Nonlinear parametrization of geological fields using ...projet.ifpen.fr/Projet/upload/docs/application/pdf/2019...A non-cooperative game between two players, the generator G and the

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.

[email protected]

MASCOT–NUM 2019 annual conference,IFPEN Rueil–Malmaison – France,

18–20 March 2019

Ahmed Elsheikh (HWU, UK) parametrization using GANs 20 March, 2019 1 / 46

mailto:[email protected]

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


Outline







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


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


How to efficiently solveUQ, IUQ and robust optimization problems?


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.


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.


Learn a compact representationof stochastic fields


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


Parameterization using PCA

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First 8 eigen modes of the search space


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)


PCA is so great: Do we need another parametrizationtechniques?


MultiPoint Geostatistics representation

(a) Conceptual images


Geological model parametrization

Well data,analogs,TIs, etc.

MPS · · ·

· · ·1 2 N

Gbuild

G (z)z · · ·

z ∼ pz

low dimension high dimension

“synthetic” samples


Outline







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


Likelihood as a classification problem (another CNN)

D(y) := fn(fn−1(· · · (f1(y)))), where fl(x) = σl(Uly + cl)


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.


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


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


Outline







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


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)


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


log(1− Dψ(y))}

Step B:minθ

maxψ{ E

y∼Py


log(1− Dψ(y))}


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]).


Dataset

(a) Semi-straight channels

(b) Meandering channels

(c) Conceptual images


Generated realizations: visual comparison

(a) Original realizations

(b) Realizations generated using GAN

(c) Realizations generated using PCA


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

Generated realizations: permeability histogram

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log-permeability

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(b) Meandering pattern


Practical advantages of WGAN – Stability

0 10000 20000 30000

iteration

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GAN

loss

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loss

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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.


Outline







Forward uncertainty propagation study

Water injection in oil-filled reservoir

Quarter-five spot problem

Using 5000 permeability realizations, Estimate:

saturation statistics

water breakthrough times


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



0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000


0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000


0.0 0.5 1.0 0.00 0.05 0.10 −75 0 75 −10 2495 5000




0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500


0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500


0.0 0.5 1.0 0.00 0.05 0.10 −20 25 70 −10 2245 4500



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

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2.0

2.5

freq

uenc

y

Data

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GAN

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saturation

(b) Meandering pattern


Water breakthrough times ,

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

PVI

0

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dens

ity

DataGANPCA


Water breakthrough times ,

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

PVI

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dens

ity

DataGANPCA


Outline







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.)


Thank you


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.


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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Nonlinear parametrization of geological fields using ...projet.ifpen.fr/Projet/upload/docs/application/pdf/2019...A non-cooperative game between two players, the generator G and the