[PPT] - Machine learning for bounce calculation Ryusuke Jinno (IBS-CTPU) PowerPoint Presentation

SLIDE 1

Machine learning for bounce calculation

Based on 1805.12153 2018/12/10 @ ICTP workshop on machine learning landscape

01 / 29

Ryusuke Jinno (IBS-CTPU)

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Ryusuke Jinno / 29

1805.12153

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Ryusuke (隆介) Jinno (神野)

SELF INTRODUCTION

2016/3 : Ph.D. @ Univ. of Tokyo
2016/4-8 : KEK, Japan
2016/9- : IBS-CTPU, Korea
2019/4- : DESY, Germany (planned)

Research interests & recent works

Machine learning : Application of machine learning to QFT tunneling problem
Gravitational waves : Analytic approach to GW production in phase transitions
Inflation : Hillcliming inflation (an inflationary attractor)

Hillclimbing Higgs inflation (new realization of Higgs inflation with hillclimbing scheme)

(P)reheating : Preheating in Higgs inflation (discovery of “spike preheating” phenomena)

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Ryusuke Jinno / 29

1805.12153

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Ryusuke (隆介) Jinno (神野)

SELF INTRODUCTION

2016/3 : Ph.D. @ Univ. of Tokyo
2016/4-8 : KEK, Japan
2016/9- : IBS-CTPU, Korea
2019/4- : DESY, Germany (planned)

Research interests & recent works

Machine learning : Application of machine learning to QFT tunneling problem
Gravitational waves : Analytic approach to GW production in phase transitions
Inflation : Hillcliming inflation (an inflationary attractor)

Hillclimbing Higgs inflation (new realization of Higgs inflation with hillclimbing scheme)

(P)reheating : Preheating in Higgs inflation (discovery of “spike preheating” phenomena)

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Ryusuke Jinno / 29

1805.12153

TALK PLAN

1. Machine learning : lightning introduction
2. Machine learning meets tunneling problem in QFT
3. Data taking / Machine setup / Training process / Results
4. Summary

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MACHINE LEARNING: LIGHTNING INTRODUCTION

[ https://www.slideshare.net/awahid/big-data-and-machine-learning-for-businesses ]

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Ryusuke Jinno / 29

1805.12153 [ https://www.slideshare.net/awahid/big-data-and-machine-learning-for-businesses ]

Can I apply this technique to problems in high-energy physics?

MACHINE LEARNING: LIGHTNING INTRODUCTION

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Terminology?

[ J. McCarthy ]

Machines that can perform tasks that are characteristic of human intelligence A way of achieving AI: lerning without being explicitly programmed

[ https://medium.com/iotforall/the-difference-between-artificial-intelligence-machine-learning-and-deep-learning-3aa67bff5991 ] [ https://en.wikipedia.org/wiki/Artificial_intelligence ] Note : the speaker’s major is not machine learning !! e.g.

Artificial intelligence (AI) Machine learning (ML) Neural network (NN) Deep neural network Deep learning Machine learning with artificial neurons (→ later) Neural network with deep (= many) layers of neurons

x1 xn x2 ... w1 w2 wn

MACHINE LEARNING: LIGHTNING INTRODUCTION

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Linear problem

: # of "discount" in the email x1 : # of "luxuary" in the email x2

05

: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

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Linear problem

05

: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question "Linear problem" → Good solution found easily Answer

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

: # of "luxuary" in the email x2 : # of "discount" in the email x1

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Nonlinear problem

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: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

: # of "luxuary" in the email x2 : # of "discount" in the email x1

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Ryusuke Jinno / 29

1805.12153

Nonlinear problem

: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

06

: # of "luxuary" in the email x2 : # of "discount" in the email x1

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Ryusuke Jinno / 29

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Nonlinear problem

: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

06

: # of "luxuary" in the email x2 : # of "discount" in the email x1

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Ryusuke Jinno / 29

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Nonlinear problem

: spam : not spam Your data Find such that a, b x2 = ax1 + b is the boundary

f spam & not spam

Question "Nonlinear problem" → No good solution Answer

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

06

: # of "luxuary" in the email x2 : # of "discount" in the email x1

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Nonlinear problem

r θ With some effort, you may find & useful: r = q x2

1 + x2 2

θ = arctan(x2/x1)

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

07

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Nonlinear problem

r θ With some effort, you may find & useful: r = q x2

1 + x2 2

θ = arctan(x2/x1) "Feature engineering"

In this specific case you are successful
But you cannot always find such good quantities
Any good strategy to this kind of problem?

(i.e. to capture nonlinearity) Good solution found, but...

LINEAR VS. NONLINEAR: SPAM EMAIL EXAMPLE

07

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NEURAL NETWORK ?

Biological neuron

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[ https://medium.com/autonomous-agents/ mathematical-foundation-for-activation-functions-in-artificial-neural-networks-a51c9dd7c089 ]

synapse axon synapse electric signal neuron

1. Each neuron collects electric signals through synapses
2. When the total signal exceeds a threshold,

electric signal is sent to next neuron through axon

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NEURAL NETWORK ?

Artificial neuron mimics biological neuron

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x1 x2 xn wn w1 w2 X xiwi f z

input weight nonlinear function

utput

sum z = f ⇣X xiwi + b ⌘ f(y) y Diagramatic notation Equation f : ReLU (rectified linear unit) wi b : weight : bias

⇢

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NEURAL NETWORK ?

Neural network = network of artificial neurons

= =

neuron neural network

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NEURAL NETWORK ?

Neural network = network of artificial neurons

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x1 = f (W1xin + b1) xn = f (Wnxn−1 + bn) xout = WoutxN + bout

⇢

(2 ≤ n ≤ N)

f(y) = f(y1) f(y2) f(yn)

· · ·

       

Note1 : Wn

xn

bn

= matrix / = vector / = vector Note2 :

Wnxn−1 + bn = (Wn)ij(xn−1)j + (bn)i

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NEURAL NETWORK: SUPERVISED LEARNING

How to train the neural network with "supervised learning"

b → b − α∂E ∂b

W → W − α ∂E ∂W

Training of neural network = update of weights and biases using
Then we can define "how poorly the machine predicts"

Note : there are more sophisticated algorithms, e.g. AdaGrad, Adam, ...

Error function E =

e.g.

E α : constant

Suppose we have many data of (xin, x(true)
ut

) X

data

X

i:component

(xout)i − (x(true)
ut

)i

12

b

W

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NEURAL NETWORK: HOW POWERFUL?

Ability of neural network to capture nonlinearity

My data : 100 points sampled from nonlinear function
My neural network : 2 layers, 20 neurons per each, trained for 10 sec.

(xin, x(true)

ut

) = (xin, xin(xin − 0.3)(xin − 0.6)(xin − 0.9))

0.2 0.4 0.6 0.8 1.0x

0.01

0.01 0.02 0.03 Pred 0.2 0.4 0.6 0.8 1.0x

0.01

0.01 0.02 0.03 Ans

xin xout x(true)

ut
r

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NEURAL NETWORK: HOW POWERFUL?

Ability of neural network to capture nonlinearity

My data : 100 points sampled from nonlinear function
My neural network : 2 layers, 20 neurons per each, trained for 10 sec.

(xin, x(true)

ut

) = (xin, xin(xin − 0.3)(xin − 0.6)(xin − 0.9))

0.2 0.4 0.6 0.8 1.0x

0.01

0.01 0.02 0.03 Pred 0.2 0.4 0.6 0.8 1.0x

0.01

0.01 0.02 0.03 Ans

xin xout x(true)

ut
r

13

Neural network is extremely useful in capturing nonlinearity

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NEURAL NETWORK: IMAGE RECOGNITION

cat dog

M

machine

1

label for cat label for dog

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Image classifier: nolinear relation btwn. input (= image) and output (= label)

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NEURAL NETWORK: IMAGE RECOGNITION

1

0.8 0.1 0.7 Input layer Output layer

Note : precisely, output layer is log-odds log [P(cat)/P(dog)] Note : actual image recognition is not this simple, e.g. CNN

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Image classifier: nolinear relation btwn. input (= image) and output (= label)

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TALK PLAN

1. Machine learning : lightning introduction
2. Machine learning meets tunneling problem in QFT
3. Data taking / Machine setup / Training process / Results
4. Summary

✔

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TUNNELING PROBLEM IN QFT

−V

φ

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WHY TUNNELING PROBLEM IN QFT?

Gravitational waves (GWs) from BH & NS binaries have been detected

15

Tunneling & bubble dynamics in the early Universe also produce GWs Such GWs may be detected in (near) future

Pulsar timing arrays Space Ground 0.01-1Hz 10 Hz

2

10 Hz

8

[http://rhcole.com/apps/GWplotter/]

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WHY TUNNELING PROBLEM IN QFT?

Gravitational waves (GWs) from BH & NS binaries have been detected

15

Tunneling & bubble dynamics in the early Universe also produce GWs Such GWs may be detected in (near) future

Pulsar timing arrays Space Ground 0.01-1Hz 10 Hz

2

10 Hz

8

[http://rhcole.com/apps/GWplotter/]

signal

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TUNNELING PROBLEM IN QFT

Quantum tunneling in vacuum in 1+3 dim.

Nucleation rate is dominantly determined by

Γ

with boundary conditions

d¯ φ dr (r = 0) = 0

¯ φ(r = ∞) = 0

,

Γ ∝ e−SE[ ¯

φ]

SE[¯ φ] = Z dtE Z d3x 1 2(∂E ¯ φ)2 + V (¯ φ)

,

−V

φ

rate Γ

Bounce configuration : solution of EOM with inverted potential −V

¯ φ ¯ φ

Euclidean action of "bounce configuration" −V

φ

r = 0

r = ∞

16

[ Coleman '77 ]

d2 ¯ φ dr2 + 3 r d¯ φ dr − dV d¯ φ = 0

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TIME-CONSUMING CALCULATION

−V

φ

vershoot

undershoot

Calculation of requires many times of iterations

¯ φ

Every time we have new particle physics setup, we re-calculate the EOM.

Note : there are many approaches, e.g.

17

cat-dog classifier does not have to recognize them as humans do

Can we construct a machine which gives for input potential ?

SE V

Advantages: 1. faster than any other method / 2. we can share the trained machine

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TALK PLAN

1. Machine learning : lightning introduction
2. Machine learning meets tunneling problem in QFT
3. Data taking / Machine setup / Training process / Results
4. Summary

✔ ✔

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DATA TAKING PROCESS

We use 3 classes of potentials C1-C3:

Bounce action is calculated with traditional overshoot/undershoot method

C1 C2 C3 Potential Bounce action

V (φ) =

7

X

n=1

a(1)

n φn+1

Class 1 (C1) :

V (φ) =

7

X

n=1

a(2)

n φ2n

Class 2 (C2) :

V (φ) = a(3)

1 φ2 + 7

X

n=2

a(3)

n φ2n−1

Class 3 (C3) :

                          

Coefficients generated randomly

a(i)

n

(→ backup)

Each class contains 10,000 sets of potential and bounce action

20

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Test dataset : used to check that there is no overfitting
Application dataset : machine is finally applied to this

e.g. 8,000 data from C1 2,000 data from C1 10,000 data from C1 Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Application dataset : machine is finally applied to this

e.g. 8,000 data from C2 10,000 data from C2 Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

Test dataset : used to check that there is no overfitting

2,000 data from C2

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Application dataset : machine is finally applied to this

e.g. 8,000 data from C3 10,000 data from C3 Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

Test dataset : used to check that there is no overfitting

2,000 data from C3

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Test dataset : used to check that there is no overfitting
Application dataset : machine is finally applied to this

e.g. 24,000 data from C1+C2+C3 6,000 data from C1+C2+C3 30,000 data from C1+C2+C3 Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Test dataset : used to check that there is no overfitting
Application dataset : machine is finally applied to this

e.g. 16,000 data from C2+C3 4,000 data from C2+C3 10,000 data from C1 Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Test dataset : used to check that there is no overfitting
Application dataset : machine is finally applied to this

e.g. Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

16,000 data from C3+C1 4,000 data from C3+C1 10,000 data from C2

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TRAINING & TEST & APPLICATION DATASET

We construct training & test & application dataset

Test dataset : used to check that there is no overfitting
Application dataset : machine is finally applied to this

e.g. Application t Test Training &

M

Step1 Step2

Training dataset : used for training (→ next slide)

21

16,000 data from C1+C2 4,000 data from C1+C2 10,000 data from C3

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MACHINE SETUP

N = 2 We try a simple machine :

M

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In the following, & are understood as rescaled

MACHINE SETUP

xin = ⇢

⇢ V (φsample)

φsample = 1

16, · · · , 15 16

⊕

⇢ V 0(φsample)

φsample = 1

16, · · · , 15 16

⊕

⇢ V 00(φsample)

φsample = 0

16, · · · , 16 16

Input : sampled values of potential & its derivatives

(xin)i ! (xin)i h(xin)ii σ(xin)i xout ! xout hxouti σxout

Note : implicit rescaling of input & output

...

V φ

Output : predicted value of logarithmic bounce action

xin xout hi σ & : mean & variance calculated over training & test dataset

xout = ln S(pred)

4

23

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TRAINING PROCESS

Error function = how poorly the machine predicts

x(true)

ut

= ln S(true)

4

xout = ln S(pred)

4

: predicted value of logarithmic bounce action : true value of logarithmic bounce action

Training = update of weights and biases using error function

Note : In the actual training we use a slightly more sophisticated algorithm Adam

E = 1 (# of data passed to the machine) X

data

xout − x(true)
ut
24

b → b − α∂E ∂b

W → W − α ∂E ∂W

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DETAILS OF TRAINING PROCESS

Mini-batch training

We feed the machine with 1/10 of the training data (= mini-batch) for one time
10 times of this process use the whole training data = 1 epoch
We train the machine for 10,000 epochs

Implementation

Above process is implemented with TensorFlow (r1.17)

training dataset € € € € € € € € € €

M

25

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DETAILS OF TRAINING PROCESS

Mini-batch training

We feed the machine with 1/10 of the training data (= mini-batch) for one time
10 times of this process use the whole training data = 1 epoch
We train the machine for 10,000 epochs

Implementation

Above process is implemented with TensorFlow (r1.17)

M

training dataset € € € € € € € € € 1 update

25

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DETAILS OF TRAINING PROCESS

Mini-batch training

We feed the machine with 1/10 of the training data (= mini-batch) for one time
10 times of this process use the whole training data = 1 epoch
We train the machine for 10,000 epochs

Implementation

Above process is implemented with TensorFlow (r1.17)

M

training dataset € € € € € € € € 2 update

25

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DETAILS OF TRAINING PROCESS

Mini-batch training

We feed the machine with 1/10 of the training data (= mini-batch) for one time
10 times of this process use the whole training data = 1 epoch
We train the machine for 10,000 epochs

Implementation

Above process is implemented with TensorFlow (r1.17)

M

training dataset € € € € € € € 3 update

25

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DETAILS OF TRAINING PROCESS

Mini-batch training

We feed the machine with 1/10 of the training data (= mini-batch) for one time
10 times of this process use the whole training data = 1 epoch
We train the machine for 10,000 epochs

Implementation

Above process is implemented with TensorFlow (r1.17)

Average of 10 times trial Scatter plot for machine’s performance

26

Training : 16,000 data from C3+C1 Test : 4,000 data from C3+C1 Application : 10,000 data from C2 /

Case C : training & test over 2 classes / application to the other 1 class

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RESULTS

Average of 10 times trial Scatter plot for machine’s performance

26

Training : 16,000 data from C1+C2 Test : 4,000 data from C1+C2 Application : 10,000 data from C3 /

Case C : training & test over 2 classes / application to the other 1 class

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DISCUSSION

How much precision can we expect in practical use? How much is the speedup?

Overshoot/undershoot typically takes O(1-10) sec in my code

Potential shapes in particle physics are not that many → If we train with such potentials, the resulting precision will be C1+C2+C3 or better

Other approaches take e.g. O(10 ) sec
2

[Guada et al. '18, "Polygonal bounces" (private communication)]

Our machine takes O(10) sec for training,
4

while after training it takes O(10 ) sec to calculate the bounce

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DISCUSSION

Generalizations?

Different spacetime dimensions → trivial
Multidimensional transitions → needs good ideas

2 dim. : convolutional neural network (CNN) may help ML may be used for 1 dim. part in existing multidimensional public codes e.g. 1) 2) ML may also be used for "initial position suggestor" in such public codes 3) by identifying the output as the initial position

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Different spacetime dimensions → trivial

DISCUSSION

Generalizations?

Multidimensional transitions → needs good ideas

input hidden1 input hidden1 input hidden1

2 dim. : convolutional neural network (CNN) may help ML may be used for 1 dim. part in existing multidimensional public codes e.g. 1) 2) ML may also be used for "initial position suggestor" in such public codes 3) by identifying the output as the initial position

28

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Different spacetime dimensions → trivial

DISCUSSION

Generalizations?

Multidimensional transitions → needs good ideas

input hidden1 input hidden1 input hidden1

2 dim. : convolutional neural network (CNN) may help ML may be used for 1 dim. part in existing multidimensional public codes e.g. 1) 2) ML may also be used for "initial position suggestor" in such public codes 3) by identifying the output as the initial position

28

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Different spacetime dimensions → trivial

DISCUSSION

Generalizations?

Multidimensional transitions → needs good ideas

input hidden1 input hidden1 input hidden1 input hidden1 input hidden1 input hidden1

2 dim. : convolutional neural network (CNN) may help ML may be used for 1 dim. part in existing multidimensional public codes e.g. 1) 2) ML may also be used for "initial position suggestor" in such public codes 3) by identifying the output as the initial position

28

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SUMMARY

Calculation of quantities from scalar potential can be regarded as image recognition problem We proposed using machine learning technique for such calculations, and demonstrated its usefulness in one-dim. transition We explained possible ideas for generalization to multi-dimensional transitions

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SLIDE 64

Backup

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V 00

is sampled from (flat distribution in log space)

Vmax

10−2 ≤ Vmax ≤ 10−0.5

DETAILS ABOUT POTENTIAL GENERATING PROCESS

Random seeds generation (Vmax, φ0, φ1−, φ1+, φ2)

4 numbers are generated in , and identified with

[0, 1]

r (probability 0.5 for each)

φ1+ < φ0 < φ2 < φ1− φ1+ < φ2 < φ0 < φ1−

V

V 0

Added to data if there is no local maximum/minimum other than φ = φ0, 0, 1 Coefficients are determined so that

a(i)

n

φ = 1 ⇢

takes local minimum @ φ = φ2
takes local

minimum or or

φ = 0

Vmax

⇢

@ maximum

φ = φ0

−1

φ = φ1+ φ = φ1−

takes local

minimum @ maximum

⇢ ⇢ ⇢ ⇢ ⇢ ⇢