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PSFC/RR-19-6
Confinement regime identification on Alcator C-Mod using supervised machine learning methods
A. Mathews, J.W. Hughes, A.E. Hubbard, D.G. Whyte, S.M. Wolfe,T. Golfinopoulos, D. Brunner, R.S. Granetz, C. Rea,
A.E. White, Alcator C-Mod Team
April 2019
Plasma Science and Fusion Center Massachusetts Institute of Technology
Cambridge MA 02139 USA
This work was supported by the U.S. Department of Energy (DOE) Office of Science Fusion Energy Sciences program under contracts DE-FC02-99ER54512, DE-SC0014264, and the Joseph P. Kearney Fellowship. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.
PSFC Report: PSFC/RR-19-6
CONFINEMENT REGIME IDENTIFICATION ON ALCATOR C-MOD USING SUPERVISED MACHINE
LEARNING METHODS
A. Mathews, J.W. Hughes, A.E. Hubbard, D.G. Whyte, S.M. Wolfe, T. Golfinopoulos D. Brunner, R.S.
Granetz, C. Rea, A. E. White, and the Alcator C-Mod Team
Plasma Science and Fusion Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, US
ABSTRACT
Automating confinement regime identification could provide enhanced capability for controlling toka-
maks to optimize power output. Distinguishing features between fusion plasma confinement regimes
are explored via machine learning methods to analyze experimental data from the compact, high-field
Alcator C-Mod tokamak. Four supervised machine learning techniques (Gaussian naıve Bayes, logistic
regression, multilayer perceptron, and random forest) are employed to identify accessibility conditions
for a confinement database with over 200 distinct plasma discharges consisting of approximately 400
L-, 200 H-, and 100 I-mode periods. These algorithms each demonstrate overall identification accu-
racy exceeding 90%, and can be employed concurrently during operation and establish boundaries
in confinement regimes based on past runs while providing areas for focused exploration to further
understand transition physics.
1. INTRODUCTION
Accessing high confinement plasmas is key to optimizing
energy gain, which is an essential component in devel-
oping economical nuclear fusion reactors. The discovery
of H-modes [26] significantly improved energy confine-
ment across a range of fusion devices and led to numer-
ous studies of factors for H-mode access such as plasma
shaping, toroidal field, particle density, divertor geome-
try, and wall materials [8, 11, 12, 20, 22]. Future burn-
ing plasma devices are largely relying on achieving this
higher confinement, but reduced performance from im-
purity accumulation due to increased overall particle con-
finement and the possibility of damaging edge localized
modes (ELMs) are causes for worry. ELM-suppressed
high confinement regimes such as I-mode [27] offer a
viable alternative scenario to H-mode in burning plas-
mas. I-modes are robustly observed at higher magnetic
field strength, which presently operating tokamaks rarely
achieve, yet could be crucial to sustainable and econom-
ical reactor operation [24]. Therefore, there is a need
to improve understanding of transition physics for the
scoping of confinement to future fusion devices.
Machine learning is being widely applied in nuclear fusion
on several topics including disruption prediction, opti-
mizing integrated modelling, and L-H transition physics
[1, 2, 13, 14, 21, 25]. With extensive data available on fu-
sion machines spanning several campaigns, delving into
robust techniques to study intricate and/or poorly un-
derstood mechanisms provides a useful additional tool
to further compare and validate existing models. One
2 A. Mathews
crucial element to remember when applying supervised
models is that utility of their outputs is limited by the
quality of inputs. The machine learning methods utilized
in this report are general and previously deployed in a
plethora of applications ranging from medical research
to image recognition to text identification [10, 16, 28].
Their apparent robustness in various settings serves as
a motivation to assess their accuracy in identifying con-
finement regimes in fusion plasmas. A goal of this work
is to derive a probabilistic model whereby a real-time
automated plasma control system can gain knowledge of
the likelihood of entering (or staying) in a particular con-
finement mode. Learning optimal accessibility conditions
for I-modes, for example, could help tokamaks access this
regime’s favourable properties.
2. METHODS
2.1. Confinement Regime Database
Different confinement regimes exhibit distinct properties.
The evolution of density and temperature pedestals in
the plasma edge region near the last closed flux surface
is used to classify plasma behaviour. These boundaries
can be blurred as transition regions are approached and
parameters (e.g. input heating power) are varied on a
continuum. Thus definitions of labels are critical when
applying supervised learning techniques. These are data-
driven methods and performance is limited by the con-
sistency of given data. Characterizations of L-, H-, and
I-modes are thus provided below as the applied defini-
tions in this study and akin to [27]:
H-mode: formation of both a strong edge ne and Te
pedestal along with reduction in broadband edge fluc-
tuations (. 100 kHz).
I-mode: formation of a strong edge Te pedestal with-
out significant change in ne, and appearance of high fre-
quency edge fluctuations (& 100 kHz).
L-modes are a third regime typically exhibiting relatively
weak edge pressure gradients and defined as not being H-
or I-modes, including discharge periods without auxiliary
power classically considered ohmic. Other signals such as
sharp changes in Dα and radiated power are also helpful
in diagnosing transitions but not necessary markers. A
plasma was considered to be present in Alcator C-Mod
and possibly included in the database if its plasma cur-
rent was exceeding 100 kA. Transition periods are typi-
cally events with finite durations that can vary on the or-
der of milliseconds. In the database, a constant time in-
terval (i.e. a single millisecond) was used to separate each
sample of data to classify the bifurcation, although the
observed characteristics around transition regions can be
relatively smeared and intermediate to either mode.
A database was initially developed by manually identi-
fying signals as either L-, H-, or I-mode and their associ-
ated (forward and backward) transitions. I-modes were
almost exclusively verified using an existing database de-
veloped by A.E. Hubbard and D.G. Whyte, while L- and
H-modes were obtained from a variety of sources using
the logbook primarily. It should be noted that misclas-
sifications of L-, H-, and I-mode are penalized equally,
yet the database is imbalanced as it currently consists
of manual shot selections with a frequency of particular
modes not necessarily representative of all shots histor-
ically. The true ratio of regime durations during oper-
ation is not known with exactness, but is expected to
be dominated by L-mode plasmas. The database de-
veloped and analyzed here consists of over 200 shots
with the periods spanning L-, H-, and I-modes com-
posing approximately 80%, 10%, and 10%, respectively.
Included are discharges from 1995 to 2016 on Alcator
C-Mod. The below results are based on predominantly
D-D plasmas with typically trace amounts of minority
and impurity species present. The database spans a
Confinement Regime Identification on C-Mod 3
fairly vast space where volume-averaged density ranged
from 4 × 1019 − 5 × 1020 m−3, toroidal magnetic field
from 3 − 8 T, plasma current up to 1.5 MA, auxiliary
power reaching nearly 6 MW, and shots consisting of
lower single null, upper single null, and double null X-
point configurations running the ion ~∇B drift in both the
favourable and unfavourable directions. There are over
40 readily available measurements including quantities
derived from equilibrium fits to the 2-dimensional ideal
magnetohydrodynamics (Grad-Shafranov) equation, us-
ing the EFIT code. A full list of variables available in
the database with a corresponding brief description is
provided in the Appendix.
Figure 1 : Correlation of selected features to indicate
possible redundancy. Shot number is also incorporated
for reference.
The confinement database consists of 0-D data that
does not account for measurement errors nor causal ef-
fects in time. Codes were created to automatically run
magnetics-based EFIT and populate the table with rel-
evant quantities from MDSplus, and these data were
obtained using a sampling rate of 1 millisecond on a
129 × 129 grid (based on a custom version generously
created by S. Wolfe). The goal is to ensure this method
can be validated, so training our classifiers with routinely
available measurements will assist in its repeatability
across experiments. The variables used are not exhaus-
tive but comprise a significant number of experimentally-
relevant quantities. The inputs provided to the classifiers
are selected to avoid high cross-correlation. For example,
the Shafranov shift is a relatively good input variable,
but radial displacement of flux surfaces is highly corre-
lated with fundamental features such as plasma pressure
which may be considered instead.
2.2. Machine Learning Algorithms
The current supervised methods applied from the Python
package scikit-learn are: Gaussian naıve Bayes, logistic
regression, random forests, and a class of feedforward
neural networks known as a multilayer perceptron [18].
These are trained for the classification problem of deter-
mining the class, K (which may be L, H, or I).
Gaussian naıve Bayes is a conditional probability model
based on Bayes’ theorem which states:
posterior=prior×likelihood
evidence⇔ p(K |x) = p(K)p(x|K)
p(x)
Two major assumptions are inherent to this specific clas-
sifier. The first is that each of the features (xi) are nor-
mally distributed within each confinement regime:
p(xi|K) =1
2πσ2K
exp[− (xi − µK)2
2σ2K
] (1)
The second is a naıve yet computationally expedient as-
sumption of independence:
p(K |x1, . . . , xn) ∝ p(K)p(x1 |K) · · · p(xn |K) (2)
Finally, the decision rule is based on selecting the class
that maximizes the posterior probability:
y = argmaxK∈{L,H,I}
p(K)
n∏i=1
p(xi | K) (3)
4 A. Mathews
Figure 2 : Observed distribution of βp in the entire
database based upon confinement regime identification.
It should be noted these assumptions are not necessarily
valid since, for example, βp is dependent on ne. Simi-
larly, the input features are not necessarily normally dis-
tributed. The degree to which these assumptions are
violated appears to worsen the output of this classifier
and constrain suitable inputs.
A binary (i.e. yn,K equals 0 or 1) logistic regression
model is trained by applying a ‘one vs. rest’ approach
(i.e. H-mode vs. not H-mode) which essentially reduces
the number of comparisons from NK(NK − 1)/2 to NK ,
where NK is the number of classes and equals 3 in
the present case of L-, H-, and I-modes. A ‘one vs.
one’ approach (i.e. H-mode vs. I-mode which is na-
tively handled by multinomial logistic regression) dif-
fers since it compares all classes individually to one an-
other as opposed to simply comparing versus the null
case. For NK = 3, the two approaches are equivalent
in their number of evaluations, although quickly diverge
if NK 6= 3 and further confinement modes are employed
(e.g. EDA H-mode, ohmic plasma). The probability of
the nth sample input feature vector, xn, belonging to
a particular class, K, is given by the logistic function,
pn,K(yn,K = 1|xn) = 11+e−βK ·xn , where βK is a set of
weights and bias terms optimized by minimizing a cost
function comprising logistic loss (i.e. cross-entropy) and
Tikhononv regularization:
− 1
N
N∑n=1
[yn,K log pn,K + (1− yn,K) log(1− pn,K)
]+λ‖βK‖22
(4)
The linear predictor here is linked to the response vari-
able by a logit function, βK · xn = logit(pn,K) =
ln(
pn,K1−pn,K
). This is similar to other generalized linear
models such as the probit link function where βK · xn =
Φ−1(pn,K), and Φ is the normal cumulative distribution
function. Outlining the treatments given in [6, 15, 17],
binary choice models can also be understood from their
utility by studying a response, yn,K , to a latent variable,
Un,K = βK ·xn−en, where βK is a set of regression coeffi-
cients, xn is a sample feature vector, and en is a random
variable specifying “noise” or “error” in the response,
assumed to be distributed according to some symmetric
distribution (e.g. logistic, normal) centered at 0, such
that
yn,K =
1, if Un,K > 0
0, if Un,K ≤ 0(5)
Now we can denote the cumulative distribution function
(CDF) of en as Fe, and the quantile function (inverse
CDF) of en as F−1e . Note that
Pr(yn,K = 1) = Pr(Un,K > 0) (6)
= Pr(βK · xn − en > 0) (7)
= Pr(−en > −βK · xn) (8)
= Pr(en ≤ βK · xn) (9)
= Fe(βK · xn) (10)
Since yn,K is a Bernoulli trial, where E[yn,K ] =
Pr(yn,K = 1), we have E[yn,K ] = Fe(βK · xn), or, equiv-
alently, F−1e (E[yn,K ]) = βK · xn
Confinement Regime Identification on C-Mod 5
If en ∼ Logistic(0, 1), i.e. distributed as a standard logis-
tic distribution with mean 0 and scale parameter 1, then
the corresponding quantile function is the logit function,
and logit(E[yn,K ]) = βK · xn, which is exactly a logit
model:
E[yn,K = 1 | xn] = pn,K = logit−1(βK · xn) =1
1 + e−βK ·xn
(11)
To further outline the statistics involved in this model,
the probability density function of the logistic distribu-
tion is given by:
f(x;µK , sK) =e−( x−µKsK
)
sK
(1 + e
−( x−µKsK))2 =
1
4sKsech2
(x− µK
2sK
)(12)
which has greater peaking and heavier tails (or higher
kurtosis) relative to the normal distribution. The CDF
of the logistic distribution is itself in the family of logis-
tic functions, which can be represented as a hyperbolic
tangent:
Fe(x;µK , sK) =1
1 + e−( x−µKsK
)=
1
2+
1
2tanh
(x− µK
2sK
)(13)
In these equations, x is the random variable, µK is the
mean, and sK is a scale parameter linearly proportional
to the standard deviation (σK = πsK√3
). Here a general-
ization of the logit function is given by the inverse CDF
of the logistic distribution:
F−1e (pn,K ;µK , sK) = µK + sK ln
(pn,K
1− pn,K
), (14)
F−1e′(pn,K ; sK) =
sKpn,K(1− pn,K)
(15)
The simplicity of the binary logistic regression applied
here assists in interpretation. For example, the odds of
a particular outcome can be defined as
pK(x)
1− pK(x)= eβK ·x (16)
and the odds ratio as
odds(xi + 1)
odds(xi)=eβK,i(xi+1)
eβK,ixi= eβK,i (17)
This physically means the odds of being in class K mul-
tiply by eβK,i for every 1-unit increase in the ith com-
ponent of the feature vector. This assessment does not
hold perfectly because the four features applied are not
wholly independent. Quantitative interpretation of the
parameters as effect measures is further muddled by un-
observed variables that have not been accounted so pre-
cise numerical values may not be exactly representative.
Nevertheless, these coefficients are meaningful within the
confines of the model and quantify general trends.
Random forest [3] is an ensemble algorithm which trains
independent decision trees and aggregates their individ-
ual responses to classify an input. This is accomplished
by the random forest providing a randomly selected
subset with repeated sampling of the original training
database to each of the trees, and only a subset of the
total input features can be used at each node. The deci-
sion trees themselves try to split the sample into different
branches trying to minimize an impurity measure known
as the Gini impurity, IG =∑K
pK(1 − pK), where pK
is the fraction of samples in class K. In the particular
model trained, log2N features are randomly provided for
each node in each tree, where the number of features is
N = 4. Based on runs, a random forest with approxi-
mately 50 fully grown trees was found to be optimal.
Multilayer perceptron is a feedforward artificial neural
network which consists of an input layer with N nodes
for the feature vector, one or more hidden layers, and an
output layer with NK nodes as an output. A deep fully
6 A. Mathews
connected network with 10×10×10×10 hidden layers
appeared optimal after multiple manual runs. Increas-
ing complexity of the model architecture beyond this
structure did not increase accuracy and actually wors-
ened performance on the testing set despite higher accu-
racy on the training data—a basic demonstration of high
variance resulting in poor generalization to unseen exam-
ples. This applied model utilizes a rectified linear unit
activation function, f(z) = max(0, z), at each neuron,
where z is the combined set of weighted inputs commu-
nicated from the nodes in the preceding layer. The high
degrees of freedom aids in reducing bias, and the univer-
sal approximation theorem states that multilayer feedfor-
ward neural networks with appropriate activation func-
tions and even a single hidden layer with finite neurons
can approximate any continuous function in the feature
space [4, 7], although there are no guarantees on gen-
eralized learning on untrained domains and the network
architecture may be impractically large.
The measured clock times for the trained classifiers to
make a single identification were sub-millisecond (the
only exception was the random forest algorithm). As
will be demonstrated, these methods provide an active
tool to investigate distinct confinement regimes and de-
marcate expected transition boundaries.
2.3. Evaluating Data
Based on the observed distribution of raw data, it is
found that points pertaining to particular regimes tend
to cluster together in certain ranges of parameter space.
This is extremely important when utilizing training data
that is multidimensional and sparse due to lack of data
and limitations in operation space as opposed to physics-
based constraints. Selecting the optimum number and
type of features for the confinement regime identifica-
tion problem is critical. Principle component analysis
was considered but this transformation alters the phys-
ical dimensions of the problem and can cloud interpre-
tation. The four quantities found to be optimal based
upon the mean decrease impurity via the random for-
est algorithm are: dimensionless normalized internal in-
ductance (li), poloidal beta (βp), total volumetric heat-
ing power (Pinput), and volume-averaged electron density
(ne). These quantities compose an input feature vector
x = (x1, . . . , xn) and are defined as [5]:
li =Li
2πR0
4π
µ0=〈B2
θ 〉B2θ (a)
, (18)
βp =〈p〉
Bθ(a)2/2µ0, (19)
Pinput = (Pohm + Paux)/VLCFS , (20)
where Li is the internal inductance, R0 is the major ra-
dius, Bθ is the poloidal magnetic field, Pohm is the ohmic
heating power, Paux is the total ion cyclotron resonance
heating power (without any heating inefficiencies consid-
ered and consists of primarily minority heating), VLCFS
is the volume enclosed by the last closed flux surface,
and Ip is the plasma current. The ohmic heating power
is calculated by finding the resistive voltage and multi-
plying it by the total current: Pohm = (Vloop −Li dIpdt )Ip.
The line-integrated plasma electron density is measured
by a two-color interferometer along ten vertical chords,
and this is used to construct ne.
Confinement Regime Identification on C-Mod 7
Figure 3 : Relative feature importance based on mean
decrease impurity via random forest.
1. βp (0.383± 0.075)
2. ne (0.243± 0.027)
3. Pinput (0.238± 0.075)
4. li (0.139± 0.019)
Normalization of the data was applied consistently dur-
ing preprocessing by subtracting the mean and divid-
ing the standard deviation of each feature before feeding
data to all the classifiers. This step is unnecessary in the-
ory but practically essential for the multilayer perceptron
when updating parameters for numerical purposes.
The four supervised learning techniques are trained on
a sample that consists of 80% of the unique shots in the
database (of which 80% of the time slices are randomly
sorted for training and 20% for validation), and tested on
a set containing the remaining 20% of unique shots. The
validation set contains time slices derived from discharges
included in the training set, while the testing set shares
no shots with the training set. This process is repeated
for 100 cycles, and the mean and standard deviation for
different accuracy metrics are computed based on binary
confusion matrices which can be created for multiclass
problems via the ‘one vs. rest’ approach.
3. RESULTS AND OVERVIEW
For logistic regression, the linear predictors (i.e. product
of coefficient and input feature vectors) were found to be:
βL · x = 2.79 + 0.12ne − 3.25βp + 1.81li − 0.29Pinput
(21)
βH · x = −3.89 + 1.17ne + 2.33βp − 1.72li − 1.18Pinput
(22)
βI · x = −4.53− 2.39ne + 1.68βp − 0.83li + 1.87Pinput
(23)
As expected, there is a bias towards L-mode originat-
ing from the imbalance of modes in the database. Low
values of li indicate a broad current profile, and the
negative coefficients for both H- and I-modes demon-
strates a greater prevalence of broad currents in these
two confinement regimes. Based on the above param-
eters, high βp is associated with both H- and I-modes,
although they have contrasting dependencies on both ne
and Pinput which indicate past accessibility conditions.
When applying normalization by the Greenwald density
(nG[1020m−3] = Ip[MA]/(πa0[m]2)) instead of using ne,
the accuracy of the classifiers reduced, possibly indicat-
ing nG is relatively poorer at distinguishing confinement
regimes. This is consistent with [11] where transition
power threshold was not strongly correlated with nG.
Shot 1160930033 on Alcator C-Mod achieved a record
plasma pressure on any magnetically confined fusion ex-
periment at approximately 2 atmospheres [9], and this is
essential to maximizing the triple product. No shots from
this final run day were included in the entire database,
and provides an unseen test case for the trained algo-
rithms which pushes the boundaries of the training set
itself. The four inputs to the different classifiers are
plotted, which essentially contain all the information the
classifiers utilize to determine the plasma’s mode. This
8 A. Mathews
(a) ne (b) βp
(c) li (d) Pinput
Figure 4 : Inputs given to the trained classifiers for shot
1160930033 which is an unseen test case.
shot provides a basic extrapolation test case trying to in-
crease the stored energy in a high-field, diverted tokamak
plasma. Based on the quantities in Figure 4 being the
input feature vector for each time slice, the probability
of being in an H-mode was computed using the trained
classifiers. (Probabilities are calibrated to add up to 1
and the I-mode probability remained relatively low dur-
ing the shot, therefore an L-mode can be assumed to be
identified if not an H-mode in Figure 5.) The triple prod-
uct was also calculated for the entire shot to indicate the
quality of confinement. The dark orange shaded region
indicates a manually identified region where a stable en-
hanced Dα (EDA) H-mode appears approximately 15 ms
after auxiliary power is turned on, and this is confirmed
by the electron density and temperature profiles from
core and edge Thomson scattering diagnostics. While
the triple product does not return to its initial L-mode
value immediately after exiting the dark orange shaded
region, which ends approximately when auxiliary power
is turned off, it does degrade. This reduction in confine-
ment coincides with oscillations in the classification prob-
ability as the triple product and edge pedestal are dimin-
ishing yet an ohmic H-mode with current ramping down
is still present in this region in which energy is escap-
ing the core plasma increasingly fast. Variations in the
probability correspond to variations in confinement, and
provide insight into the plasma’s expected behaviour.
Figure 5 : Classification probability of H-mode for shot
1160930033 based on input features given in Figure 4.
Probability contours in the 4-dimensional feature space
can be identified to aid experiments in operation. For
simplicity both li and Pinput are kept constant while plot-
ting, but one may seek a global or local minimum path
to access different modes in the full feature space. The
current method does not impose causal nor operational
bounds for operation (e.g. Greenwald limit, disruption
prediction algorithm), which would be essential during
real-time operation to provide forbidden regions in the
feature space. Nevertheless, boundaries in confinement
regime are produced which indicate possible critical bi-
furcation regions. Computing the optimal path between
the initial and final state can be accomplished by ap-
plying global optimization methods, although local tech-
niques may be preferable from a control standpoint. The
results appear qualitatively similar to plots produced by
D. Brunner, although the boundary regions vary consid-
erably with Pinput, and to a lesser extent with li. Tran-
sitions are also noted to be influenced by factors such as
probe plunges and wall condition, which are not explic-
itly accounted for in the present analysis.
Confinement Regime Identification on C-Mod 9
Figure 6 : Probability contours determined by each ma-
chine learning method for accessing different confine-
ment regimes on Alcator C-Mod with volumetric heating
power and normalized internal inductance kept constant.
Differences in the relative accuracy of these different
models when applied to separate machines could also
signify underlying distributions and accessibility path-
ways. For example, logistic regression improves in per-
formance when confinement regimes can be partitioned
into single continuous regions (as displayed in Figure 6)
as opposed to multiple fragments with several pockets
of distinct modes, and this provides an analytic bound-
ary. Visual inspection of the neural network probabil-
ity contours in Figure 6 indicates possible over-fitting
based on sharp gradients and isolated pockets. The ran-
dom forest exhibits a similar sharpness. This could be
a consequence of the imperfect data and may warrant
further regularization techniques to reduce the degrees
of freedom these algorithms have when generalizing to
novel scenarios despite fairly good accuracy on Alcator
C-Mod. It should be acknowledged these methods are
largely constrained by the size and quality of available
data in representing scenarios of interest. Final results
comparing the four different classifiers on both validation
and test sets are provided in the Appendix. These involve
averaging over 100 independent runs, and all four clas-
sifiers perform with above 90% total accuracy, although
with some notable differences as exemplified by Tables
2 and 3. Overall accuracy was optimal with the neural
network based on several reported metrics. The devel-
opment of this exploratory tool can now establish a new
transitions database arising from the automatic charac-
terization of time slices from tens of thousands of dis-
charges conducted over the span of decades on Alcator
C-Mod. In particular, there was data readily available
for 30175 shots comprising a total of 2177026 data points
at 20 millisecond intervals which resulted in the following
overall percentages for expected mode prevalence:
Random Forest: 86.6% L, 10.7% H, 2.7% I
Gaussian naıve Bayes: 83.6% L, 11.7% H, 4.7% I
Neural Network: 85.5% L, 10.9% H, 3.6% I
Logistic Regression: 85.9% L, 10.7% H, 3.4% I
Time slices from these shots can be directly assessed
to locate atypical experimental inputs that permit or
prevent specific confinement regimes. For example, a
user can check the signals “ssep” and “btor” (see Ta-
ble 4 in the Appendix) to locate all time slices where
both favourable ion ~∇B drift exists and an I-mode is
expected. Depending on the quantity of interest (e.g.
divertor strike point position, edge ion heat flux), associ-
ations between controlled variables and expected modes
can be systematically studied on a large scale to compre-
hensively document causal factors and improve predic-
tion capabilities.
This work presents an initial approach towards the goal
of L-, H-, and I-mode prediction. The machine learn-
ing methods applied here permit data-driven real-time
guidance for experiments and indicate regions for con-
tinued exploration. For example, by controlling parame-
ters this permits active feedback to help optimally access
10 A. Mathews
H-mode from L-mode or stay in I-mode while presently
in that regime. Incorporating a regression task going
forward to capture associated relevant values such as
energy confinement time or even pedestal pressure pro-
files directly would augment the current algorithm. This
would provide a quantitative performance measure to
differentiate confinement regimes because two plasmas
in the same mode are themselves not necessarily identi-
cal. Including input features involving time-dependent
and spatial quantities will likely increase accuracy con-
siderably as transition physics is believed to be largely
dictated by edge processes and hysteresis. A systematic
routine automating database development could enable
studies in a wider range of operational space and cross-
validation across machines. Pedestal density and tem-
perature profiles define confinement regimes and regres-
sion techniques could expedite construction of predictive
methods, which will be explored in upcoming work.
4. CONCLUSION
Distinguishing features between fusion plasma confine-
ment modes are explored via machine learning methods
to analyze experimental data from the compact, high-
field Alcator C-Mod tokamak. Supervised learning tech-
niques with zero-dimensional data and time-independent
quantities are employed which increases the transferabil-
ity of this approach for real-time confinement mode iden-
tification across separate fusion devices. Multiclass clas-
sification utilizing Gaussian naıve Bayes, logistic regres-
sion, multilayer perceptron (i.e. feedforward neural net-
works), and random forests performed similarly and ob-
tained an average accuracy above 90% using the plasma’s
normalized internal inductance, poloidal beta, total heat-
ing power (ohmic and ICRH), and volume-averaged den-
sity as inputs. Development of a new training confine-
ment database with over 200 distinct shots consisting
of approximately 400 L-, 200 H-. and 100 I-mode peri-
ods extends upon previous databases and the new clas-
sifiers open pathways to explore large-scale comparative
studies, guided experimentation, and increased control
of confinement regimes in fusion plasmas. Cross-machine
validation is important to understand generalizability in
accessing different plasma behaviour, and further explo-
ration of statistical methods and spatiotemporal data are
expected to improve insights into transition physics.
5. ACKNOWLEDGEMENTS
Supported by the U.S. Department of Energy (DOE) Of-
fice of Science Fusion Energy Sciences program under
contracts DE-FC02-99ER54512, DE-SC0014264, and the
Joseph P. Kearney Fellowship.
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075005.[14] O. Meneghini et al 2015 Nucl. Fusion 55 083008.[15] C. Mood 2010 Euro. Soc. Rev. 26 1.[16] K. Nigam et al 2000 Machine Learning 39 2-3.[17] S. O’Halloran 2005 Econometrics for Sustainable Development
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Confinement Regime Identification on C-Mod 11
APPENDIX
Accuracy metrics included in the report are: true positive rate (TPR), true negative rate (TNR), positive pre-
dictive value (PPV), negative predictive value (NPV), and Matthews correlation coefficient (MCC). These are
based on the outcomes of a confusion matrix, i.e. true positive (TP), true negative (TN), false negative
(FN), and false positive (FP), which are produced by applying the ‘one vs. rest’ approach for each L-, H-
, and I-modes. This is an inherently multiclass problem, but can be simply converted into a binary compar-
ison for simpler analysis of accuracy metrics across modes. When dealing with imbalanced datasets, the to-
tal accuracy can misrepresent the overall strength of a classifier, in which case an alternative measure such as
the Matthews correlation coefficient (MCC) which accounts for unevenness in the database may be preferred.
TPR =TP
TP + FN(sensitivity or recall) (A.1)
TNR =TN
TN + FP(specificity) (A.2)
PPV =TP
TP + FP(precision) (A.3)
NPV =TN
TN + FN(A.4)
ACC =TP + TN
TP + TN + FP + FN(A.5)
MCC =TP× TN− FP× FN√
(TP + FP)(TP + FN)(TN + FP)(TN + FN)(A.6)
The results for the different classifiers after 100 cycles are reported below. The results in the main report are based on
utilizing the trained classifiers from 1 of the cycles. The total accuracy (ACC) is based on the cumulative correct and
incorrect classifications for all 3 modes combined. A metric known as the area under curve (AUC) is also reported.
AUC is a measure of the area under a curve known as the receiver operating characteristic (ROC) which is a plot
of the true positive rate as a function of the false positive rate at different threshold cut-off points for a particular
classification decision. A higher AUC indicates the algorithm is learning quicker to distinguish between groups.
TP1
1
FN
0
FP0 TN
Actu
al
Prediction
Table 1: Example of a binary confusion matrix.
12 A. Mathews
GNB LR MLP RFL
-mode
PPV 0.959± 0.003 0.943± 0.004 0.971± 0.004 0.989± 0.001
TPR 0.925± 0.004 0.959± 0.003 0.970± 0.004 0.991± 0.001
TNR 0.867± 0.011 0.808± 0.013 0.906± 0.015 0.964± 0.003
NPV 0.776± 0.009 0.855± 0.010 0.902± 0.011 0.971± 0.002
AUC 0.957± 0.003 0.971± 0.002 0.989± 0.001 0.998± 0.000
MCC 0.763± 0.010 0.783± 0.013 0.875± 0.009 0.957± 0.003
H-m
ode
PPV 0.773± 0.014 0.833± 0.013 0.861± 0.015 0.953± 0.004
TPR 0.775± 0.017 0.752± 0.017 0.841± 0.024 0.936± 0.006
TNR 0.966± 0.003 0.977± 0.002 0.980± 0.003 0.993± 0.001
NPV 0.966± 0.003 0.964± 0.002 0.977± 0.003 0.991± 0.001
AUC 0.953± 0.005 0.966± 0.003 0.986± 0.002 0.997± 0.000
MCC 0.740± 0.014 0.762± 0.014 0.830± 0.013 0.937± 0.005
I-m
ode
PPV 0.723± 0.013 0.817± 0.014 0.913± 0.014 0.983± 0.003
TPR 0.894± 0.019 0.814± 0.021 0.943± 0.012 0.986± 0.002
TNR 0.954± 0.004 0.976± 0.002 0.988± 0.002 0.998± 0.000
NPV 0.985± 0.002 0.975± 0.003 0.992± 0.002 0.998± 0.000
AUC 0.973± 0.003 0.982± 0.002 0.997± 0.001 0.999± 0.000
MCC 0.775± 0.015 0.791± 0.018 0.918± 0.009 0.983± 0.002
ACC 0.903± 0.004 0.917± 0.004 0.951± 0.003 0.984± 0.001
Table 2: Accuracy metrics for validation set.
Confinement Regime Identification on C-Mod 13
GNB LR MLP RFL
-mode
PPV 0.955± 0.015 0.941± 0.016 0.954± 0.014 0.941± 0.015
TPR 0.927± 0.018 0.958± 0.014 0.955± 0.013 0.956± 0.011
TNR 0.855± 0.047 0.803± 0.051 0.848± 0.044 0.799± 0.046
NPV 0.779± 0.046 0.853± 0.043 0.852± 0.043 0.843± 0.039
AUC 0.951± 0.019 0.970± 0.010 0.970± 0.011 0.959± 0.014
MCC 0.758± 0.039 0.777± 0.041 0.804± 0.036 0.769± 0.036
H-m
ode
PPV 0.770± 0.055 0.830± 0.054 0.814± 0.052 0.777± 0.061
TPR 0.774± 0.064 0.751± 0.062 0.851± 0.029 0.763± 0.059
TNR 0.964± 0.010 0.976± 0.008 0.972± 0.009 0.966± 0.010
NPV 0.965± 0.011 0.962± 0.011 0.969± 0.009 0.964± 0.010
AUC 0.952± 0.018 0.965± 0.013 0.967± 0.011 0.949± 0.020
MCC 0.735± 0.046 0.758± 0.047 0.807± 0.035 0.734± 0.046
I-m
ode
PPV 0.720± 0.081 0.807± 0.082 0.831± 0.081 0.826± 0.078
TPR 0.864± 0.064 0.802± 0.077 0.843± 0.073 0.755± 0.076
TNR 0.956± 0.015 0.975± 0.012 0.978± 0.010 0.980± 0.010
NPV 0.981± 0.010 0.974± 0.013 0.979± 0.012 0.968± 0.013
AUC 0.961± 0.031 0.981± 0.008 0.981± 0.022 0.970± 0.019
MCC 0.757± 0.059 0.777± 0.059 0.815± 0.064 0.762± 0.061
ACC 0.900± 0.016 0.914± 0.015 0.923± 0.013 0.909± 0.013
Table 3: Accuracy metrics for testing set.
14 A. Mathews
Variable Description Units Source
shot shot number in Alcator C-Mod logbook - -
id arbitrary unique identification number - -
present mode current mode (L, H, or I) - -
next mode mode at next transition (L, H, I, or end) - -
time current time s -
time at transition time at which next transition occurs s -
btor toroidal magnetic field T MAGNETICS
ip plasma current A MAGNETICS
i beam neutral beam A DNB
p lh lower hybrid power W LH
p icrf net ion cyclotron radio frequency power W RF
p icrf d D-port ion cyclotron radio frequency power W RF
p icrf e E-port ion cyclotron radio frequency power W RF
p icrf j3 J3-port ion cyclotron radio frequency power W RF
p icrf j4 J4-port ion cyclotron radio frequency power W RF
freq icrf d frequency of ICRH at D-port Hz RF
freq icrf e frequency of ICRH at E-port Hz RF
freq icrf j frequency of ICRH at J-port (both J3 and J4) Hz RF
beta N normalized beta - EFIT
beta p poloidal beta - EFIT
beta t toroidal beta - EFIT
kappa elongation (vertical) - EFIT
triang l lower triangularity - EFIT
triang u upper triangularity - EFIT
triang overall triangularity = 0.5 (triang l + triang u) - EFIT
li normalized internal inductance - EFIT
psurfa surface area of plasma on last closed flux surface m2 EFIT
areao cross-sectional area of last closed flux surface m2 EFIT
vout volume of last closed flux surface m3 EFIT
aout minor radius of last closed flux surface m EFIT
rout major radius of last closed flux surface m EFIT
zout z-position of last closed flux surface m EFIT
zmag z-position of magnetic axis m EFIT
rmag r-position of magnetic axis m EFIT
lgap inner gap to primary separatrix m EFIT
rgap outer gap to primary separatrix m EFIT
zsep lower z-position of lower x-point m EFIT
zsep upper z-position of upper x-point m EFIT
p rad radiated power (2π foil) W SPECTROSCOPY::TWOPI FOIL
Confinement Regime Identification on C-Mod 15
p rad core core radiated power W SPECTROSCOPY::BOLOMETER
rsep lower r-position of lower x-point m EFIT
rsep upper r-position of upper x-point m EFIT
zvsin z-position of inner strike point m EFIT
rvsin r-position of inner strike point m EFIT
zvsout z-position of outer strike point m EFIT
rvsout r-position of outer strike point m EFIT
upper gap top gap to primary separatrix m EFIT
lower gap lower gap to primary separatrix m EFIT
q0 safety factor at centre - EFIT
qstar safety factor in cylindrical approximation - EFIT
q95 edge safety factor (at 95% poloidal flux surface) - EFIT
V loop efit loop voltage V EFIT
V surf efit surface voltage V EFIT
Wmhd stored plasma energy J EFIT
dWmhddt time derivative of stored plasma energy W EFIT
cpasma calculated plasma current A EFIT
ssep midplane separation between separatrices m EFIT
P ohm ohmic heating power W -
Dalpha Dα(n = 3 to n = 2) W/(m2 · sr) SPECTROSCOPY
Halpha Hα(n = 3 to n = 2) W/(m2 · sr) SPECTROSCOPY
HoverHD Hα/(Hα +Dα) - SPECTROSCOPY
nLave 04 line-averaged electron density (chord 4) m−3 ELECTRONS:TCI
NL 04 line-integrated electron density (chord 4) m−2 ELECTRONS:TCI
nebar efit volume-averaged electron density m−3 ELECTRONS:TCI
b bot mks B-port divertor pressure mTorr EDGE
e bot mks E-port divertor pressure mTorr EDGE
g side rat G-port midplane external pressure mTorr EDGE
update time time/date at which the row is last updated - -
Table 4: Variables presently available in confinement database.
Feature Mean (µ) Variance (σ2)
βp 0.256 0.015
li 1.366 0.038
ne (m−3) 1.547× 1020 4.158× 1039
Pinput (W/m3) 2.194× 106 2.156× 1012
Table 5: Transformation of input features (i.e. x−µσ ) for applied machine learning method based on training set.