nonlinear parametrization of geological fields using...
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Nonlinear parametrization of geological fields usingGenerative Adversarial Networks (GANs)
Shing Chan, PhD student,Ahmed H. Elsheikh
School of Energy, Geoscience, Infrastructure and Society,Heriot-Watt University, United Kingdom.
MASCOT–NUM 2019 annual conference,IFPEN Rueil–Malmaison – France,
18–20 March 2019
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Uncertainty propagation for subsurface reservoir models
Subsurface reservoir modelsReservoir management
• Computationally expensive models Predictive modeling for decision support – challenges
Computationally expensive
Ahmed H. Elsheikh (HWU, Edinburgh, UK) 9–11th of April 2014 4 / 23
Ahmed Elsheikh (HWU, UK) Data-Driven MsFV May 2017 4 / 37
Monte-Carlo approach for uncertainty propagationBackground: Uncertainty Quantification
Modelsettings
Ahmed Elsheikh (HWU, UK) Data-Driven MsFV May 2017 5 / 37
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Forward model – Two-phase porous media flow
Combine mass conservation and Darcy’s law
−∇ · (Kλt(Sw )∇p) = q → Pressure Equation
Water saturation equation only: ( So + Sw = 1)
φ∂Sw∂t
+∇ · (f (Sw ) vt) =Qw
ρw→ Saturation Equation
λw (Sw ) =(Snw )2
µw, λo(Sw ) =
(1− Snw )2
µo, Snw =
Sw − Swc
1− Sor − Swc
Swc , Sor is the irreducible saturationsµw , µo are the fluid viscosities, ρw , ρo are the fluid densitiesf (Sw ) = λw/λt is the fractional flow functionsK is the permeability tensor,φ is the porosityp = po = pw is the pressureq = Qo/ρo + Qw/ρw is the normalized source or sink term
λt(Sw ) = λw (Sw ) + λo(Sw ) is the total mobility
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How to efficiently solveUQ, IUQ and robust optimization problems?
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Learn an efficient emulator I
Regression models, commonly you start by dimension reduction thenbuild a regressor (non-intrusive PC, GPR, NN, RF)
I Elsheikh, Ahmed H; Hoteit, I; Wheeler, Mary F; Efficient Bayesianinference of subsurface flow models using nested sampling and sparsepolynomial chaos surrogates, CMAME 2014.
Learn a map from low fidelity models (fast to run) to high fidelitymodels (slow to run)
I Josset, Laureline; Demyanov, Vasily; Elsheikh, Ahmed H; Lunati,Ivan; Accelerating Monte Carlo Markov chains with proxy and errormodels, Computers and Geosciences 2015.
I Kopke, Corrina; Irving, James; Elsheikh, Ahmed H; Accounting formodel error in Bayesian solutions to hydrogeophysical inverse problemsusing a local basis approach, ADWR 2018.
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Learn an efficient emulator II
Learn to upscale, e.g. exploit locality properties of some multi-scalemethods
I Chan, Shing; Elsheikh, Ahmed H; A machine learning approach forefficient uncertainty quantification using multiscale methods, JCP 2018.
Learn reduced order models, i.e. simplified dynamical system usingglobal basis functions (e.g. POD, DEIM, etc.)
I Kani, Nagoor J; Elsheikh, Ahmed H; Reduced-order modeling ofsubsurface multi-phase flow models using deep residual recurrent neuralnetworks, TIPM 2019.
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Learn a compact representationof stochastic fields
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Parameterization using PCA
Given a set of realizations y1, y2, · · · , yN , (yi ∈ RM),let Y = [y1; y2; · · · ; yN ] and C = 1
N YYT (the covariance matrix).
PCA parametrization is:
y = UΛ1/2ξ
= ξ1
√λ1u1 + · · ·+ ξM
√λMuM
where:
U = [u1; · · · ; uM ] matrix of eigenvectors of C
Λ = diag(λ1, · · · , λM) diagonal matrix of eigenvalues of C
ξ = (ξ1, · · · , ξM) is a noise vector
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Parameterization using PCA
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First 8 eigen modes of the search space
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Parametrization – General form
Let y ∈ RM be the random vector representing an unknown, and z ∈ Rm,z ∼ pz a noise vector with a known distribution pz (e.g. uniform, normal,etc.). We want:
A functional relationship y = G (z)
m� M
G fast evaluation.
G differentiable wrt. z
Standard Parametrization approaches:
G (z) := Az + b (e.g. PCA)
G (z) := Φ−1(Az + b) (e.g. kPCA)
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PCA is so great: Do we need another parametrizationtechniques?
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MultiPoint Geostatistics representation
(a) Conceptual images
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Geological model parametrization
Well data,analogs,TIs, etc.
MPS · · ·
· · ·1 2 N
Gbuild
G (z)z · · ·
z ∼ pz
low dimension high dimension
“synthetic” samples
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Generation using Convolutional Neural Networks (CNN)
G (z) := fn(fn−1(· · · (f1(z)))), where fl(x) = σl(Wlx + bl)
Figure from “Unsupervised representation learning with deep convolutional generativeadversarial networks”, Radford et. al., 2016
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Likelihood as a classification problem (another CNN)
D(y) := fn(fn−1(· · · (f1(y)))), where fl(x) = σl(Uly + cl)
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NN concepts: Convolutional Neural Networks
fl(x) = σl(Wlx + bl)u1
u2
u3
u4
v1
v2
v3
w11
w21
w34
W =
w11 w12 w13 w14
w21 w22 w23 w24
w31 w32 w33 w34
(a) A fully connected layer.
u1
u2
u3
u4
v1
v2
v3
w1
w2
w1
w2
w1
w2
W =
w1 w2 0 00 w1 w2 00 0 w1 w2
(b) A convolutional layer.
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NN concepts: Loss function for classification
In binary classification, where the number of classes equals 2, themean square error could be simply defined as:
1
n
n∑1
(yi − pi )2
I y is a binary indicator (0 or 1) depending on the class label cI p is the predicted probability that an observation o is of class c
Another smoother loss function is the cross-entropy loss:
−n∑1
(yi log(pi ))
I log - the natural log
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NN concepts: Cross-entropy loss function
In binary classification, where the number of classes equals 2, thecross-entropy loss function can be calculated as:
−(∑
c=1
y log(p) +∑c=0
(1− y) ∗ log(1− p)
)where:
I log - the natural logI y is a binary indicator (0 or 1) depending on the class label cI p is the predicted probability that an observation o is of class c
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Generative adversarial networks (GAN)
A non-cooperative game between two players, the generator G and thediscriminator D
D(y)
G (z)z
Training Dataset
Real or Fake
feedback
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Solving the minmax game
Let Dψ : Y → [0, 1] be the discriminator network parametrized by weightsψ to be determined. The training of the generator and discriminator usesthe following loss function:
L(ψ, θ) := Ey∼Py
logDψ(y) + Ey∼Pθ
log(1− Dψ(y)) (1)
where y = Gθ(z) ∼ Pθ. In effect, this loss is the classification score of thediscriminator, therefore we train Dψ to maximize L, and Gθ to minimize L:
minθ
maxψL(ψ, θ) (2)
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Solving the minmax game
We know how to solve a min problem and a max problem by alternatingbetween:Step A:
minθ
maxψ{ E
y∼Py
logDψ(y) + Ey∼Pθ
log(1− Dψ(y))}
Step B:minθ
maxψ{ E
y∼Py
logDψ(y) + Ey∼Pθ
log(1− Dψ(y))}
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Wasserstein GAN
Optimization of the GAN minmax game is unstable (objectivefunction is not monotonically decreasing)
Wasserstein formulation of GAN (WGAN) tries to optimize a differentloss function
L(ψ, θ) := Ey∼Py
Dψ(y)− Ey∼Pθ
Dψ(y) (3)
and a constraint in the search space of Dψ,
Training goal is to solve the following minmax problem:
minθ
maxψ:Dψ∈D
L(ψ, θ) (4)
where now Dψ : Y → R and D is the set of 1-Lipschitz functions(loosely enforced by constraining the weights ψ to a compact space,e.g. by clipping the values of the weights in an interval [−c , c]).
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Dataset
(a) Semi-straight channels
(b) Meandering channels
(c) Conceptual images
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Generated realizations: visual comparison
(a) Original realizations
(b) Realizations generated using GAN
(c) Realizations generated using PCA
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Generated realizations: visual comparison
(a) Original realizations
(b) Realizations generated using GAN
(c) Realizations generated using PCAAhmed Elsheikh (HWU, UK) parametrization using GANs 20 March, 2019 29 / 46
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Generated realizations: permeability histogram
0.0 0.5 1.00
5
10
15
20
25
freq
uenc
y
Data
0.0 0.5 1.00
5
10
15
20
25GAN
−2 −1 0 1 20.0
0.5
1.0
1.5PCA
log-permeability
(a) Semi-straight pattern
0.0 0.5 1.00
5
10
15
20
25
freq
uenc
y
Data
0.0 0.5 1.00
5
10
15
20
25GAN
−2 −1 0 1 20.0
0.5
1.0
1.5PCA
log-permeability
(b) Meandering pattern
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Practical advantages of WGAN – Stability
0 10000 20000 30000
iteration
0.00
0.02
0.04
0.06
0.08W
GAN
loss
1
2
34
5 6 7
12
34
56
7
0 10000 20000 30000
iteration
0.00
0.05
0.10
0.15
0.20
0.25
GAN
loss
1 2
3
45
6
7
12
34
56
7
Convergence curves of a WGAN model (top) and a standard GAN model (bottom). On the right, we show samples along thetraining of the corresponding models. We see that GAN loss is uninformative regarding sample quality.
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Forward uncertainty propagation study
Water injection in oil-filled reservoir
Quarter-five spot problem
Using 5000 permeability realizations, Estimate:
saturation statistics
water breakthrough times
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Saturation statistics ,mean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.08 0.16 −70 0 70 −10 2420 4850
(a) Statistics based on original realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.08 0.16 −70 0 70 −10 2420 4850
(b) Statistics based on GAN realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.08 0.16 −70 0 70 −10 2420 4850
(c) Statistics based on PCA realizations
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Saturation statistics ,mean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000
(a) Statistics based on original realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000
(b) Statistics based on GAN realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000
(c) Statistics based on PCA realizations
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Saturation statistics ,mean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500
(a) Statistics based on original realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500
(b) Statistics based on GAN realizationsmean variance skewness kurtosis
0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500
(c) Statistics based on PCA realizations
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Saturation histogram
0.0 0.5 1.00
1
2
3
4fr
eque
ncy
Data
0.0 0.5 1.0
GAN
0.0 0.5 1.0
PCA
saturation
(a) Semi-straight pattern
0.0 0.5 1.00.0
0.5
1.0
1.5
2.0
2.5
freq
uenc
y
Data
0.0 0.5 1.0
GAN
0.0 0.5 1.0
PCA
saturation
(b) Meandering pattern
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Water breakthrough times ,
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
PVI
0
2
4
6
8
10
12
dens
ity
DataGANPCA
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Water breakthrough times ,
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
PVI
0
2
4
6
8
10
12
14
dens
ity
DataGANPCA
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Outline
1 Introduction and background
2 The unreasonable effectiveness of deep neural networks
3 Generative adversarial networks
4 Numerical evaluation for subsurface flow problems
5 Conclusions and outlook
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Conclusions
Positive findings
WGANs generate visually plausible realizations
Generated realizations preserve the flow statistics
WGANs is far more stable than GANs
Introduced a powerful point based conditioning by learning aninference network (arXiv:1807.05207v1)
Warnings !!
Finding the equilibrium of the min-max game is challenging – (seearXiv preprint arXiv:1809.07748)
Fast evolving field and new methods are proposed everyday(GLO,VAE-GAN, etc.)
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Thank you
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Main References
Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, DavidWarde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.Generative adversarial nets. In Advances in neural information processingsystems, pages 26722680, 2014.
Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein GAN.arXiv preprint arXiv:1701.07875, 2017.
Shing Chan and Ahmed H Elsheikh. Parametrization and generation ofgeological models with generative adversarial networks. arXiv preprintarXiv:1708.01810, 2017.
Shing Chan and Ahmed H Elsheikh. Parametric generation of conditionalgeological realizations using generative neural networks. arXiv preprintarXiv:1807.05207v1, 2018.
Shing Chan and Ahmed H Elsheikh. Exemplar-based synthesis of geologyusing kernel discrepancies and generative neural networks. arXivpreprint arXiv:1809.07748, 2018.
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Main Reference
Lukas Mosser, Olivier Dubrule, and Martin J Blunt. Reconstruction ofthree-dimensional porous media using generative adversarial neuralnetworks. arXiv preprint arXiv:1704.03225, 2017.
Lukas Mosser, Olivier Dubrule, and Martin J Blunt. Conditioning ofthree-dimensional generative adversarial networks for pore andreservoir-scale models. arXiv preprint arXiv:1802.05622, 2018.
Eric Laloy, Romain Herault, Diederik Jacques, and Niklas Linde. Efficienttraining-image based geostatistical simulation and inversion using aspatial generative adversarial neural network. arXiv preprintarXiv:1708.04975, 2017.
Emilien Dupont, Tuanfeng Zhang, Peter Tilke, Lin Liang, and WilliamBailey. Generating realistic geology conditioned on physicalmeasurements with generative adversarial networks. arXiv preprintarXiv:1802.03065, 2018.
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