dataset and training: Training and validation RMSE was targeted at 3–4 meV/atom, matching the accuracy level of the DFT energies in the database.PINN and mathematical NN models were constructed with comparable fitting capacity and approximately equal training and validation errors.
Energy accuracy: Both potentials correlate excellently with DFT energies across a 7 eV/atom energy range, with the largest errors arising for highly expanded crystal structures.The error distributions are near-Gaussian and centered at zero.
Physical properties: PINN agrees more accurately with most reported DFT and experimental aluminum properties and correctly predicts the [110] dumbbell as metastable.The NN instead predicts spontaneous rotation to the [100] orientation for that configuration.
Extrapolation: PINN more accurately predicts unknown configurations than NN in most testing cases, including high-temperature dislocation and HCP aluminum structures.The testing dataset contained 26,425 supercells and 2,376,388 atoms.
Extrapolation: Under strong compression, PINN reproduces the rapidly increasing DFT energy and newly computed DFT points, while NN develops wiggles and deviates from repulsive behavior.The difference appears when extrapolating beyond the training volume interval.
Force predictions: RMSE was approximately 0.1 eV/Å for PINN forces versus approximately 0.2 eV/Å for NN forces on validation data.For testing, PINN RMSE remained approximately 0.1 eV/Å, compared with 0.4 eV/Å for dislocation and 0.6 eV/Å for HCP cases using NN.
Comparison with traditional potentials: Fixed-parameter traditional models become less accurate as structures depart from equilibrium, whereas PINN’s local-environment adaptivity improves accuracy.The comparison included an EAM aluminum potential and a fixed-parameter BOP with the same functional form.

IV. DISCUSSION AND CONCLUSIONS

PINN potentials retain high DFT-interpolation accuracy while making extrapolation more reliable through a physical model of interatomic bonding. The approach improves Al force transferability and supports adaptable model complexity, although efficiency enhancements and multicomponent extensions remain future work.

Accuracy and transferability: PINN potentials achieve high DFT-interpolation accuracy while providing more reliable extrapolation than purely mathematical neural-network potentials.The Al PINN potential also demonstrates better interpolation accuracy and significantly improved transferability than a regular NN potential with about the same number of parameters.
Accuracy and transferability: PINN potentials are especially advantageous for atomic forces, which are important for molecular-dynamics simulations of mechanical behavior and other Al processes.The paper reports that the Al potential could be used for accurate simulations of these processes.
Future directions: Physics-based ML potentials can use regression methods beyond neural networks and vary bonding-model sophistication to balance modeling detail and computational speed.The authors state that the bonding model may be made more sophisticated or simpler in the interest of speed.
Future directions: Separating locally adjustable parameters from globally fixed parameters is proposed as a future modification to improve computational efficiency.The proposed split assigns local parameters to individual environments and global parameters to all environments.
Future directions: Extending PINN potentials to binary and multicomponent systems is identified as another major task for future work.The current discussion presents this extension as unfinished.
Computational cost: PINN potentials are about a factor of two slower than regular NN potentials with the same number of parameters, due to the additional BOP calculation.They remain orders of magnitude faster than straight DFT calculations but slower than traditional potentials.

METHODS

The PINN method represents local atomic environments with structural descriptors and uses a neural network to adjust parameters of a physics-based bond-order potential during simulations.

Local structural descriptors: Local structural parameters G_i encode each atom’s environment for the PINN model.The selected descriptors include angular information through Legendre polynomials and radial information from truncated Gaussians.
Local structural descriptors: The descriptor set uses polynomials with m = 0, 1, 2, 4, 6 for 12 r0 values, producing k = 60 structural parameters per atom.Polynomials up to degree m = 6 were found sufficient to represent diverse atomic environments.
Bond-order potential: The BOP energy sums pair interactions within cutoff radius rc and incorporates bond-order and screening factors into the atomic energy.The bond-order factor weakens bonds as the number of neighboring bonds increases, while screening accounts for surrounding atoms.
Bond-order potential: The BOP includes promotion or embedding-energy terms, allowing the model to represent both covalent and metallic systems.The promotion term represents electronic-structure change costs in covalent materials, while the embedding interpretation connects the model to metallic systems.
Model parameters: The BOP contains 8 adjustable parameters, with cutoff parameters fixed at rc = 6 Å and d = 1.5 Å.The eight parameters are Ai, Bi, αi, βi, ai, hi, σi and λi.
Training and optimization: The feedforward PINN network uses a 60 × 15 × 15 × 8 architecture, while its NN comparison uses 60 × 16 × 16 × 1.Hidden-layer sizes were selected to target approximately 4 meV/atom accuracy without overfitting.
Training and optimization: Training compares predicted and DFT energies after uniformly shifting DFT values by 0.38446 eV/atom to match Al’s experimental cohesive energy of 3.36 eV/atom.The objective includes weight and bias regularization and penalizes deviations of PINN parameters from database-averaged values.