Physically informed artificial neural networks for atomistic modeling of materialsphysics.gmu.edu/~ymishin/resources/Nat_Comm_PINN.pdf · 2019. 6. 1. · using high-dimensional nonlinear

ARTICLE

Physically informed artificial neural networks foratomistic modeling of materialsG.P.Purja Pun1, R. Batra2, R. Ramprasad 2 & Y. Mishin1

Large-scale atomistic computer simulations of materials heavily rely on interatomic potentials

predicting the 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 inter-

polation within 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. This approach,

called the physically informed neural network (PINN) potential, is demonstrated by devel-

oping a general-purpose PINN potential for Al. We suggest that the development of physics-

based ML potentials is the most effective way forward in the field of atomistic simulations.

https://doi.org/10.1038/s41467-019-10343-5 OPEN

1 Department of Physics and Astronomy, MSN 3F3, George Mason University, Fairfax, VA 22030, USA. 2 School of Materials Science and Engineering, GeorgiaInstitute of Technology, Atlanta, GA 30332, USA. Correspondence and requests for materials should be addressed to Y.M. (email: [email protected])

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Large-scale molecular dynamics (MD) and Monte Carlo(MC) simulations of materials are traditionally implementedusing classical interatomic potentials predicting the potential

energy and Newtonian forces acting on atoms. Computationswith such potentials are very fast and afford access to systemswith millions of atoms and MD simulation times up to hundredsof nanoseconds. Such simulations span a wide range of time andlength scales and constitute a critical component of the multiscaleapproach in materials modeling and computational design.

Several functional forms of interatomic potentials have beendeveloped over the years, including the embedded-atom method(EAM)1–3, the modified EAM (MEAM)4, the angular-dependentpotentials5, the charge-optimized many-body potentials6, reactivebond-order potentials7–9, and reactive force fields10 to name afew. These potentials address particular classes of materials orparticular types of applications. Their functional forms depend onthe physical and chemical models chosen to describe interatomicbonding in the respective class of materials.

A common feature of all traditional potentials is that theyexpress the potential energy surface (PES) of the system, E=E(r1, ..., rN, p), as a relatively simple function of atomic coordi-nates (r1, ..., rN), N being the number of atoms (Fig. 1a). Knowingthe PES, the forces acting on the atoms can be computed bydifferentiation and used in MD simulations. The potential func-tions depend on a relatively small number of fitting parametersp= (p1, ..., pm) (typically, m= 10–20) and are optimized (trained)on a relatively small database of experimental data and first-principles density functional theory (DFT) calculations. The

traditional potentials are, of course, much less accurate than DFTcalculations. Nevertheless, many of them demonstrate a reason-ably good transferability to atomic configurations lying welloutside the training dataset. This important feature owes itsorigin to the incorporation of at least some basic physics in thepotential form. As long as the nature of chemical bondingremains the same as assumed during the potential development,the potential can predict the system energy adequately even fornew configurations not seen during the training process. Unfor-tunately, the construction of good quality potentials is a long andpainful process requiring personal experience and intuition and ismore art than science8,11. In addition, the traditional potentialsare specific to a particular class of materials and cannot be easilyextended to other materials or improved in a systematic manner.

During the past decade, a new direction has emerged whereininteratomic potentials are developed by employing machine-learning (ML) methods12–22. The idea was originally conceived inthe chemistry community in the 1990s in the effort to improvethe accuracy of inter-molecular force fields23,24, an approach thatwas later adopted by the physics and materials science commu-nities. The general idea is to forego the physical insights andreproduce the PES by interpolating between DFT data pointsusing high-dimensional nonlinear regression methods such as theGaussian process regression19,25–27, interpolating moving leastsquares28, kernel ridge regression12,20,21, compressed sensing29,30,gradient-domain machine-learning model31, or the artificialneural network (NN) approach13–18,32–38. If properly trained, aML potential can predict the system energy with a nearly DFTaccuracy (a few meV/atom). ML potentials are not specific to aparticular class of materials or type of chemical bonding. Theycan be improved systematically if weaknesses are discovered ornew DFT data become available. The training process can beimplemented on-the-fly by running ab initio MD simulations26.

A major weakness of ML potentials is their poor transferability.Being purely mathematical constructions devoid of any physicalmeaning, they can accurately interpolate the energy between thetraining configurations but are generally incapable of properlyextrapolating the energy to unknown atomic environments. As aresult, the performance of ML potentials outside the trainingdomain can be very poor. There is no reason why a purelymathematical extrapolation scheme would deliver physicallymeaningful results outside the training database. This explainswhy the existing ML potentials are usually (with rare excep-tions39) narrowly focused on, and only tested for, a particulartype of physical properties. This distinguishes them from thetraditional potentials which, although less accurate, are designedfor a much wider range of applications and diverse properties.

In this work we propose a new approach that can drasticallyimprove the transferability of ML potentials by informing them ofthe physical nature of interatomic bonding. We focus on NNpotentials as an example, but the approach is general and can bereadily extended to other methods of nonlinear regression. Likeall ML potentials, the proposed physically informed NN (PINN)potentials are trained using a large DFT dataset. However, bycontrast to the existing, mathematical NN potentials, the PINNpotentials incorporate the basic physics and chemistry of atomicinteractions leveraged by the extraordinary adaptivity and train-ability of NNs. The PINN potentials thus strike a golden com-promise between the two extremes represented by the traditional,physics-guided interatomic potentials, and the mathematical NNpotentials.

The general idea of combining traditional interatomic poten-tials with NNs was previously discussed by Malshe et al.40, whoconstructed an adjustable Tersoff potential41–43 for a Si5 cluster.Other authors have also applied machine-learning methods toparameterize physics-based models of molecular interactions,

Atom iNeuralnetwork

Otheratoms

Σ PES

Otheratoms

Atom i Neuralnetwork

Interatomicpotential Ei

Ei

Ei

Σ PES

Localparameters(ai1, ..., aiλ)

Globalparameters(b1, ...,b�)

Neighboringatoms (r i1, ..., r in)

Atom i Neuralnetwork

Potentialparameters(pi1, ..., pim)

Otheratoms

Interatomicpotential

Σ PES

Neighboringatoms (r1, ..., rn)

b

c

d

Atom i

Otheratoms

Interatomicpotential Ei

Σ PES

a

Potentialparameters(pi1, ..., pim)

Neighboringatoms (r i1, ..., r in)

Local structural parameters(Gi

1,Gi2, ..., Gi

k)


1,Gi2, ..., Gi

k)


1,Gi2, ..., Gi

k)

Fig. 1 Flowcharts of the development of atomistic potentials. a Traditionalinteratomic potential. b Mathematical NN potential. c Physically informedNN (PINN) potential with all-local parameters. d PINN potential withparameters divided into local and global. The dashed rectangle outlines theobjects requiring parameter optimization. PES is the potential energysurface of the material

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primarily in the context of broad exploration of the compositionalspace of molecular (mostly organic) matter44–46. Glielmo et al.47

recently proposed to construct n-body Gaussian process kernelsto capture the n-body nature of atomic interactions in physicalsystems. The PINN potentials proposed in this paper are inspiredby such approaches but extend them to (1) more advancedphysical models with a broad applicability, and (2) large-scalesystems by introducing local energies Ei linked to local structuralparameters Gl

i. The focus is placed on the exploration of theconfigurational space of defected solids and liquids in single-component and, in the future, binary or multicomponent sys-tems. The main goal is to improve the transferability of intera-tomic potentials to unknown atomic environments while keepingthe same level of accuracy of training as normally achieved withmathematical machine-learning potentials.

ResultsPhysically informed neural network potentials. The currentlyexisting, mathematical NN potentials13–18,32–36 partition the totalenergy E into a sum of atomic energies, E ¼Pi Ei. A single NN isconstructed to express each atomic energy Ei as a function of a setof local fingerprint parameters (also called symmetry para-meters13) ðG1

i ;G2i ; :::;G

ki Þ. These parameters encode the local

environments of the atoms. The network is trained by minimizingthe error between the energies predicted by the NN and therespective DFT total energies for a large set of atomic config-urations. The flowchart of the method is depicted in Fig. 1b.

The proposed PINN model is based on the followingconsiderations. A traditional, physics-based potential can alwaysbe trained to reproduce the energy of any given atomicconfiguration with any desired accuracy. Of course, this potentialwill not work well for other configurations. Imagine, however,that the potential parameters have been trained for a large set ofreference structures, one structure at a time, each time producinga different parameter set p. Suppose that, during the subsequentsimulations, we have a way of identifying, on the fly, a referencestructure closest to any current atomic configuration. Then theaccuracy of the simulation can be drastically improved bydynamically choosing the best set of potential parameters forevery atomic configuration accoutered during the simulation.Now, since the atomic energy Ei only depends on the localenvironment of atom i, the best parameter set for computing Eican be chosen by only examining the local environment of thisatom. The energies of different atoms are then computed by usingdifferent, environment-dependent, parameter sets while keepingthe same, physics-motivated functional form of the potential.

Instead of generating and storing a large set of discretereference structures, we can construct a continuous NN-basedfunction mapping the local environment of every atom on aparameter set of the interatomic potential optimized for thatparticular environment. Specifically, the local structural para-meters (fingerprints) Gl

i (l = 1, ..., k) of every atom i are fed intothe network, which then maps them to the optimized parameterset pi appropriate for atom i. Mathematically, the local energytakes the functional form

Ei ¼ Ei ri1; :::; rin; pi Gliðri1; :::; rinÞ

� �� ; ð1Þ

where (ri1, ..., rin) are atomic positions in the vicinity of atom i.In comparison with the direct mapping Gl

i 7!Ei implementedby the mathematical NN potentials, we have added anintermediate step: Gl

i 7!pi 7!Ei. The first step is executed by theNN and the second by a physics-based interatomic potential. Aflowchart of the two-step mapping is shown in Fig. 1c. It isimportant to emphasize that this intermediate step does notdegrade the accuracy relative to the direct mapping, because a

feedforward NN can always be trained to execute any real-valuedfunction48,49. Thus, for any functional form of the potential, theNN can always adjust its architecture, weights and biases toachieve the same mapping as in the direct method. However,since the chosen potential form captures the essential physics ofatomic interactions, the proposed PINN potential will display abetter transferability to new atomic environments. Even if thepotential parameters predicted by the NN for an unknownenvironment are not very accurate, the physics-motivatedfunctional form will ensure that the results remain at leastphysically meaningful. This physics-guided extrapolation is likelyto be more reliable than the purely mathematical extrapolationinherent in the existing NN potentials. Obviously, the samereasoning applies to the interpolation process as well, which canalso be more accurate.

The functional form of the PINN potential must be generalenough to be applicable across different classes of materials. Inthis paper we chose a simple analytical bond-order potential(BOP)50–52 that must work equally well for both covalent andmetallic materials. For a single-component system, the BOPfunctions are specified in the Methods section. They capture thephysical and chemical effects such as the pairwise repulsionbetween atoms, the angular dependence of the chemical bondstrength, the bond-order effect (the more neighbors, the weakerthe bond), and the screening of chemical bonds by surroundingatoms. In addition to being appropriate for covalent bonding, theproposed BOP form reduces to the EAM formalism in the limit ofmetallic bonding.

Example: PINN potential for Al. To demonstrate the PINNmethod, we have constructed a general-purpose potential foraluminum. The training and validation datasets were randomlyselected from a pre-existing DFT database20,21. Some additionalDFT calculations have also been performed using the samemethodology as in refs. 20,21. The selected DFT supercellsrepresent seven crystal structures for a large set of atomic volumesunder isotropic tension and compression, several slabs with dif-ferent surface orientations, including surfaces with adatoms, asupercell with a single vacancy, five different symmetrical tiltgrain boundaries, and an unrelaxed intrinsic stacking fault on the(111) plane with different translational states along the [211]direction. The database also includes several isolated clusters withthe number of atoms ranging from 2 (dimer) to 79. The ground-state face centered cubic (FCC) structure was additionally subjectto uniaxial tension and compression in the [100] and [111]directions at 0 K temperature. Most of the atomic configurationswere snapshots of DFT MD simulations in the microcanonical(NVE) or canonical (NVT or NPT) ensembles for several atomicvolumes at several temperatures. Some of the high-temperatureconfigurations were part-liquid, part crystalline. In total, thedatabase contains 3649 supercells (127592 atoms). More detailedinformation about the database can be found in the Supple-mentary Tables 1 and 2. To avoid overfitting or selection bias, the10-fold cross-validation method was used during the training.The database was randomly partitioned in 10 subsets. One ofthem was set aside for validation and the remaining data was usedfor training. The process repeated 10 times for different choices ofthe validation subset.

The local structural parameters Gli chosen for Al are specified

in the Methods section. The NN contained two hidden layerswith the same number of nodes in each. This number wasincreased until the training process produced a PINN potentialwith the root-mean-square error (RMSE) of training andvalidation close to 3–4 meV per atom, which was set as our goal.This is the level of accuracy of the DFT energies included in the

NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-10343-5 ARTICLE




database. For comparison, a mathematical NN potential wasconstructed using the same methodology. The number of hiddennodes of the NN was adjusted to give about the same number offitted parameters and to achieve approximately the same RMSE oftraining and validation as for the PINN potential. Table 1summarizes the training and validation errors averaged over the10 cross-validation runs. One PINN and one NN potential wereselected for a more detailed examination reported below.

Figure 2 and Supplementary Fig. 1 demonstrate excellentcorrelation between the predicted and DFT energies over a 7 eVper atom wide energy range for both potentials. The errordistribution has a near-Gaussian shape centered at zero.Examination of errors in individual groups of structures(Supplementary Fig. 2) shows that the largest errors originatefrom the crystal structures (especially FCC, HCP, and simplehexagonal) subjected to large expansion.

Table 2 summarizes some of the physical properties of Alpredicted by the potentials in comparison with DFT data fromthe literature. There was no direct fit to any of these properties,although atomic configurations most relevant to some of theproperties were represented in the training dataset. While bothpotentials agree with the DFT data well, the PINN potential tends

to be more accurate for most properties. For the [110] self-interstitial dumbbell, the NN potential predicts an unstableconfiguration that spontaneously rotates to the [100] orientation,whereas the PINN potential correctly predicts such configura-tions to be metastable. Figure 3 shows the linear thermalexpansion factor as a function of temperature predicted by thepotentials in comparison with experimental data. The PINNpotential displays good agreement with experiment without directfit, whereas the NN potential overestimates the thermal expansionat high temperatures. (The discrepancies at low temperatures aredue to the quantum effects that are not captured by classicalsimulations.) As another test, the radial distribution function andthe bond angle distribution in liquid Al were computed at severaltemperatures for which experimental and/or DFT data areavailable (Supplementary Figs 4 and 5). In this case, bothpotentials were found to perform equally well. Any smalldeviations from the published DFT calculations are within theuncertainty of the different DFT flavors (exchange-correlationfunctionals).

For testing purposes, we computed the energies of theremaining groups of structures that were part of the originalDFT database20,21 but were not used here for training orvalidation. The full information about the testing dataset(26,425 supercells containing a total of 2,376,388 atoms) can befound in the Supplementary Table 3. For example, Fig. 4compares the energies predicted by the potentials with DFTenergies from high-temperature MD simulations for a supercellcontaining an edge dislocation or HCP Al. In both cases, thePINN potential is obviously more accurate. The remaining testingcases are presented in the Supplementary Figs. 6–10. Althoughthere are cases where both potentials perform equally well, inmost cases the PINN potential predicts the energies of unknownatomic configurations more accurately than the NN potential.

For further testing, the energies of the crystal structures of Alwere computed for atomic volumes both within and beyond thetraining interval. Both potentials accurately reproduce the DFTenergy–volume relations for all volumes spanned by the DFT

Table 1 Fitting and validation errors of the straight NN andPINN models

Model NNarchitecture

Number ofparameters

RMSE oftraining (meVper atom)

RMSE ofvalidation (meVper atom)

NN 60 × 16 ×16 × 1

1265 3.36 3.85

NN′ 47 × 18 ×18 × 1

1225 3.62 3.54

PINN 60 × 15 ×15 × 8

1283 3.46 3.59

4a

3

2

1

0

–4–4

–3

–3

–2

–2

–1

–1

0

DFT energy (eV/atom)

Com

pute

d en

ergy

(eV

/ato

m)

1 2 3 4

35

30

25

20

Per

cent

age

15

10

5

0–20 –15 –10 –5 0

EPINN – EDFT (meV/atom)

5 10 15 20

b

3

2

1

0

–4–4

–3

–3

–2

–2

–1

–1

0DFT energy (eV/atom)

Com

pute

d en

ergy

(eV

/ato

m)

1 2 3

c35

30

25

20

Per

cent

age

15

10

5

0–20 –15 –10 –5 0

EPINN – EDFT (meV/atom)

5 10 15 20

d

Fig. 2 Accuracy of the PINN potential. a, c Energies of atomic configurations in the a training and c validation datasets computed with the PINN potentialversus DFT energies. The straight line represents the perfect fit. b, d Error distributions in the b training and d validation datasets




database (Fig. 5 and Supplementary Fig. 3). However, extrapola-tion to larger or smaller volumes reveals significant differences.For example, the PINN potential correctly predicts that thecrystal energy continues to rapidly increase under strongcompression (repulsive interaction mode). In fact, the extra-polated PINN energy goes exactly through the new DFT pointsthat were not included in the training or validation datasets, seeexamples in Fig. 6. By contrast, the energy predicted by the NNmodel immediately develops wiggles and strongly deviates fromthe physically meaningful repulsive behavior. Such artifacts werefound for other structures as well.

To demonstrate that the unphysical behavior exhibited by theNN potential is not a specific feature of our structural parametersGli or the training method, we constructed another NN potential

using a third-party NN-training package PROPhet53. Thispotential, which we refer to as NN′, uses the Behler-Parrinellosymmetry functions13, which are different from our structuraldescriptor Gl

i. The NN-training algorithm is also different. A 47 ×18 × 18 × 1 network containing 1225 fitting parameters was

trained on exactly the same DFT database to about the sameaccuracy as the NN and PINN potentials (Table 1). Figure 6shows that the NN′ potential behaves in a similar manner as ourNN potential, closely following the DFT energies within thetraining/validation domain and becoming unphysical as soon aswe step outside this domain.

While the atomic forces were not used for either trainingor validation, they were compared with the DFT forces oncethe training was complete. For the validation dataset,this comparison probes the accuracy of interpolation, whereasfor the testing dataset the accuracy of extrapolation. As expected, forthe validation dataset the PINN forces are in better agreement withDFT calculations than the NN forces (RMSE ≈ 0.1 eVÅ−1 versus≈0.2 eVÅ−1) as illustrated in Fig. 7a, b. For the testing dataset,the advantage of the PINN model in force predictions iseven more significant. For example, for the dislocation andHCP cases discussed above, the PINN potential provides moreaccurate predictions (RMSE ≈ 0.1 eVÅ−1) than the NN potential(RMSE ≈ 0.4 eVÅ−1 for the dislocation and 0.6 eVÅ−1 for theHCP case) (Fig. 7c, f). This advantage persists for all other groups ofstructures from the testing database.

It was also interesting to compare the PINN potential withtraditional, parameter-based potentials for Al. One of them wasthe widely accepted EAM Al potential54 that had been fitted to amix of experimental and DFT data. The other was a BOPpotential of the same functional form as in the PINN model. Itsparameters were fitted in this work using the same DFT databaseas for the PINN/NN potentials and then fixed once and for all.Figure 8 compares the DFT energies with the energies predictedby the EAM and BOP models across the entire set of referenceconfigurations. The PINN predictions are shown for comparison.The plots demonstrate that the traditional, fixed-parametermodels generally follow the correct trend but become increasinglyless accurate as the structures deviate from the equilibrium, low-energy atomic configurations. The adaptivity to the local atomicenvironments built into the PINN potential greatly improves theaccuracy.

Table 2 Aluminum properties predicted by the PINN and NN potentials

Property DFT NN PINN

E0 (eV per atom) −3.7480a −3.3606 −3.3609a0 (Å) 4.039a,d; 3.9725–4.0676c 4.0409 4.0396B (GPa) 83a; 81f 80 79c11 (GPa) 104a; 103–106d 108 117c12 (GPa) 73a; 57–66d 66 60c44 (GPa) 32a; 28–33d 25 32γs(100) (J m−2) 0.92b 0.897 0.899γs(110) (J m−2) 0.98b 0.986 0.952γs(111) (J m−2) 0.80b 0.837 0.819Efv (eV) 0.665–1.346c; 0.7e 0.640 0.678Efv (eV) unrelaxed 0.78e 0.71 0.77Emv (eV) 0.304−0.621c 0.627 0.495EfI (Td) (eV) 2.200–3.294c 2.683 2.840EfI (Oh) (eV) 2.531–2.948c 1.600 2.367EfI ⟨100⟩ (eV) 2.295–2.607c 1.529 2.246EfI ⟨110⟩ (eV) 2.543–2.981c 1.529* 2.713EfI ⟨111⟩ (eV) 2.679–3.182c 2.631 2.815γSF (mJ m−2) 134i; 146g; 158h 128 121γus (mJm−2) 162j; 169i; 175h 143 132

The potential predictions are compared with DFT calculations from the literatureE0 equilibrium cohesive energy, a0 equilibrium lattice parameter, B bulk modulus, cij elastic constants, γs surface energy, Efv vacancy formation energy, Emv vacancy migration barrier, EfI interstitial formationenergy for the tetrahedral (Td) and octahedral (Oh) positions and split dumbbell configurations with different orientations, γSF intrinsic stacking fault energy, γus unstable stacking fault energy. All defectenergies are statically relaxed unless otherwise indicatedaRef. 61; bref. 62; cref. 63; dref. 64; eref. 65; fref. 66 ; gref. 67; href. 68; iref. 69; jref. 70

*Unstable and flips to the ⟨100⟩ dumbbell orientation

–1.0

–0.5

0.0

0.5

1.0

1.5

2.0

2.5

0 200 400 600 800 1000

Line

ar th

erm

al e

xpan

sion

(%

)

Temperature (K)

NN

PINN

Experiment

Fig. 3 Linear thermal expansion of Al relative to room temperature (295 K)predicted by the PINN and NN potentials in comparison with experiment60





DiscussionThe proposed PINN potential model is capable of achieving thesame high accuracy in interpolating between DFT energies on thePES as the currently existing mathematical NN potentials. The

construction of PINN potentials requires the same type of DFTdatabase, is equally straightforward, and does not heavily rely onhuman intuition. However, extrapolation outside the domain ofatomic configurations represented in the training database is nowbased on a physical model of interatomic bonding. As a result, theextrapolation becomes more reliable, or at least more failure-proof, than the purely mathematical extrapolation. The accuracyof interpolation can also be improved for the same reason. As anexample, the PINN Al potential constructed in this paperdemonstrates better accuracy of interpolation and significantlyimproved transferability than a regular NN potential with aboutthe same number of parameters. The advantage of the PINNpotential is especially strong for atomic forces, which areimportant for molecular dynamics. The potential could be usedfor accurate simulations of mechanical behavior and other pro-cesses in Al. Construction of general-purpose PINN potentials forSi and Ge is currently in progress.

We believe that the development of physics-based ML poten-tials is the best way forward in this field. Such potentials need notbe limited to NNs or the particular BOP model adopted in thispaper. Other regression methods can be employed and theinteratomic bonding model can be made more sophisticated, orthe other way round, simpler in the interest of speed.

Other modifications are envisioned in the future. For example,not all potential parameters are equally sensitive to local envir-onments. To improve the computational efficiency, the para-meters can be divided into two subsets40: local parameters ai=(ai1, ..., aiλ) adjustable according to the local environments asdiscussed above, and global parameters b= (b1, ..., bμ) that arefixed after the optimization and used for all environments (as inthe traditional potentials). The potential format now becomes

Ei ¼ Ei ri1; :::; rin; ai Gliðri1; :::; rinÞ

� �; b

� �: ð2Þ

During the training process, the global parameters b and thenetwork weights and biases are optimized simultaneously, as

a

–3.3–3.3

–3.2

–3.2

–3.1

–3.1

Dislocation


PIN

N e

nerg

y (e

V/a

tom

)

b

–3.3–3.3

–3.2

–3.2

–3.1

–3.1

Dislocation


NN

ene

rgy

(eV

/ato

m)

c

HCP

–0.2

–0.3

–0.4

–0.5

–0.6

–0.7

–0.8

–0.9

–1.0–1.0 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2


d

–1.0 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 –0.2

–0.2

HCP–0.3

–0.4

–0.5

–0.6

–0.7

–0.8

–0.9

–1.0

DFT energy (eV/atom)N

N e

nerg

y (e

V/a

tom

)

PIN

N e

nerg

y (e

V/a

tom

)

Fig. 4 Testing of the NN and PINN potentials. a, b Energy of an edge dislocation in Al in NVE MD simulations starting at 700 K. c, d Energy of HCP Al inNVT MD simulations at 1000, 1500, 2000, and 4000 K. The energies predicted by the PINN (a, c) and NN (b, d) potentials are compared with DFTcalculations from20,21. The straight lines represent the perfect fit

a

10–4.0

–3.0

–2.0

–1.0

0.0

1.0

2.0

3.0

4.0

20 30 40 50

Volume (Å3/atom)

60 70 80 90 1000

Ene

rgy

(eV

/ato

m)

HCP

BCC

SC

b

4020

Volume (Å3/atom)

60 80 120 1401000–4.0

–3.0

–2.0

–1.0

0.0

1.0

2.0

3.0

4.0

Ene

rgy

(eV

/ato

m)

A15

SH

DC

Fig. 5 Energy–volume relations for Al crystal structures. Comparison of theenergies predicted by the PINN potential (lines) and by DFT calculations(points). a Hexagonal close-packed (HCP), body-centered cubic (BCC), andsimple cubic (SC) structures. b A15 (Cr3Si prototype), simple hexagonal(SH), and diamond cubic (DC) structures




shown in Fig. 1d. Extension of PINN potentials to binary andmulticomponent systems is another major task for the future.

All ML potentials are orders of magnitude faster than straightDFT calculations but inevitably much slower than the traditionalpotentials. Preliminary tests indicate that PINN potentials areabout 25% slower than the regular NN potentials for the samenumber of parameters, the extra overhead being due to the BOPcalculation. However, the computational efficiency depends onthe parallelization method and computer architecture. All com-putations reported in this paper utilized in-house software par-allelized with MPI for training and with OpenMP for MD andMC simulations (see example in Supplementary Fig. 14). Colla-borative work is underway to develop highly scalable HPC soft-ware packages for physically informed ML potential training andMD/MC simulations using multiple CPUs or GPUs, or both. Theresults will be reported in a forthcoming paper.

MethodsLocal structural parameters. There are many possible ways of choosing localstructural parameters13–18,34,36. After trying several options, the following set ofGli ’s was selected. For an atom i, we define

gðmÞi ¼

Xj;k

Pm cos θijk� �

f ðrijÞf ðrikÞ;m ¼ 0; 1; 2; :::; ð3Þ

where rij and rik are distances to atoms j and k, respectively, and θijk is the anglebetween the bonds ij and ik. In Eq. (3), Pm(x) is the Legendre polynomial of order

m and

f ðrÞ ¼ 1σ3

e�ðr�r0Þ2=σ2 fcðrÞ ð4Þ

is a truncated Gaussian of width σ centered at point r0. The truncation functionfc(r) is defined by

fcðrÞ ¼ðr�rcÞ4

d4þðr�rcÞ4r � rc

0; r � rc:

(ð5Þ

This function and its derivatives up to the third go to zero at a cutoff distance rc.The parameter d controls the truncation range.

For example, P0(x)= 1 and gð0Þi characterizes the local atomic density near atom

i. Likewise, P1(x)= x and gð1Þi can be interpreted as the dipole moment of a set ofunit charges placed at the atomic positions j and k. As such, this parametermeasures the degree of local deviation from spherical symmetry in the

environment (gð1Þi ¼ 0 for spherical symmetry). For m= 2, we have P2(x)= (3x2−1)/2 and gð2Þi is related to the quadrupole moment of a set of unit charges placed atthe atomic positions around atom i. We found that polynomials up to degree m=6 should be included to accurately represent the diverse atomic environment. Each

gðlÞi is computed for several values of σ and r0 spanning a range of interatomic

distances. For each atom, the set of k gðmÞi ’s obtained is arranged in a one-

dimensional array ðG1i ;G

2i ; :::;G

ki Þ. In this work we chose σ= 1.0 and used

polynomials with m= 0, 1, 2, 4, 6 for 12 r0 values, giving a total of k= 60 Gli ’s.

The BOP potential. In the BOP model adopted in this work, the energy of an atomi is postulated in the form

Ei ¼12

Xj≠i

eAi�αi rij � SijbijeBi�βi rij

h ifcðrijÞ þ EðpÞ

i ; ð6Þ

where rij is the distance between atoms i and j and the summation is over all atom jother than i within the cutoff radius rc. The bond-order parameter bij is taken in theform

bij ¼ ð1þ zijÞ�1=2; ð7Þwhere

zij ¼ a2iXk≠i;j

Sikðcosθijk þ hiÞ2fcðrikÞ ð8Þ

represents the number of chemical bonds (other than ij) formed by atom i. Largerzij values (more bonds) lead to a smaller bij and thus weaker ij bond.

The screening factor Sij reduces the strength of bonds by surrounding atoms.For example, when counting the bonds in Eq. (8), we screen them by Sik, so thatstrongly screened bonds contribute less to zij. The screening factor Sij is given by

Sij ¼Yk≠i;j

Sijk; ð9Þ

where the partial screening factor Sijk represents the contribution of a neighboringatom k (different from i and j) to the screening of the bond ij. Sijk is given by

Sijk ¼ 1� fcðrik þ rjk � rijÞe�λ2i ðrikþrjk�rijÞ: ð10ÞIt has the same value for all atoms k located on the surface of an imaginary

spheroid whose poles coincide with the atoms i and j. For all atoms k outside thiscutoff spheroid, on which rik+ rjk− rij= rc, we have Sijk= 1 — such atoms are toofar away to screen the bond. If an atom k is placed on the line between the atoms iand j, we have rik+ rjk− rij= 0 and Sijk is small — the bond ij is strongly screened(almost broken) by the atom k. This behavior reasonably reflects the nature ofchemical bonding.

Finally, the promotion energy EðpÞi is taken in the form

EðpÞi ¼ �σ i

Xj≠i

SijbijfcðrijÞ !1=2

: ð11Þ

For a covalent material, EðpÞi accounts for the energy cost of changing the

electronic structure of a free atoms before it forms chemical bonds. For example,for group IV elements, this is the cost of the s2p2→ sp3 hybridization. On the other

hand, EðpÞi can be interpreted as the embedding energy

Fð�ρiÞ ¼ �σ i �ρi� �1=2 ð12Þ

appearing in the EAM formalism1,2. Here, the host electron density on atom i isgiven by �ρi ¼

Pj≠i Sijbij fcðrijÞ. Due to this feature, this BOP model can be applied

to both covalent and metallic systems.The BOP functions depend on eight parameters Ai, Bi, αi, βi, ai, hi, σi, and λi,

which constitute the parameter set (p1, ..., pm) with m= 8. The cutoff parameterswere fixed at rc= 6 Å and d= 1.5 Å.

a

Simple cubic structure

0

–4

–2

0

2

4

6

8

10

12

14

5 10 15 20 25 30 35 40 45 50

Training set

Test set

Validation set

PINN

NN

NN’

BOP

EAM

Volume (Å3/atom)

Ene

rgy

(eV

/ato

m)

b

Diamond cubic structure

Training set

Test set

Validation set

PINN

NN

NN’

BOP

EAM

0 5 10 15 20 25 30 35 40 45 50

Volume (Å3/atom)

–4

–2

0

2

4

6

8

10

12

14

Ene

rgy

(eV

/ato

m)

Fig. 6 Zoom into the repulsive part of the energy–volume relationspredicted by the PINN, NN, NN′, EAM, and BOP potentials (curves) andDFT calculations (points)





The neural network and training procedures. The feedforward NN containedtwo hidden layers and had the 60 × 15 × 15 × 8 architecture for the PINN potentialand 60 × 16 × 16 × 1 for the NN potential. The number of nodes in the hiddenlayers was chosen to reach the target accuracy of about 3-4 meV/atom withoutoverfitting.

The training/validation database consisted of DFT total energies for a set ofsupercells. The DFT calculations were performed using projector-augmented wave(PAW) pseudopotentials as implemented in the electronic structure Vienna Abinitio Simulation Package (VASP)55,56. The generalized gradient approximation(GGA) was used in conjunction with the Perdew, Burke, and Ernzerhof (PBE)density functional57,58. The plane-wave basis functions up to a kinetic energy cutoffof 520 eV were used, with the k-point density chosen to achieve convergence to afew meV per atom level. Further details of the DFT calculations can be found inrefs. 20,21. The energy of a given supercell s, Es ¼Pi E

si , predicted by the potential

was compared with the DFT energy EsDFT. Note that the original E

sDFT values were

not corrected to remove the energy of a free atom. To facilitate comparison with

literature data, prior to the training all DFT energies were uniformly shifted by0.38446 eV per atom to match the experimental cohesive energy of Al, 3.36 eV peratom59. The NN was trained by adjusting its weights wεκ and biases bκ to minimizethe objective function

E ¼Xs

Es � EsDFT

� �2 þ τXϵκ

wϵκj j2 þXκ

bκj j2 !

þ γXη

pη � �pη

�� 2 !

: ð13Þ

The second term was added to avoid overfitting by controlling the magnitudesof the weights and biases. The parameter τ controls the degree of regularization.The third term ensures that the variations of the PINN parameters relative to theirdatabase-averaged values �pη remain small. The minimization of E wasimplemented by the Davidson–Fletcher–Powell algorithm of unconstrainedoptimization. The optimization was repeated several times starting from differentrandom states and the solution with the smallest E was selected as final. The PINNand NN forces were computed by the finite-difference method.

aValidation set

15

10

PIN

N F

X (

eV/Å

) 5

0

–5

–10

–15

DFT FX (eV/Å)

151050–5–10–15

cDislocation

PIN

N F

X (

eV/Å

)

4

3

2

1

0

–1

–2

–3

–4

DFT FX (eV/Å)

–4 –3 –2 –1 0 1 2 3 4

PIN

N F

X (

eV/Å

)

15

15

10

10

HCP

5

5

0

0

–5

–5

–10

–10–15

–15

DFT FX (eV/Å)

e

b 15

Validation set10

NN

FX (

eV/Å

)

DFT FX (eV/Å)

5

0

151050

–5

–5

–10

–10–15

–15

dDislocation

NN

FX (

eV/Å

)

4

3

2

1

0

–1

–2

–3

–4

DFT FX (eV/Å)

–4 –3 –2 –1 0 1 2 3 4

HCP

PIN

N F

X (

eV/Å

)

15

10

5

0

–5

–10

–15151050–5–10–15

DFT FX (eV/Å)

f

Fig. 7 Testing of atomic force predictions. The x-component of atomic forces for a, b validation database, c, d edge dislocation in NVE MD simulationsstarting at 700 K, and e, f HCP Al in NVT MD simulations at 300, 600, 1000, 1500, 2000, and 4000 K. The forces predicted by the PINN (a, c, e) and NN(b, d, f) potentials are compared with DFT calculations from refs. 20,21. The straight lines represent the perfect fit. See Supplementary Figs 11–13 for allcomponents of the forces




Data availabilityAll data that support the findings of this study are available in the SupplementaryInformation file or from the corresponding author upon reasonable request.

Received: 7 September 2018 Accepted: 26 April 2019

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AcknowledgementsWe are grateful to Dr. James Hickman for performing some of the additional Al DFTcalculations used in this work. We are also grateful to Dr. Vesselin Yamakov fornumerous helpful discussions, the development of a software package for PINN-basedsimulations, and for benchmarking the computational speed of the method. The authorsacknowledge support of the Office of Naval Research under Awards No. N00014-18-1-2612 (G.P.P.P. and Y.M.) and N00014-17-1-2148 (R.B. and R.R.). This work was alsosupported in part by a grant of computer time from the DoD High PerformanceComputing Modernization Program at ARL DSRC, ERDC DSRC and Navy DSRC.

Author contributionsY.M. developed the PINN theory and initiated this research project. G.P.P.P. wrote thecomputer software for the NN and PINN potential training, validation and testing underY.M.’s direction and supervision. He also created the Al NN and PINN potentialsreported in this paper and tested their properties. R.B. generated much of the DFT datafor Al used in this work under R.R.’s advise and supervision. Y.M. wrote the initial draftof the manuscript. All co-authors were engaged in discussions, contributed ideas at allstages of the work, participated in the manuscript editing, and approved its final version.

Additional informationSupplementary Information accompanies this paper at https://doi.org/10.1038/s41467-019-10343-5.

Competing interests: The authors declare no competing interests.

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