unsupervised identification of the phase transition …...• 1 not so interesting: no phase...

UNSUPERVISED RECOGNITION OFPHASE TRANSITION ON THE LATTICE

ANDREAS ATHENODOROU

MARIE SKLODOWSKA CURIE INDIVIDUAL POSTDOCTORAL FELLOW

INFN, UNIVERSITY OF PISA & THE CYPRUS INSTITUTE

8TH INTERNATIONAL CONFERENCE ON

NEW FRONTIERS IN PHYSICS (ICNFP 2019),

KOLYMBARI, 27TH OF AUGUST 2019

BASED ON ARXIV:1903.03506 AND UNPUBLISHED WORK

IN COLLABORATION WITH: C. ALEXANDROU, A. APSEROS, C. CHRYSOSTOMOU, C. HAVADJIA, S. SHIAKAS, S. PAUL, D. VADACCHINO

MACHINE LEARNING IN STATISTICAL SYSTEMS

Neural networks can be used to identify phases and phase transitions in condensed matter systems via

supervised machine learning. Readily programmable through modern software libraries, we show that

a standard feed-forward neural network can be trained to detect multiple types of order parameter

directly from raw state configurations sampled with Monte Carlo………………………………………

We show that this classification occurs within the neural network without knowledge of the Hamiltonian

or even the general locality of interactions. These results demonstrate the power of machine learning

as a basic research tool in the field of condensed matter and statistical physics.

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MACHINE LEARNING IN STATISTICAL SYSTEMS

TempoIn

tensi

ty

Like

Dislike

fastrelaxed

light

soa

ring

A

B

C

A

B

C

K-nearest neighbors

MACHINE LEARNING

Supervised learning

Unsupervised learning

Reinforcement learning

Label 1 2 3

2

(With some high probability)

Classification

Cat

Dog

x

MACHINE LEARNING

1. Input layer

2. Hidden layer

3. Hidden layer

4. Hidden layer

5. .

6. .

7. .

8. .

9. .

……..

10. Output layer

𝑥1

𝑥2

𝑥3

𝑥4

𝑥5

𝑥6

𝑥7

𝑥8

𝑦𝑗 = 𝑓

1

8

𝑤𝑖𝑗𝑥𝑖 + 𝑏𝑗

Ԧ𝑦

1

2

n

Ԧ𝑧

1

2

m

𝑧𝑘 = 𝑓

1

𝑛

𝑤𝑖𝑘 𝑥𝑖 + 𝑏𝑘

Ԧ𝑒

1

2

l

𝑧𝑟 = 𝑓

1

𝑚

𝑤𝑖𝑟 𝑥𝑖 + 𝑏𝑟

NEURAL NETWORKS

• 1𝐷 Not so interesting: No phase transition

(never magnetised)

• 2𝐷 more interesting: There is a phase

transition

• Simplest Description of Ferromagnetism

• Hamiltonian:

• 2𝐷 Ising model is the simplest non trivial

statistical system one can investigate

• It is in the same Universality Class with the 𝑍2 Gauge Theory as well as the 𝑆𝑈(𝑁) for 𝑁 = 2,3,4.

• It is exactly solvable for zero external magnetic field ℎ (Lars Onsager 1944)

• Critical Temperature:

Nearest neighbors

THE ISING MODEL


(never magnetised)


transition


• Hamiltonian:

Nearest neighbors

Observables:

• Magnetization is the order parameter:

𝑚 =1

𝑁σ𝑖=1𝑁 𝑠𝑖

The 2D Ising model has a second order phase

transition (magnetization is continuous)

• Magnetic susceptibility

𝜒 =𝑁

𝑇𝑚2 − 𝑚 2

• Heat Capacity

𝐶 =𝜕 𝐸

𝜕𝑇

THE ISING MODEL


(never magnetised)


transition


• Hamiltonian:

Nearest neighbors

Observables (near criticality ~𝑇𝑐):

• Magnetization is the order parameter:

𝑚 =1

𝑁σ𝑖=1𝑁 𝑠𝑖 𝑚(𝑇)~ 𝑇 − 𝑇𝑐

𝑏

The 2D Ising model has a second order phase

transition (magnetization is continuous)

• Magnetic susceptibility

𝜒 =𝑁

𝑇𝑚2 − 𝑚 2 χ(𝑇)~ 𝑇 − 𝑇𝑐

−𝛾

• Heat Capacity

𝐶 =𝜕 𝐸

𝜕𝑇χ(𝑇)~ 𝑇 − 𝑇𝑐

−𝛼

THE ISING MODEL

𝐿

• The autoencoder aims to define a representation (encoding) for our assemblage of data, by

performing dimensionality reduction:

• It encodes the input data ({𝑋}) from the input layer into a latent dimension ({𝑧})• Then it uncompresses that latent dimension into an approximation of the original data ({X})

• It consists of two components, the encoder function 𝑔𝜑 and a decoder function 𝑓𝜃 and the

reconstructed input is 𝑋 = 𝑓𝜃(𝑔𝜑(𝑥))

• The autoencoder learns the parameters 𝜑 and 𝜃 together

• 𝑋 = 𝑓𝜃(𝑔𝜑(𝑥)) can approximate an identity function.

• Criterion: Minimize the Mean Square Error (MSE):

THE ISING MODEL IN THE AUTOENCODER

• Eight layers

• Fully connected (Dense)

• Single latent dimension

• Through experimentation

Dropout, reduces overfitting G. E. Hinton et al, arXiv:1207.0580.


• Eight layers

• Fully connected (Dense)

• Single latent dimension

• Through experimentation

Implementation:

• Using Keras (F. Chollet et al., “Keras.” https://keras.io, 2015)

• Using Tensorflow (M. Abadi et al OSDI, vol. 16, pp. 265–283, 2016.)

• 66.66.% of data used for training

• 33.33.% of data used for validating

• Training is performed for 2000 iterations

• Example:


• Each configuration is re-expressed in the form of a vector:

(-1, 1, -1,1,1, -1,1,-1,1,-1,1,1,-1,1,-1,1,1,1,1,-1)

Input into the encoder

𝐿

𝐿𝐿2

Latent Dimension

• In other words, each configuration is

assigned a number, the latent

dimension, which includes all the

physically necessary information so

that the decoder re-creates the

actual configuration

PROCEDURE

• Each configuration is re-expressed in the form of a vector:

(-1, 1, -1,1,1, -1,1,-1,1,-1,1,1,-1,1,-1,1,1,1,1,-1)

Input into the encoder

𝐿

𝐿𝐿2

• In other words, each configuration is assigned a number, the latent dimension, which includes all the

physically necessary information so that the decoder re-creates the actual configuration

• Τhe autoencoder receives information for only one lattice volume, and thus, it "knows" nothing about

configurations produced for other volume sizes

PROCEDURE

Attempt to identify signals of the phase structure of the 2D-Ising model

Question: How the latent dimension behaves as a function of the temperature T for each configuration.

• We produce in total 40000 configurations

• 200 configurations for every single temperature 𝑇.

• Temperature range 𝑇 = 1 − 4.5

• We make sure that we cover the whole range of temperatures between the two extreme cases

of the Ising behaviour:

• the nearly "frozen" at 𝑇 = 1• the disordered at 𝑇 = 4.5• We assume that we have no prior knowledge on what is happening in-between

• Wait! Isn’ t there a degree of supervision?

• We could have chosen different temperature ranges that cover all possible phase regions

For instance, we could choose instead 𝑇 = 0.01 − 1000 (computationally more expensive)

PROCEDURE

THE LATENT DIMENSION PER CONfiGURATION

We plot the latent dimension for each different configuration, as a function of the temperature 𝑇, for

four different lattice sizes, 𝐿 = 25, 35, 50, 150.



For low temperatures we obtain two plateaus, one located at 𝑧 = 1 and one at 𝑧 = −1.

• Corresponds to two distinct states that are not connected through any kind of transformation

• This reflects the spontaneously broken Ζ2 ≡ {−1,1} global symmetry group.

• One can interpret these two plateaus as the two cases where all spins are up or down.

• Is this true?

• We test the correlation coefficient between the latent dimension to magnetization

Correlation Coefficient between

magnetization and latent dimension




For low temperatures we obtain two plateaus, one located at 𝑧 = 1 and one at 𝑧 = −1.

• Corresponds to two distinct states that are not connected through any kind of transformation

• This reflects the spontaneously broken Ζ2 ≡ {−1,1} global symmetry group.

• One can interpret these two plateaus as the two cases where all spins are up or down.

• Is this true?

• We test the correlation coefficient between the latent dimension to magnetization

• The latent dimension −1 and 1 corresponds to the two orientations of the spins.

• Finally, the two plateaus become more distinct as the lattice size increases.

• At some temperature range 𝛥𝛵𝑡𝑟𝑎𝑛𝑠. the aforementioned behaviour collapses to one state,

which is located around 𝑧 = 0. This reflects the restoration of Ζ2 symmetry.

• In other words, it corresponds to the case where all the spins are disoriented.

• Critical point: as the lattice size increases the width of this transition decreases and this step

becomes steeper and steeper. At 𝐿 → ∞ 𝑇𝑐(𝐿) is localised right on the critical temperature 𝑇𝑐


Hence:

The autoencoder predicts the two phases

It provides to a good approximation the

critical temperature!

In fact from 𝐿 = 150, by fitting a line we

get 𝑇 ≅ 2.28(4).

Already consistent with analytical value

𝑇 ≅ 2.269… .

Of course this does not tell us what happens

at 𝐿 → ∞


What happens within different "temperature windows"

What if we use a temperature window within the range 𝑇 = 1 − 2 and apply the autoencoder

• Two ordered states are visible without the presence of a critical point

• Since there is no visible signal for a phase transition behaviour within this range of 𝑇 it is

reasonable to use another temperature window

What if we choose 𝑇 = 3 − 4.5• No particular pattern is observed

𝑇 = 1 − 2

𝑇 = 3 − 4.5

DIFFERENT TEMPERATURE WINDOWS

Next step would be to go to 𝑇 = 2 − 3

First: Average latent dimension 𝑧 as a function of the Temperature 𝑇

𝐿 = 150

For low temperatures the latent dimension is, in a good approximation, equally distributed

between the values of 𝑧 = 1 and 𝑧 = −1

Safe to consider absolute value of the latent dimension

CAN WE DEFINE MACROSCOPIC QUANTITIES?

CAN WE DEFINE MACROSCOPIC QUANTITIES?

Since the latent dimension per configuration is symmetric with respect to the 𝑇 axis, it would be

reasonable to define the average absolute latent dimension

The latent dimension resembles the behaviour of the magnetization per spin configuration as a

function of the temperature:

𝑚 =1

𝑁conf

𝑖=1

𝑁conf

𝑠𝑖

THE LATENT SUSCEPTIBILITY AND THE CRITICAL TEMPERATUREWe print the average absolute latent dimension as a function of the temperature

Compare with magnetization:

Second order As if First order

Absolute average latent dimension can be used as an order parameter to identify the critical

temperature, but cannot capture the right order of the phase transition

We print the average absolute latent dimension as a function of the temperature

Compare with magnetization:

Becomes

Steeper

𝑇𝑐(𝐿) extracted from the autoencoder data will suffer less from finite size scaling effects

THE LATENT SUSCEPTIBILITY AND THE CRITICAL TEMPERATURE


We define the latent susceptibility as

Like the magnetic susceptibilityTheir peaks determine

More points are requiredNo overfitting!!!!

(Using the extracted weights)

We extract the latent susceptibility as well as the magnetic susceptibility:

From the peaks we extract

Does it extrapolate to the right value


Results obtained using the latent susceptibility

suffer less from finite-size scaling effects as

compared to those when using the magnetic

susceptibility.

We present 𝑇𝑐(𝐿) extracted from fitting the

latent susceptibility and the magnetic

susceptibility as a function of 1/𝐿.

We adopt the standard finite scaling behavior:

Analytically extracted value:

For magnetic susceptibility 𝜈 = 1


• Apply deep learning Autoencoder on Configurations of the 2D Ising model at zero magnetic field

• No prior knowledge on the system

• We determine both phase regions and the critical temperature

• We define the average absolute latent dimension as a quasi-order parameter

• We define the latent susceptibility

• We observe the spontaneously broken Ζ2 ≡ {−1,1} global symmetry group.

• We can determine the critical temperature 𝑇𝑐 to an adequate accuracy

• We can extract 𝑇𝑐(𝐿) as a function of 1/𝐿 with small finite scaling effects

• Autoencoder does not know about the order of the phase but instead recognizes the phase sector as

the important feature of the Ising model configurations

• WHY DOES THIS WORK???

CONCLUSIONS

FURTHER WORK: 3D-ISING MODEL

Late

nt d

imension

L=10 L=12

𝐿 = 16 𝐿 = 24 𝐿 = 32 𝐿 = 36

• 3D-Ising Model

Critical Temperature 𝑇𝐶 = 4.511… (Talapov and Blöte (1996))

Second Order phase transition

Late

nt dim

ensi

on

Linea

r Activa

tion

Tanh

Activa

tion

• 4D-Ising Model

Critical Temperature 𝑇𝐶 = 6.65 (P. H. Lundow, K. Markström 2012)

3 states

Late

nt d

imension

L=10 L=12 L=16

𝑇𝑇 𝑇

FURTHER WORK: 4D-ISING MODEL

• POTTS MODEL

The Potts (1952) model is a generalization of the Ising model such that each spin can have

𝑞 values.

Interesting because it is a multistate model with

2nd order phase transition for 𝑞 ≤ 4 1st order phase transition for 𝑞 > 4

3 states 4 states

Late

nt d

imensi

on

Late

nt d

imensi

on

FURTHER WORK: 2D-POTTS

Autoencoders can “see” the broken Center Symmetry of the Underlying Group!

𝑍(𝐺) = {𝑧 ∈ 𝐺 ∣ ∀𝑔 ∈ 𝐺, 𝑧𝑔 = 𝑔𝑧}

Very Interesting to see what happens with field theories:

𝑈(1) – Infinite order phase transition with continuous center symmetry

𝑆𝑈(𝑁) – 𝑍𝑁 center symmetry

𝑍(2) 𝑍(3) 𝑍(4) 𝑐𝑜𝑛𝑡.

2D-Ising Model 2D Q=3 Potts 2D Q=4 Potts 2D XY-Model

FURTHER WORK:

• Unsupervised investigation of the phase transition in 2D-Potts and XY-models

• Unsupervised investigation of the phase transition in field theories such as

• Compact 𝑈(1) (infinite order phase transition)

• 𝑆𝑈(𝑁) (second order for small 𝑁 first order as we increase 𝑁)

• Supervised/unsupervised investigation of Topological field objects in statistical systems and

quantum field theories (Instantons, Nielsen-Olesen vortices, Kosterlitz-Thouless vortices)

• Production of configurations using generative algorithms

FURTHER WORK

UPCOMING WORKSHOPMachine Learning for High Energy Physics, on and off the Lattice

ECT* Trento, 23 – 27 March 2020

Organising Committee:

• Andreas Athenodorou (Pisa)

• Dimitrios Giataganas (NKUA)

• Enrico Rinaldi (RIKEN)

• Biagio Lucini (Swansea)

• Kyle Cranmer (NYU)

• Constantia Alexandrou (Cyprus)

unsupervised identification of the phase transition …...• 1 not so interesting: no phase...

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