development of physically informed neural network (pinn) … · 2019-09-09 · c(r) = cutoff...
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Speaker: J. Hickman*
LAMMPS Workshop and Symposium
August 13-15, 2019
* National Institute of Standards and Technology (NIST) Materials measurement laboratory (MML)
Supported by NRC postdoctoral fellowship
Development of physically informed neural network (PINN) interatomic potentials
G. Purja Pun**, F. Tavazza*, Y. Mishin**
** George Mason University: Department of Physics
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Motivation
Ab-initio/DFT Classical (MD/MC)
• Slow • Size limited (N~ ) • 0K or short timescales • Very accurate
Better potential models
image source: http://evolution.skf.com/us/bearing-research-going-to-the-atomic-scale
Speed Accuracy Scalability
Transferability
Compromises
Classical MD/MC
102
i
• Computationally Fast • Larger systems (N~ ) • Kinetic phenomena/long simulations • Accuracy depends on approximation of
the potential energy surface (PES)
107
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Interatomic potential model typesTraditional interatomic potential:
“Mathematical” or “straight” NN potentials:
Physically informed neural network (PINN):
Pros Cons
• Fast • Decent extrapolation • Physically inspired
• Fast relative to DFT • DFT level accuracy
(~1-5 mEv) within training set
• Relatively straight forward/routine to train/fit
• Systematic improvement (add more data)
• Slower than traditional potentials
• Bad extrapolation
• Difficult to train/fit • Hard to improve
upon once finalized
• Accuracy limitations
• Slower than traditional potentials
• Same as straight NN • Decent
extrapolation • Physically inspired
• LJ, EAM, ADP, Tersoff, REAX … etc
• Machine learning potentials • Gaussian process regression • Interpolating moving least squares
• Kernel ridge regression • Compressed sensing • ANN potentials
local parameterization Physically informed artificial neural networks for atomistic modeling of materials GPP Pun, R Batra, R Ramprasad, Y Mishin - Nature communications, 2019
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Artificial neural networks
sigmoid
transfer functions:
Artificial neural network
p2⋮
input layerhidden layers output
layer
⋮ ⋮ ⋮⋮
G1
G2
GN
p1
pM
w11
w12
w13
b1
b2
wN1 bk
inpu
t vec
tor
outp
ut v
ecto
r 60 16 16 8
The weights and bias are the NN’s fitting parameters (~1500 parameters)
data:
test
validationtraining
Activation function
Activation function
Artificial neural networks
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training data
Neural network prediction
-sin(x)
Training region unable to extrapolate
excellent interpolation
Train NN to reproduce provided data
Toy example:
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Traditional BOP potential cutoff function
bond order parameterpromotional
energy
Ep = − σo (Σj≠iSijbij fc(rij))12
Model can be used for Metallic
or covalent systems
Physically informed artificial neural networks for atomistic modeling of materials GPP Pun, R Batra, R Ramprasad, Y Mishin - Nature communications, 2019
Ei =12 (Σj≠ieAo−αorij − SijbijeBo−βorij) fc(rij) + Ep
bij = (1 + zij)− 12
Sijk = 1 − fc(rik + rjk − rij)e−λo(rik+rjk−rij) screening
zij = aoΣk≠i,jSik(cos(θijk + ho)2 fc(rik)Sij = ∏
k≠i,j
Sijk
(Ao, Bo, αo, βo, ao, ho, λo, σo)Fit a baseline traditional potential via 8 adjustable
step-0:
i
j
krik
rij✓ijk
E1E2
EN
⋮
Structures DFT energies Model energiesE1E2
EN
⋮
structure-1structure-2
structure-N⋮
“Training set”
RMSE = ( Σs(Es − Es)2
N )12
Model errorTraining process:pytorch
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Local structure parameters
Gm,ni = ∑
j,k
Pm(cos(θijk))f(rij)f(rik)
f(r) =1rn
e− (r − rn)2
σ2 fc(r)
i
i
j
k
rik
rij✓ijk
Structure parameters:
Gi’s act as “fingerprints” of the local atomic
environment
Angular term:Pm(cos(θ)) m = 0,1,2,4,6 (Legendre polynomials)
Radial term: fc(r) = Cutoff function
r1 r2rr12. . .rmin rc
choose 12 values for rn
8 rn with 5Pm
for each atom
concentric shells
Gm,ni = [40 × 1]
Physically informed artificial neural networks for atomistic modeling of materials GPP Pun, R Batra, R Ramprasad, Y Mishin - Nature communications, 2019
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PINN potential model
i
(r1, r2 . . . rn)
Energy of atom-i neighboring atom
positions
Local structure
parameter calculation
G1 = G1(rij, θikj)G2 = G2(rij, θikj)
(G1, G2 . . . GM)
structural “fingerprints”
Structure’s energy
BOP Model Ei
atomic energy
A(x) =1
e−x + 1−
12
δA
δα
δλ
perturbations to baseline parameters
RMSE = ( Σs(Es − Es)2
N )12
= NN(w, b)
(Ao, Bo, αo, βo, ao, ho, λo, σo)
(Ai, Bi, αi, βi, ai, hi, λi, σi) atom-i parameterization
=
baseline parameters+
NN perturbation(δA, δB, δα, δβ, δa, δh, δλ, δσ)
Training:
Find the NN weights and bias’s which minimize the RMSE
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Volume-energy curves
FCC
SC
BC8
diamond
wurtzite
diamond
thermally perturbed structures
ZOOMSingle layer silicene:Tetrameter
Bulk phases:Combined
NN: 60x32x8
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Reproduction of DFT energy landscape
Distribution of training error Distribution of validation error
(untrained)
y=x Equality with DFT
Training data Validation data
RMSE~3.5 meV
RMSE~ 4 meV
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Si PINN comparison with traditional potentialPINN:
RMSE (meV/atom)~ 3.5PINN
700X and 300X improvement respectively
zoom inPINN
SW
MOD
y=x Equality with DFT
±200 meV
Modified Tersoff
RMSE (meV/atom)
12812591Stillinger-Weber (1985)
Traditional potentials:
Pun, GP Purja Physical Review B 95.22 (2017): 224103.
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unphysical extrapolation!!
Excellent interpolation
physical extrapolation
physical extrapolation
zoom in
Training region
Diamond
ANN vs PINN Extrapolation
unphysical extrapolation!!
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ANN vs PINN Extrapolation
BCC (Si)
Simple Cubic (Si)Wurtzite (Si)
Simple Cubic (Al)
Physically informed artificial neural networks for atomistic modeling of materials GPP Pun, R Batra, R Ramprasad, Y Mishin - Nature communications, 2019
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Computational efficient
Paragrand MC:https://software.nasa.gov/software/LAR-18773-1Source: Vesselin Yamakov (NASA)
100
2.5
50
NOTE: results sensitive to hyper parameter choice
4500
273250
PINN vs EAM
NN vs EAM serial: 120X slower 40 CPU: ~30X slower GPU: ~15X slower
serial: 166X slower 40 CPU: ~40X slower GPU: ~20X slower
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Thermal properties (preliminary)
Paragrand MC:https://software.nasa.gov/software/LAR-18773-1
PINN-A PINN-B
Thermal expansion
Liquid RDF
PINN-A
Traditional potentials
MODC Pun, GP Purja Physical Review B 95.22 (2017): 224103.
syst
emat
ic im
prov
emen
t
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Future work•Get PINN_BOP working in LAMMPS •Hyper-parameter tune (speed/accuracy) •Explore other Chemical systems: •Ge, Pt, Cu
•Binary: SiGe, SiAl, …. etc •Possibly explore other traditional potentials formats •(PINN_EAM, PINN_ADP, etc )
•Applications •study thermal properties of 2D structures
•Si ,Ge, SiGe
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Conclusions•Developed a new silicon interatomic potential using the new PINN potential format •Even in preliminary stage we are obtaining excellent agreement with the DFT energies •Current potential reproduces DFT data around ~500x better than current traditional potentials •Better transferability than ANN potentials •Investigating methodological considerations to streamlining the fitting procedure for faster future development
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Acknowledgements
•Ganga Purja Pun •Vesselin Yamakov •Francesca Tavazza •Yuri Mishin •Adam Robinson •GMU,NRC,NIST
• https://www.tf.uni-kiel.de/matwis/amat/iss/kap_5/backbone/r5_3_3.html • https://www.wordstream.com/blog/ws/2017/07/28/machine-learning-applications • https://www.researchgate.net/figure/268158499_fig1_Figure-1-Phase-diagram-of-SiGe-alloys-showing-separation-of-the-solidus-and-liquidus • http://evolution.skf.com/us/bearing-research-going-to-the-atomic-scale • /https://www.chegg.com/homework-help/questions-and-answers/consider-concentric-metal-sphere-spherical-shells-shown-figure--innermost-solid-sphere-rad-
q4808250
Image references:
• Stillinger, Frank H., and Thomas A. Weber. "Computer simulation of local order in condensed phases of silicon." Physical review B 31.8 (1985): 5262. • (“In review”) G. P. Purja Pun(1), R. Batra(2), R. Ramprasad (3) and Y. Mishin(1): (1) George Mason Univ., (2) Univ. Connecticut, (3) Georgia Tech • Pun, GP Purja, and Y. Mishin. "Optimized interatomic potential for silicon and its application to thermal stability of silicene." Physical Review B 95.22 (2017): 224103.
References:
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