Statistical analysis of heat flux in NSTX
using Neural Network
Jung-Kyun Park, Y. S. Na*, D. H. Na, J. W. Park
Department of Nuclear Engineering, Seoul National University, Seoul, Korea
*[email protected]
2017.02.16
5th A3 foresight workshop on spherical torus
Contents
I.
Motivation
II. Neural Network
III. Statistical analysis with NSTX data
-
I.
Summary & Future Work
-
2
Analysis setup
Neural network result
Interpretation of neural network
Preliminary result using sensitivity scan
Summary
Future Work
Seoul National University
Fusion and Plasma Application Laboratory
Motivation
• Understanding of plasma transport in tokamak plasma is one of the most important
issues.
• Neoclassical and gyrokinetic simulation codes are developed to describe transport
phenomena.
• Because of computationally challenging, reduced theory-based models(ex. TGLF ..)
are developed (But it is still computationally expensive for full transport simulations).
• Conventional empirical models and power scaling laws are easy to interpret and
implement, but due to limited operational regime, cannot fully explain transport.
•
3
Neural network is applied to develop new transport models.
Seoul National University
Fusion and Plasma Application Laboratory
Neural Network
Basic neural network
https://deeplearning4j.org/neuralnet-overview
For single perceptron(neuron)
y
𝑦 = 𝑓(
𝑤𝑖 𝑥𝑖 + 𝑏𝑖 )
(𝑤𝑖 ∶ 𝑤𝑒𝑖𝑔ℎ𝑡, 𝑏𝑖 ∶ 𝑏𝑖𝑎𝑠)
(𝑓 ∶ 𝑎𝑐𝑡𝑖𝑣𝑎𝑡𝑖𝑜𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛)
Feedforward
- Input data go forward to output.
Backpropagation
- Output errors go back to adjust weights.
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http://www.frank-dieterle.de/phd/2_7_1.html
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Fusion and Plasma Application Laboratory
Neural Network
EX) O. Meneghini et al., PoP 21, 060702 (2014)
Applying neural network to transport modeling
successfully in DIII-D(conventional tokamak)
 Can be applied in Spherical Torus too?
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Seoul National University
Fusion and Plasma Application Laboratory
Analysis setup
6
Target Machine
NSTX (Spherical Torus)
Layers & Neurons
1-3 layers, 5-15 neurons
Shot conditions
No (odd) MHD (<1),
neglect sawtooth, ELMs, TAEs etc.
Input data mining
18 radial points (0.05 ≤ 𝑟 𝑎 ≤ 0.9, ±0.05) at a single time point
averaged over radial (±0.05) & time (±15 ms)
Total datasets
~680 shots, ~680 sets
(50% for training, 30% for validation, 20% for TEST)
Input parameters
𝑅/𝐿𝑛𝑒 , 𝑅/𝐿𝑛𝑖 , 𝑅/𝐿 𝑇𝑒 , 𝑅/𝐿 𝑇𝑖 , 𝑅/𝐿𝑃𝑓𝑎𝑠𝑡 , 𝑛𝑖 /𝑛𝑒 , 𝑇𝑖 /𝑇𝑒 , 𝑃𝑓𝑎𝑠𝑡 /𝑃,
𝑞, 𝑠, 𝑅𝛻𝑉𝜙 /cs , 𝑉𝜙 /cs , 𝜅, δ, 𝑍𝑒𝑓𝑓 , 𝜌∗ , 𝜈𝑒𝑖 /(𝑐𝑠 𝑎), 𝛽𝑒, 𝑙𝑜𝑐𝑎𝑙 ,
𝛼𝑀𝐻𝐷 (19 parameters)
Target parameters
𝑄𝑒,𝑁 , 𝑄𝑖,𝑁
(normalized with gyro-Bohm scaling, 𝑄𝐺𝐵 = 𝑛𝑒 𝑇𝑒 𝜌∗2 𝑐𝑠 )
NN Training algorithm
Levenberg-Marquardt
(Most generally used in nonlinear least squares)
Seoul National University
Fusion and Plasma Application Laboratory
30
6
25
5
20
4
15
3
Qi,N
Qe,N
Neural network result
10
5
1
0
0
0.0
0.2
0.4
0.6
0.8
(=r/a)
7
2
1.0
0.0
0.2
0.4
0.6
0.8
1.0
(=r/a)
Normalized 𝑄𝑒 profiles for #141647
Normalized 𝑄𝑖 profiles for #141647
Blue : from TRANSP
Blue : from TRANSP
Red : from Neural-Network
Red : from Neural-Network
Seoul National University
Fusion and Plasma Application Laboratory
Neural network result
𝑹𝟐 with NN
𝑸𝒆,𝑵
𝑸𝒊,𝑵
1.0
0.9
0.8
0.8
0.7
0.7
R
2
0.9
R
2
1.0
0.6
0.6
0.5
0.5
0.4
0.4
0.0
0.2
0.4
0.6
0.8
1.0
0.0

0.2
0.4
0.6
0.8
1.0

* 𝑅2 : determination coefficient  shows regression’s accuracy
(𝑅2
2
𝑖 𝑒𝑖
= 1 − (𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎), 𝑒𝑖 : residual)
Neural network result shows high accuracy at outer region
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Seoul National University
Fusion and Plasma Application Laboratory
Neural network result
• Neural network result shows high accuracy.
• Neural network is mostly used in tokamak for
-
Response model development for control algorithm
Scenario development
Prediction for future device (ex. NSTX-U)
• In this research, with high accuracy, physics interpretation is performed.
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Fusion and Plasma Application Laboratory
Interpretation of Neural Network Result
• Neural Network is a ‘black-box’ by itself.
• There are several ways to interpret Neural Network results,
1) Visualization of Neural Network [J. Wejchert et al., NIPS (1989)]
 connect lines between neurons with different color or thickness based on
weighting factor
2) Variance based sensitivity analysis [Nasir BILAL, 22nd ICEC (2014)]
 Calculate the main effects of a given parameter and all the interactions involving
that parameter using variance
 Widely used in nonlinear, non-monotonic input-output relationship
• Here, we will use variance based sensitivity analysis.
10
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Fusion and Plasma Application Laboratory
Preliminary result using sensitivity scan
Sensitivity scan in 𝝆 = 𝟎. 𝟖𝟎 for 𝑸𝒆,𝑵 , 𝑸𝒊,𝑵
0.45
0.55
0.40
0.50
0.45
sensitivity (arbitrary)
sensitivity (arbitrary)
0.35
0.30
0.25
0.20
0.15
0.10
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.05
0.00
0.00
1
2
3
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7
8
9 10 11 12 13 14 15 16 17 18 19
1
2
3
4
input parameters
6
7
8
9 10 11 12 13 14 15 16 17 18 19
input parameters
𝑸𝒆,𝑵
11
5
𝑸𝒊,𝑵
#
Var.
#
Var.
#
Var.
#
Var.
1
𝐑/𝑳𝒏𝒆
6
𝒏𝒊 /𝒏𝒆
11
𝐑𝛁𝑽𝝓 /𝒄𝒔
16
𝝆∗
2
𝐑/𝑳𝒏𝒊
7
𝑻𝒊 /𝑻𝒆
12
𝑽𝝓 /𝒄𝒔
17
𝝂𝒆𝒊 /(𝒄𝒔 𝒂)
3
𝐑/𝑳𝑻𝒆
8
𝑷𝒇𝒂𝒔𝒕 /𝐏
13
𝛋
18
𝜷𝒆,𝒍𝒐𝒄𝒂𝒍
4
𝐑/𝑳𝑻𝒊
9
𝐪
14
𝜹
19
𝛼𝑀𝐻𝐷
5
𝐑/𝑳𝑷𝒇𝒂𝒔𝒕
10
𝒔
15
𝒁𝒆𝒇𝒇
Seoul National University
Fusion and Plasma Application Laboratory
Preliminary result using sensitivity scan
Dominant parameters
𝝆
dominant
parameters
0.50
0.65
𝑄𝑒,𝑁
𝑄𝑖,𝑁
𝑅 𝐿 𝑇𝑒 , 𝑹 𝑳𝑻𝒊 ,
𝜌∗ , 𝜈𝑒𝑖 /(𝑐𝑠 𝑎)
𝑃𝑓𝑎𝑠𝑡 𝑃 ,
𝑍𝑒𝑓𝑓
𝑄𝑒,𝑁
0.80
𝑄𝑖,𝑁
𝑹 𝑳𝒏𝒆 , 𝑹 𝑳𝑻𝒆 ,
𝑞,
𝑉 𝑐 ,𝜌 ,
𝑅𝛻𝑉𝜙 /𝑐𝑠 𝜙 𝑠 ∗
𝛽𝑒,𝑙𝑜𝑐𝑎𝑙
𝑄𝑒,𝑁
𝑄𝑖,𝑁
𝑅 𝐿𝑛𝑒 ,
𝑃𝑓𝑎𝑠𝑡 𝑃, 𝜌∗
q
• Some interesting results are observed.
- At 𝜌 = 0.50, electron heat flux is dominated by ion temperature scale length
- At 𝜌 = 0.65, ion heat flux is dominated by electron density, temperature scale length
• Further interpretation is on-going.
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Fusion and Plasma Application Laboratory
Summary
• Neural network is applied for modeling transport phenomena in Spherical Torus.
• Neural Network result shows high accuracy.
 This can be used in many problems(control, prediction, etc.).
• Neural network is used for physical approach, shows some interesting results.
 Electron heat flux is dominated by ion-related parameters, and opposite case is also
found.
• Physical interpretation of plasma transport using neural network is on-going.
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Fusion and Plasma Application Laboratory
Future work
W. Guttenfelder et al., NF 53, 093022 (2013)
Applying neural network for each microinstability
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Fusion and Plasma Application Laboratory
Future work
0.14
0.12
0.10
e, local
0.08
0.06
0.04
0.02
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
ei/(cs/a)
Separate plasma regimes with various parameters
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Fusion and Plasma Application Laboratory
Backup
Input parameter
16
#
Var.
#
Var.
1
𝐑/𝑳𝒏𝒆
Electron density scale length
11
𝐑𝛁𝑽𝝓 /𝒄𝒔
2
𝐑/𝑳𝒏𝒊
Ion density scale length
12
𝑽𝝓 /𝒄𝒔
3
𝐑/𝑳𝑻𝒆
Electron temperature scale length
13
𝛋
Elongation
4
𝐑/𝑳𝑻𝒊
Ion temperature scale length
14
𝜹
Triangularity
5
𝐑/𝑳𝑷𝒇𝒂𝒔𝒕
Fast particle pressure scale length
15
𝒁𝒆𝒇𝒇
6
𝒏𝒊 /𝒏𝒆
Ion to electron density ratio
16
𝝆∗
7
𝑻𝒊 /𝑻𝒆
Ion to electron temperature ratio
17
𝝂𝒆𝒊 /(𝒄𝒔 𝒂)
8
𝑷𝒇𝒂𝒔𝒕 /𝐏
Fast particle pressure ratio
18
𝜷𝒆,𝒍𝒐𝒄𝒂𝒍
9
𝐪
Safety factor
19
𝜶𝑴𝑯𝑫
10
𝒔
Magnetic shear
Normalized toroidal velocity shear
Toroidal Mach number
Effective atomic number
Normalized gyroradius
Normalized Electron-ion collisional
frequency
Local electron beta
MHD alpha parameter
Seoul National University
Fusion and Plasma Application Laboratory