multi-scale microstructure-property modeling of elastic...

Multi-Scale Microstructure-Property Modeling of ElasticLocalization Relationships in High Contrast Composites

MURI Team Meeting III

Ruoqian (Rosanne) Liu1, Ankit Agrawal1, Alok Choudhary1

Yuksel Yabansu2, Surya Kalidindi2

1Electrical Engineering and Computer ScienceNorthwestern University

2Materials Science and EngineeringGeorgia Institute of Technology

August 18, 2015

Rosanne Liu (Northwestern) Mining of P-S-P August 18, 2015 1 / 37

A Major Struggling of the Title

A Data Science Modeling

a Multi-Scale Behavior a Typical Material


Overview

1 What Happened in the PastThe Problem of Localization ModelingThe Data-Driven Modeling of It

2 The Same Problem with Better ResultsA Two-Scale Modeling SchemeThe Use of 2-Pt Statistics

3 The Expectation of Even Better OutcomesThe What’s Called “Deep Learning”Convolutional Learning of Images


What Happened in the Past

Outline





What Happened in the Past The Problem of Localization Modeling

The Big Picture

Keywords

material informatics; data science; predictive modeling.

Goal

combination of best known physics with data science; computationalefficiency; generalization.

Data

three dimensional (3-D); voxel based microstructure volume element(MVE); high contrast.



Physics Based vs. Data Based

Physical Based Models

Generally accomplished by solving governing field equationsnumerically (e.g., finite element models), while satisfying theappropriate material constitutive laws and the imposed boundary andinitial conditions.

Computational resource requirements are usually very high.

No systematic learning from discarded simulations and solutions.

Data Models

Distill transferable knowledge from trials, even failed ones.

Calibration is a one-time computational cost.

Knowledge obtained can be generalized to future, unseen cases.

Dramatic savings in both time and effort.



Localization

Localization: the spatial distribution of the response at the microscale foran imposed loading condition at the macroscale.

A microstructure in 3D

A process: impose a constant load of 5×10-‐4

A response of microstructure in 3D

Data Models?

Finite Element



A Closer Look at the Data

A collection (2,500) of 3-D MVEs, each of a dimension21× 21× 21, of a digitally created high contrasttwo-phase composite.

Volume fractions vary from 1.0% to 99.4%, in total100 variations.

A corresponding collection of elastic deformation fields.The response field is captured as a continuous value ineach spatial voxel.


What Happened in the Past The Data-Driven Modeling of It

Past Attempts of Data Mining

Feature Extrac+on Regression

features Input: microstructure

Output: response

Data-driven predictive modeling: single-agent

Level 1 neighbors Level 2 neighbors

Level 3 neighbors Level 4 neighbors



Results Accomplished with Single Agent Modeling

0 20 40 60 80 100

Volume Fraction (%) of each MVE

0

5

10

15

20

25

30

Indiv

idual M

ASE (

%)

Ex 3a-1: RF + 57 features, e=13.02%

Ex 3a-2: RF + 93 features, e=14.25%

Ex 3b-1: SVR + 57 features, e=16.36%

Ex 3b-2: SVR + 93 features, e=17.01%

Mean absolute strain error (MASE) e = 1N

∑N

∣∣∣ p−ppimposed

∣∣∣× 100%,

N : number of samples; p, p, pimposed: actual, predicted, imposed strain on a sample.



Results Accomplished with Single Agent Modeling

FE Ex 3a-1 Ex 3b-1

Ex 3a-1 Ex 3b-1 Rosanne Liu (Northwestern) Mining of P-S-P August 18, 2015 14 / 37

The Same Problem with Better Results

Outline





The Same Problem with Better Results A Two-Scale Modeling Scheme

Why Two-Scale?

The internal structure of a material system is hierarchical.

There are multiple length scales where information passes in between.

In our problem, the response needs to be addressed at micro-scale fora condition applied at macro-scale.

A microstructure

Macro-scale (cube-wise) Information: volume fraction, volume structure …

Micro-scale (voxel-wise) information: neighboring structure …

How to combine?



Handling Multiple Scale Lengths in Machine Learning

In machine learning, this amounts to a problem of data representation.

How to represent a voxel in a cube?

Each voxel is represented by, say, its neighboring information.

Higher level information, like volume fraction, provides voxels withdistinct learning environments, which we call contexts.

How to realize, identify, and obtain distinct contexts from a data is achallenge.

How to include the contextual information relates to the field ofrepresentation learning.



The Multi-Contextual Solution

Feature Extrac+on Regression

features Input: microstructure

Output: response

Data-driven predictive modeling: single-agent

Macro-‐Feature Extrac.on

Division/Resampling

Macro features Input: microstructure

Output: response

Data-driven predictive modeling

Local-‐feature Extrac.on

…

Regression

Regression Local-‐feature Extrac.on

Local-‐feature Extrac.on Regression

…



How to Partition Contexts

A multi-scale modeling solution:

1 Consider the “type of cube” as the context of learning.

Idea

“similar” cubes should be grouped together to create alearning context.

Question

What structural/compositional descriptors differentiatemicrostructure cubes from one another?

Answer

1 Developing lower-order geometric descriptorsNum. of clusters (connected components)Max, min, ave. size of clustersDispersion (ave. of cluster center distances)

2 Use 2-point statistics



How to Partition Contexts

A multi-scale modeling solution:1 Consider the “type of cube” as the context of learning.2 Within each context, conduct predictive modeling.

Idea

Take inputs from local neighbors to establishstatistical/mathematical, rule-based relationships.

Question

How many neighbors to include? What neighboringinformation is important? What modeling techniques to use?

Answer

1 Separate neighboring voxels into levels.2 Extract information on each level.3 Rank and select the best set of features.


The Same Problem with Better Results The Use of 2-Pt Statistics

Quick Review of 2-Point Statistics

It describes the probability of two phases separated by this particulardistance.It can be thought of as a lumpiness factor – the higher the value forsome distance scale, the more lumpy the universe is at that distancescale.It enables a quantitative understanding of the microstructure-propertyrelationship.

r



Quick Review of 2-Point Statistics

It describes the probability of two phases separated by this particulardistance.It can be thought of as a lumpiness factor – the higher the value forsome distance scale, the more lumpy the universe is at that distancescale.It enables a quantitative understanding of the microstructure-propertyrelationship.

r = sqrt(10)



2-Point Statistics in 3D MVE

A reduction from 21× 21× 21 phasevalues to 179 correlation values.

E.g. 100 functions for MVE #5, 30, 55, ..., 2480, each for one unique volume fraction.



2-Point Statistics in 3D MVE

A reduction from 21× 21× 21 phasevalues to 179 correlation values.

E.g. 25 functions for MVE #1000, 1001, ..., 1024, all with the same volume fraction.



Results

Contextual partition is conducted with three methods.

P1:Make partition of MVEs based on the volume fraction.Results in 100 groups.

A sample slice shown with VF of 50.22% (one of the hardest cubes)

FEM No PartitionAll test MASE: 17.01

Cube MASE: 25.86

Slice MASE: 27.02

P1All test MASE: 8.89

Cube MASE: 12.32

Slice MASE: 12.13



Results


P2:Make partition of MVEs based on data clustering with a set of

selected macro-features.Results in 93 groups.



Cube MASE: 25.86

Slice MASE: 27.02


Cube MASE: 12.66

Slice MASE: 12.69Rosanne Liu (Northwestern) Mining of P-S-P August 18, 2015 26 / 37


Results


P3:Make partition of MVEs based on PCA of 2-point correlation

functions.Results in 90 groups.



Cube MASE: 25.86

Slice MASE: 27.02


Cube MASE: 12.14

Slice MASE: 11.93Rosanne Liu (Northwestern) Mining of P-S-P August 18, 2015 27 / 37

The Expectation of Even Better Outcomes

Outline





The Expectation of Even Better Outcomes The What’s Called “Deep Learning”

A Revolution in Machine Learning Society

Over the last few month, the what’s called “deep learning” has producedbreakthrough results in speech, image, and natural language.

MIT Tech Reviews list of top-10 breakthroughs of 2013

improved speech recognition technology by 30%, an earthquake inthis field

caused big companies, such as Microsoft, Facebook, Google, Apple,Baidu Yahoo! and IBM to heavily invest in this technology

the perfect method to exploit the information locked away in Big Data

now fully used GPU power for a huge performance boost



Deep Learning

Learning the representation

We know the way in which data are represented can make a hugedifference in the success of a learning algorithm.

Deep learning enables the learning of multiple levels of representation,discovering more abstract features in the higher levels.

Learning as human does

Because human brains appear deep, AI-tasks require deep circuits

Because it is natural for humans to represent concepts at multiplelevels of abstractions, deep architecture makes sense.

Because human learn mostly unsupervised, only partially supervised.



Deep Learning: the Basic Recipe

Greedy Layer-Wise Learning of Representations

1 Let h0(x) = x be the lowest-level representation of the data, given bythe observed raw input x.

2 For l = 1 to LTrain an unsupervised learning model taking hl−1(x) at levell − 1 as input, and after training, producing representationshl(x) = Rl(hl−1(x)) at the next level.

Several variants from this point on.

Supervised learning with fine-tuning: most common

Unsupervised: Deep autoencoders or a Deep Boltzmann Machine



The Achievements in Image Recognition

Computer vision is where deep learning in 2006 first showed itsbreakthrough. Then, voice, natural language, drug discovery ...

Automatically learning the representation from raw pixels

Using large amounts of data

Learning very complex problems

Mimic human brain representations

Wins all competitions:

IJCNN 2011 Traffic Sign Recognition Competition

ISBI 2012 Segmentation of neuronal structures in EM stacks challenge

ICDAR 2011 Chinese handwriting recognition



Deep Learning: the Power of Handling Big Data

Amount of data

Per

form

ance

Deep learning

Most learning algorithms

Reproduced from Andrew Ng’s Invited Talk at RSS2014


The Expectation of Even Better Outcomes Convolutional Learning of Images

Convolutional Neural Networks

What makes automatic image learning possible.

A regular 3-layer neural network A ConvNet arranges its neurons in three dimensions



Convolutional Neural Networks



Design of CNN for MVE Problem

Preliminary design of CNN applied to the MVE problem:

Take a local unit structure and loop over the whole cube.

At each position feed the input microstructure into a series of neurons.

Compute the forward function, remember the error between initialpredicted response and actual FE response.

After looping aggregate the error over every voxel.

Backpropagate the error to update neuron weights.


The End

Thank You!Questions and Discussion


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