© Simon Trebst

Quantitative Modeling of Complex Systems

© Simon Trebst

Machine learning

Many computer programs/apps have machine learning algorithms built-in already.

In computer science, machine learning is concerned with algorithms that allow for data analytics, most prominently dimensional reduction and feature extraction.

spam filters face recognition voice recognition

© Simon Trebst

Machine learning

digital assistants self-driving cars

Applications of machine learning techniques are booming and poised to enter our daily lives.

© Simon Trebst

Machines at play

1996: G. Kasparow vs. IBM’s deep blue 2016: L. Sedol vs. Google’s AlphaGo

Machine learning techniques can make computers play.

A computer at play is probably one of the most striking realization of artificial intelligence.

© Simon Trebst

How do machines learn?

How do machines learn?

What is it that they can learn?Can we control what they learn?

How can we benefit from machine learning?

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x2

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biological neural network artificial neural network

© Simon Trebst

Machine�Learning�Day�2017

Organizer

Prof.�Dr.�Simon�Trebst,�Theoretical�PhysicsProf.�Dr.�Achim�Tresch,�Bioinformatics�and�Computational�BiologyDr.�Michael�von�Papen,�Competence�Area�3:�Quantitative�Modeling�of�Complex�Systems

Time Title Speaker Institute

09:00 Intro Simon�Trebst

09:15 Quantum�Machine�Learning Simon�Trebst Theoretical�Physics

09:45 kernDeepStackNet:��An�R�package�for�tuning�kernel�deep�stacking�networks

Thomas�Welchowski��and�Matthias�Schmid

IMBIE,�Bonn

10:15 coffee�break

10:45 Applying�Machine�Learning�to��Text�Mining

Jürgen�Hermes Department�of�Linguistics

11:15 Machine�learning�and�the��social�sciences

Jörn�Grahl Digital�Transformation�and�Value�Creation

11:45 lunch�break

13:15 Compressed�Sensing�for�Sparse�and�Low-Rank�Models

David�Gross Theoretical�Physics

13:45 Sparse�PCA�and�convex��optimization

Frank�Vallentin Mathematical�Institute

14:15 coffee�break

14:45 Reconstruction�of�Biological�Networks

Achim�Tresch Bioinformatics�and��Computational�Biology

15:15 Illuminating�Genetic�Networks��with�Random�Forest

Andreas�Beyer CECAD

15:45 End

Machine learning quantum phases of matter

Simon TrebstInstitute for Theoretical Physics

Machine Learning DayCologne, April 2017

25.-27.�April�2017University�of�CologneInstitute�for�Theoretical�physics

Quantum�Machine�Learning�

tate-of-the-art� machine� learning�techniques� have� recently� attracted�

avid� interest� from� the� statistical�physics�community�for�their�capacity�to�distinguish�different�phases�of�quantum�matter� in� an� automated� way.� This�two-day� mini-workshop� will� provide�

informal� setting� for� discussions�amongst�numerical�practitioners�who�

�recently�started�to�apply�machine�learning� techniques� to� quantum�statistical�physics�problems.�Besides�short�talks�there�will�be�ample�time�for�interactions�amongst�the�attendees.

Peter�Bröcker��CologneGuiseppe�Carleo��ZurichMaciej�Koch-Janusz�ZurichRoger�Melko��WaterlooTitus�Neupert��ZurichEvert�van�NieuwenburgCaltechYi�(Frank)�Zhang��Cornell

Speakers

© Simon Trebst

artificial neural networks

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Artificial neural networks mimic biological neural networks (albeit at a much smaller scale).

They allow for an implicit knowledge representation,which is infused in supervised or unsupervised learning settings.

© Simon Trebst

artificial neural networksartificial neurons

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x3

outputbw1

w2

w3

example – Should I skip the first talk (or sleep in)?

quantum stuff?

free coffee?

9am – really? +3

2-2

+2

sleep in

(binary)input ⇥ (~w · ~x� b)

0z

Θ(z)

0

1

© Simon Trebst

artificial neural networksArtificial neural networks are pretty powerful.

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3

-2

-2

NAND-gate

x1 x2

0 00 11 01 1

output

1110

Like circuits of NAND gates artificial neural networks can encode arbitrarily complex logic functions, thus allowing for universal computation.

But the power of neural networks really comes about by varying the weights such that one obtains some desired functionality.

© Simon Trebst

neural network architectures

input layer output layerhidden layers

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feedforward network

Neural networks with multiple hidden layershave been popularized as “deep learning” networks.

© Simon Trebst

How to train a neural network?

• quadratic cost function

desiredoutput

actualoutput

C(~w,~b) =1

2n

X

x

||y(x)� a(x)||2x1

x2

x3

x4

x5

perceptrons

0z

Θ(z)

01

sigmoid neurons

0z

Θ(z)1

0

Small adjustments on the level of a single neuron should result in small changes of the cost function.

© Simon Trebst

How to train a neural network?

gradient descent

• quadratic cost function

desiredoutput

actualoutput

C(~w,~b) =1

2n

X

x

||y(x)� a(x)||2

• back propagation algorithmRumelhart, Hinton & Williams, Nature (1986)

@C

@w

@C

@b

extremely efficient way to calculate all partial derivatives

needed for a gradient descent optimization.

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© Simon Trebst

pattern recognition

93106x1

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Some 60 lines of code (Python/Julia) will do this for you with >95% accuracy.

conv pool conv pool full dropout full

conventionalmatrix reductions

Much higher accuracy possible for networks with additional convolutional layers.

© Simon Trebst

convolutional neural networksConvolutional neural networks look for recurring patterns using small filters.

© Simon Trebst

convolutional neural networksConvolutional neural networks look for recurring patterns using small filters.

© Simon Trebst

convolutional neural networksConvolutional neural networks look for recurring patterns using small filters.

filter

activation map

Slide filters across image and create new image based on how well they fit.

element-wisematrix product

© Simon Trebst

GPUs & open-source codes

© Simon Trebst

Machine learningquantum phases of matter

© Simon Trebst

Quantum matter

water ice

superconductor

Bose-Einstein condensate

© Simon Trebst

Model systems

interacting many-body system

pair-wise interactionsfavor alignment of “spins”

Ising model

© Simon Trebst

Model systems

interacting many-body system

pair-wise interactionsfavor alignment of “spins”

Ising model

© Simon Trebst

Model systems

interacting many-body system

pair-wise interactionsfavor alignment of “spins”

Ising model

paradigmatic modelfor a phase transition

high temperature low temperature“critical” temperature

© Simon Trebst

Supervised learning approach

Supervised learning approach 1) train convolutional neural network on representative “images” deep within the two phases2) apply trained network to “images” sampled elsewhere to predict phases + transition

What are the right images to feed into the neural network?

phase A phase Bphase

transition

trainhere

trainhere

predict phases by applying neural network here

step 1

step 2�

General setupConsider some system, which as a function of some parameter λ exhibits a phase transition between two phases.

© Simon Trebst

classical phases of matterMachine learning of thermal phase transitionin the classical Ising model 6

2.0 2.5 3.0 3.5 4.0 4.5 5.0T

0.0

0.2

0.4

0.6

0.8

1.0

Out

putl

ayer

L = 30

T < Tc

T > Tc

FIG. 2. Detecting the critical temperature of the triangular Ising model through the crossing of the

values of the output layer vs T . The orange line signals the triangular Ising model Tc

/J = 4/ ln 3,

while the blue dashed line represents our estimate T

c

/J = 3.63581.

shown in Figure 3. In a conventional condensed-matter approach, the ground states and the

high-temperature states are distinguished by their spin-spin correlation functions: power-

law decay in the Coulomb phase at T = 0, and exponential decay at high temperature.

Instead we use supervised learning, feeding raw Monte Carlo configurations to train a fully-

connected neural network (Figure 1(A)) to distinguish ground states from high-temperature

states. Figure 3(A) and Figure 3(B) display high- and low-temperature snapshots of the

configurations used in the training of the model. For a square ice system with N = 2 ⇥

16⇥ 16 spins, we find that a standard fully-connected neural network with 100 hidden units

successfully distinguishes the states with a 99% accuracy. The network does so solely based

on spin configurations, with no information about the underlying lattice – a feat di�cult for

the human eye, even if supplemented with a layout of the underlying Hamiltonian locality.

These results indicate that the learning capabilities of neural networks go beyond the

simple ability to encode order parameters, extending to the detection of subtle di↵erences

in higher-order correlations functions. As a final demonstration of this, we examine an Ising

Pirs

a: 1

6080

016

Page

10/

33

Pirs

a: 1

6080

016

Page

10/

33

Nature Physics (2017)

© Simon Trebst

quantum phases of matterHubbard model on the honeycomb lattice

semi-metal

SDW

© Simon Trebst

quantum phase transitionGreen’s functions are ideal objects/images for machine learning based discrimination of quantum phases.

1 2 3 4 5 6 7 8

interaction U

0.0 0.0

0.25 0.25

0.5 0.5

0.75 0.75

1.0 1.0

pred

ictio

n

L = 15L = 12L = 9L = 6

U = 1.0 U = 16.0

SDWDirac

3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2

interaction U

0.0 0.0

0.25 0.25

0.5 0.5

0.75 0.75

1.0 1.0

pred

ictio

n

3.8 4.2 4.6 4.8

0.5

0.0

1.0 L = 15L = 9

© Simon Trebst

unsupervised learning

© Simon Trebst

Unsupervised learningEmploy ability to “blindly” distinguish phases to map out an entirephase diagram with no hitherto knowledge about the phases.

Example: hardcore bosons / XXZ model on a square lattice

H = �X

hi,ji

�S+i S�

j + S�i S+

j

�+�

X

hi,ji

Szi S

zj + h

X

i

Szi

VOLUME 88, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 2002

Finite-Temperature Phase Diagram of Hard-Core Bosons in Two Dimensions

Guido Schmid,1 Synge Todo,1,2 Matthias Troyer,1 and Ansgar Dorneich3

1Theoretische Physik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan

3Institut für Theoretische Physik, Universität Würzburg, 97074 Würzburg, Germany(Received 4 October 2001; published 9 April 2002)

We determine the finite-temperature phase diagram of the square lattice hard-core boson Hubbardmodel with nearest neighbor repulsion using quantum Monte Carlo simulations. This model is equivalentto an anisotropic spin-1!2 XXZ model in a magnetic field. We present the rich phase diagram with afirst order transition between a solid and superfluid phase, instead of a previously conjectured supersolidand a tricritical end point to phase separation. Unusual reentrant behavior with ordering upon increasingthe temperature is found, similar to the Pomeranchuk effect in 3He.

DOI: 10.1103/PhysRevLett.88.167208 PACS numbers: 75.10.Jm, 05.30.Jp, 67.40.Kh, 74.25.Dw

A nearly universal feature of strongly correlated systemsis a phase transition between a correlation-induced insulat-ing phase, with localized charge carriers, and an itinerantphase. High-temperature superconductors [1], manganites[2], and the controversial two-dimensional (2D) “metal-insulator transition” [3] are just a few examples of thisphenomenon. The 2D hard-core boson Hubbard modelprovides the simplest example of such a transition froma correlation-induced charge density wave insulator nearhalf filling to a superfluid (SF). It is a prototypical modelfor preformed Cooper pairs [4], spin flops in anisotropicquantum magnets [5,6], SF helium films [7], and super-solids [8,9].

In simulations of this model, which does not suffer fromthe negative sign problem of fermionic simulations, wecan investigate some of the pertinent questions about suchphase transitions: what is the order of the quantum phasetransitions in the ground state and the finite-temperaturephase transitions? Are there special points with dynami-cally enhanced symmetry [10]? Can there be coexistenceof two types of order (such as a supersolid–coexistingsolid and superfluid order)? Answers to these questionsalso provide insight into the other problems alluded toabove.

The Hamiltonian of the hard-core boson Hubbard modelwe study is

H ! 2tX

"i,j#$ay

i aj 1 ayj ai% 1 V

X

"i,j#ninj 2 m

X

ini ,

(1)

where ayi $ai% is the creation (annihilation) operator for

hard-core bosons, ni ! ayi ai is the number operator,

V is the nearest neighbor Coulomb repulsion, and m isthe chemical potential. This model is equivalent to ananisotropic spin-1!2 XXZ model with Jz ! V and jJxyj !2t in a magnetic field h ! 2V 2 m. The zero field (andzero magnetization mz ! 0) case of the spin modelcorresponds to the half filled bosonic model (densityr ! "mz# ! 1!2) at m ! 2V . Throughout this Letterwe will use the bosonic language, and refer to the corre-

sponding quantities in the spin model where appropriate.Because of the absence of efficient Monte Carlo algo-rithms for classical magnets in a magnetic field there arestill many open questions even in the classical version ofthis model, which was only studied by a local updatedmethod [11].

In Fig. 1 we show the ground-state phase diagram[6,9,12]. For dominating chemical potential m the systemis in a band insulating state (r ! 0 and r ! 1, respec-tively), while it shows staggered checkerboard chargeorder (r ! 1!2) for dominating repulsion V . These solidphases are separated from each other by a SF. Earlierindications for a region of supersolid phase between thecheckerboard solid and SF phase turned out to be due tophase separation at this transition which is of first orderat T ! 0 [6,9,12].

All of these phases extend to finite temperatures. Onthe strong repulsion side the hard-core boson Hubbardmodel is equivalent to an antiferromagnetic Ising model att ! 0, and the insulating behavior extends up to a finite-temperature phase transition of the Ising universality class

−12 −8 −4 0 4 8 12

(µ−2V)/t

0

2

4

6

V/t

empty

solid

superfluid

Heisenberg pointfull

checkerboard

ρ=1/2

ρ=0 ρ=1

FIG. 1. Ground-state phase diagram of the hard-core bosonHubbard model. The dashed line indicates the cut along whichwe calculated the finite-temperature phase diagram shown inFig. 2.

167208-1 0031-9007!02!88(16)!167208(4)$20.00 © 2002 The American Physical Society 167208-1

PRL 88, 167208 (2002)

hS+i S�

j i+ hS�i S+

j i real space hwindow in direction

learning rate 0.0001window width 0.5fully connected network, 1024 neurons

hS+i S�

j i+ hS�i S+

j i real space hwindow in direction

learning rate 0.0001window width 0.5fully connected network, 1024 neurons

© Simon Trebst

Unsupervised learningEmploy ability to “blindly” distinguish phases to map out an entirephase diagram with no hitherto knowledge about the phases.

Example: hardcore bosons / XXZ model on a square lattice

H = �X

hi,ji

�S+i S�

j + S�i S+

j

�+�

X

hi,ji

Szi S

zj + h

X

i

Szi

VOLUME 88, NUMBER 16 P H Y S I C A L R E V I E W L E T T E R S 22 APRIL 2002

Finite-Temperature Phase Diagram of Hard-Core Bosons in Two Dimensions

Guido Schmid,1 Synge Todo,1,2 Matthias Troyer,1 and Ansgar Dorneich3

1Theoretische Physik, Eidgenössische Technische Hochschule Zürich, CH-8093 Zürich, Switzerland2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan

3Institut für Theoretische Physik, Universität Würzburg, 97074 Würzburg, Germany(Received 4 October 2001; published 9 April 2002)

We determine the finite-temperature phase diagram of the square lattice hard-core boson Hubbardmodel with nearest neighbor repulsion using quantum Monte Carlo simulations. This model is equivalentto an anisotropic spin-1!2 XXZ model in a magnetic field. We present the rich phase diagram with afirst order transition between a solid and superfluid phase, instead of a previously conjectured supersolidand a tricritical end point to phase separation. Unusual reentrant behavior with ordering upon increasingthe temperature is found, similar to the Pomeranchuk effect in 3He.

DOI: 10.1103/PhysRevLett.88.167208 PACS numbers: 75.10.Jm, 05.30.Jp, 67.40.Kh, 74.25.Dw

A nearly universal feature of strongly correlated systemsis a phase transition between a correlation-induced insulat-ing phase, with localized charge carriers, and an itinerantphase. High-temperature superconductors [1], manganites[2], and the controversial two-dimensional (2D) “metal-insulator transition” [3] are just a few examples of thisphenomenon. The 2D hard-core boson Hubbard modelprovides the simplest example of such a transition froma correlation-induced charge density wave insulator nearhalf filling to a superfluid (SF). It is a prototypical modelfor preformed Cooper pairs [4], spin flops in anisotropicquantum magnets [5,6], SF helium films [7], and super-solids [8,9].

In simulations of this model, which does not suffer fromthe negative sign problem of fermionic simulations, wecan investigate some of the pertinent questions about suchphase transitions: what is the order of the quantum phasetransitions in the ground state and the finite-temperaturephase transitions? Are there special points with dynami-cally enhanced symmetry [10]? Can there be coexistenceof two types of order (such as a supersolid–coexistingsolid and superfluid order)? Answers to these questionsalso provide insight into the other problems alluded toabove.

The Hamiltonian of the hard-core boson Hubbard modelwe study is

H ! 2tX

"i,j#$ay

i aj 1 ayj ai% 1 V

X

"i,j#ninj 2 m

X

ini ,

(1)

where ayi $ai% is the creation (annihilation) operator for

hard-core bosons, ni ! ayi ai is the number operator,

V is the nearest neighbor Coulomb repulsion, and m isthe chemical potential. This model is equivalent to ananisotropic spin-1!2 XXZ model with Jz ! V and jJxyj !2t in a magnetic field h ! 2V 2 m. The zero field (andzero magnetization mz ! 0) case of the spin modelcorresponds to the half filled bosonic model (densityr ! "mz# ! 1!2) at m ! 2V . Throughout this Letterwe will use the bosonic language, and refer to the corre-

sponding quantities in the spin model where appropriate.Because of the absence of efficient Monte Carlo algo-rithms for classical magnets in a magnetic field there arestill many open questions even in the classical version ofthis model, which was only studied by a local updatedmethod [11].

In Fig. 1 we show the ground-state phase diagram[6,9,12]. For dominating chemical potential m the systemis in a band insulating state (r ! 0 and r ! 1, respec-tively), while it shows staggered checkerboard chargeorder (r ! 1!2) for dominating repulsion V . These solidphases are separated from each other by a SF. Earlierindications for a region of supersolid phase between thecheckerboard solid and SF phase turned out to be due tophase separation at this transition which is of first orderat T ! 0 [6,9,12].

All of these phases extend to finite temperatures. Onthe strong repulsion side the hard-core boson Hubbardmodel is equivalent to an antiferromagnetic Ising model att ! 0, and the insulating behavior extends up to a finite-temperature phase transition of the Ising universality class

−12 −8 −4 0 4 8 12

(µ−2V)/t

0

2

4

6

V/t

empty

solid

superfluid

Heisenberg pointfull

checkerboard

ρ=1/2

ρ=0 ρ=1

FIG. 1. Ground-state phase diagram of the hard-core bosonHubbard model. The dashed line indicates the cut along whichwe calculated the finite-temperature phase diagram shown inFig. 2.

167208-1 0031-9007!02!88(16)!167208(4)$20.00 © 2002 The American Physical Society 167208-1

PRL 88, 167208 (2002)

real space hwindow in directionhSzi S

zj i

learning rate 0.0001window width 0.5fully connected network, 1024 neurons

real space hwindow in directionhSzi S

zj i

learning rate 0.0001window width 0.5fully connected network, 1024 neurons

© Simon Trebst

summary

© Simon Trebst

Summary

Monte Carlo + machine learning approach can be used to distinguish phases of interacting many-body systems in(quantum) statistical physics.

The approach can be adapted to an unsupervised setting for quickly and semi-automatically mapping out phase diagrams of (quantum) many-body systems.

Can we identify even some of the more subtle quantum mechanical properties such as long-range entanglementand (non-local) topological order?

Thanks!