Simulation Prediction of Heat Transport with Machine Learning in Tokamak Plasmas

Hui Li; Yan-Lin Fu; Ji-Quan Li; Zheng-Xiong Wang

doi:10.1088/0256-307X/40/12/125201

PDF (1673 KB)

Simulation Prediction of Heat Transport with Machine Learning in Tokamak Plasmas

1.
Key Laboratory of Materials Modification by Laser, Ion, and Electron Beams (Ministry of Education), School of Physics, Dalian University of Technology, Dalian 116024, China
2.
State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical and Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, China
3.
Southwestern Institute of Physics, Chengdu 610041, China

More Information |

PDF Get Citation

Corresponding author:
Hui Li, E-mail: huilee@dlut.edu.cn

Zheng-Xiong Wang, E-mail: zxwang@dlut.edu.cn

Received Date: July 05, 2023

Published Date: November 20, 2023

Abstract

Abstract

Machine learning opens up new possibilities for research of plasma confinement. Specifically, models constructed using machine learning algorithms may effectively simplify the simulation process. Previous first-principles simulations could provide physics-based transport information, but not fast enough for real-time applications or plasma control. To address this issue, this study proposes SExFC, a surrogate model of the Gyro-Landau Extended Fluid Code (ExFC). As an extended version of our previous model ExFC-NN, SExFC can capture more features of transport driven by the ion temperature gradient mode and trapped electron mode, using an extended database initially generated with ExFC simulations. In addition to predicting the dominant instability, radially averaged fluxes and radial profiles of fluxes, the well-trained SExFC may also be suitable for physics-based rapid predictions that can be considered in real-time plasma control systems in the future.

FullText(HTML)

Article Text

Artificial intelligence is currently regarded as a Priority Research Opportunity (PRO) in fusion control,^[1,2] which has been effectively implemented in plasma-shape parameter reconstruction, surrogate model construction, and perturbation detection.^[3–5] As high-performance computing resources become more powerful, machine learning approaches are regarded as essential tools for modeling and big data exploitation, including analysis and operation of fusion plasmas.^[6–9] The emergence of these methods provides opportunities and applications for the exploration of complex phenomena. For instance, in micro-turbulence simulation models, the application of neural networks (NNs) for the construction of fast surrogate models has been proposed^[10,11] in recent work.

Much effort has recently been devoted to explaining the underlying physics of turbulent transport in tokamak plasmas. Direct numerical simulations, which have also become powerful tools in turbulence research, have revealed a great deal about it. Within the discharge time scales, time-evolved tokamak simulations have also been performed typically using ‘integrated modelling’.^[12] An essential part of integrated models is the prediction of turbulent fluxes, especially in the tokamak core, where plasma micro-instabilities usually dominate the transport^[13,14] process. For normal calculation of tokamak discharges, nevertheless, it requires huge computational cost to compute fluxes, especially using nonlinear models. Benefiting from the development of machine learning, these approaches have been applied in plasma research in various fields, such as the regression of energy confinement scaling nonlinearly,^[15] database construction,^[16] rapid determination of equilibria parameters,^[17] reconstruction of temperature profile,^[18] analysis of charge exchange spectrum on the JET device,^[19] classification of disruption,^[20] and development of surrogate models.^[21,22]

Motivated by the need for further prediction of transport characteristics caused by micro-turbulence with real-time character, in this work, the properties of electron heat transport, mainly in L-mode plasma conditions, are firstly introduced, using the Extended Fluid Code (ExFC). In particular, the electron heat pinch is found, which can be compared with theoretical and simulation research.^[23] In order to shorten the execution cost of ExFC, ExFC-NN^[24] is extended. Here, a surrogate model based on machine learning has been raised, which is named as SExFC. The time cost of building a machine learning-based surrogate model mainly lies in the training process, while the evaluation process is pretty rapid, which in our case only costs around several microseconds. SExFC can predict the type of dominant turbulence [including pure ion temperature gradient (ITG), pure trapped electron mode (TEM), as well as mixed ITG and TEM], radially averaged fluxes [〈Q_e〉_r, 〈Q_i〉_r and 〈Γ〉_r], and radial profiles of perturbations [ ${\tilde{T}}_{i} (r)$ , ${\tilde{T}}_{e} (r)$ , and $\tilde{n} (r)$ ], corresponding radial profiles of fluxes [Q_i(r), Q_e(r), and Γ(r)], and so on.

ExFC is a recently developed global-type transport gyro-Landau fluid code containing the 3-dimensional finite difference approach. ExFC aims to investigate the physics of flux (external source terms)/gradient-driven turbulence in tokamak plasmas with multi-scale and multi-mode properties. The functions are simulated as follows:^[25–28]

$\begin{array}{l} d_{t} n_{e} = & - ω_{d t e} (n_{0} φ - T_{e 0} n_{e} - n_{0} T_{e}) + D_{n} \nabla_{⊥}^{2} n_{e}, \end{array}$

(1)

$\begin{array}{l} d_{t} T_{e} = & - T_{e 0} ω_{d t e} [(ς - 1) (φ - τ_{e n} n_{e}) - (2 ς - 1) T_{e}] \\ - (ς - 1) \sqrt{8 m_{e} T_{e 0} / m_{i} π} | \nabla_{∥} | T_{e} + D_{Te} \nabla_{⊥}^{2} T_{e}, \end{array}$

(2)

$\begin{array}{l} d_{t} Ω = & - T_{i 0} a (\partial_{r} \ln n_{0} + \partial_{r} \ln T_{i 0}) \partial_{θ} \nabla_{⊥}^{2} φ \\ + f_{c} a \partial_{r} \ln n_{0} \partial_{θ} φ - \nabla_{∥} υ_{∥} \\ + ω_{d i} [(1 + f_{c} / τ) φ + T_{i} + f_{t} τ_{i n} n_{e}] \\ + f_{t} ω_{d t e} (φ - T_{e} - τ_{e n} n_{e}) + D_{U} \nabla_{⊥}^{2} Ω, \end{array}$

(3)

$\begin{array}{l} d_{t} υ_{∥} = & - \nabla_{∥} T_{i} - f_{t} τ_{i n} \nabla_{∥} n_{e} - (1 + f_{c} / τ) \nabla_{∥} φ \\ + D_{υ} \nabla_{⊥}^{2} υ_{∥}, \end{array}$

(4)

$\begin{array}{l} d_{t} T_{i} = & - T_{i 0} (ς - 1) \nabla_{∥} υ_{∥} + T_{i 0} ω_{d i} {(ς - 1) [(1 + f_{c} / τ) φ \\ + f_{t} τ_{i n} n_{e}] + (2 ς - 1) T_{i}} \\ - (ς - 1) \sqrt{8 T_{i 0} / π} | \nabla_{∥} | T_{i} + D_{Ti} \nabla_{⊥}^{2} T_{i}, \end{array}$

(5)

in which n_e is the normalized electron density, T_e is the electron temperature, Ω is the vorticity defined as

$Ω = [f_{c} n_{0} (φ - 〈 φ 〉)] / T_{e 0} - \nabla_{⊥}^{2} φ$ , υ_|| is the parallel ion velocity and T_i is the ion temperature. Furthermore, electrostatic potential averaged around the magnetic surface is represented by the symbol 〈φ〉. It is mentioned that vorticity contains both electron and ion temperature which means TEM and toroidal ITG modes are coupled. As a result, both ITG and TEM may be included in the multi-mode system. The gyroBohm type is applied as the normalization. The time coordinate is normalized by a/c_s0 (c_s0 = (T_i0/m_i)^1/2, ion sound speed):

$(t \frac{c_{s 0}}{a}, \frac{r}{ρ_{s 0}}, ρ_{s 0} \nabla_{⊥}, a \nabla_{∥}) \to (t, r, \nabla_{⊥}, \nabla_{∥}),$

(6)

and the spatial coordinate is normalized by

$ρ_{s 0} (ρ_{s 0} = m_{i} c_{s 0} / e_{i} B_{0}$ , ion sound gyroradius):

$\frac{a}{ρ_{s 0}} (\frac{n_{e}}{n_{0}}, \frac{T_{e}}{T_{e 0}}, \frac{e φ}{T_{e 0}}, \frac{υ_{∥}}{c_{s 0}}, \frac{T_{i}}{T_{i 0}}) \to (n_{e}, T_{e}, φ, υ_{∥}, T_{i}) .$

(7)

The q profile is displayed as an example in Fig. 1.

Fig. Fig. 1. Schematic of the q profile.

DownLoad: Full-Size Img PowerPoint

The radial profiles are formed:

$n_{0} (r) = n_{c} \exp {- (Δ_{r} / L_{n}) \tanh [(r - r_{0}) / Δ_{r}]},$

(8)

$T_{s 0} (r) = T_{sc} \exp {- (Δ_{r s} / L_{Ts}) \tanh [(r - r_{0}) / Δ_{r s}]} .$

(9)

Here, the subscript label s stands for the species of electrons and ions, the label c corresponds to the plasma center at the magnetic axis. The ratios of temperature and density are displayed as

$τ = T_{e 0} / T_{i 0}, τ_{e n} = T_{e 0} / n_{0}, τ_{i n} = T_{i 0} / n_{0} .$

(10)

The subscript 0 represents the equilibrium quantity; a is the minor radius, and R is the major radius. The inverse aspect ratio is represented as ε = a/R. Operator

$f_{t} = 2 \sqrt{2 r / R} / π$ is the fraction of trapped electrons, then the fraction of passing electrons is f_c = 1–f_t. Operators of trapped electron precession drift and ion magnetic drift are defined as

$ω_{d t e} = 2 ε λ_{t} q r^{- 1} \partial_{ϕ}$ and

$ω_{d i} = 2 ε (\cos θ r^{- 1} \partial_{θ} + \sin θ \partial_{r})$ , respectively. The precession frequency of trapped electrons is λ_t = 1/4+2s/3, in which the magnetic shear is defined as s = rdq/qdr. The other operators are expressed as follows:

$d_{t} f = \partial_{t} f + [φ, f],$

(11)

$[φ, f] = r^{- 1} (\partial_{r} φ \partial_{θ} f - \partial_{θ} φ \partial_{r} f),$

(12)

in which f represents any field variable. The adiabatic compression index is ς = 5/3. The diffusivity terms, D_n = D_Te = D_U = D_υ = D_Ti = 4.8 × 10⁻³, are included to damp smaller-scale fluctuations. Equations (1)–(5) are numerically solved with ExFC in toroidal coordinate (r,θ,ϕ) and the simulation domains are [L_r,L_θ,L_ϕ] ≈ [a,2π,2π]. More details about the code and models are included in our previous work.^[24] In the following, all the results are calculated with usages of the gradient-driven version ExFC.

In our previous research, a credible NN-based model, ExFC-NN, has been developed.^[29] A clear and detailed methodology of the ExFC-NN construction is described in our previous work (Ref. [29]). The electrostatic ITG and TEM turbulence system is taken into account in ExFC-NN. The well-trained ExFC-NN perfectly reproduces and predicts what the ExFC can do while taking less calculation time. Detailed in Table 1, the applications of previous ExFC-NN have been concluded with predictions of the type of dominant turbulence (including pure ITG, pure TEM as well as the mixed ITG and TEM), radially averaged fluxes (〈Q_e〉_r, 〈Q_i〉_r, and 〈Γ〉_r), and radial profiles of perturbations [ ${\tilde{T}}_{i} (r)$ , ${\tilde{T}}_{e} (r)$ , and $\tilde{n} (r)$ ] as well as corresponding radial profiles of fluxes (Q_i(r), Q_e (r), and Γ(r)). The average region is in [0.3, 0.8] and time is also averaged in the nonlinear steady stage. Furthermore, the ExFC-NN has been considered to realize experimental analysis of the HL-2A fusion device which may provide reference for the design of experimental parameters. In this work, more plasma parameters are contained, i.e., the database has been expanded. Additional plasma physics is taken into account as a result. The extended ExFC-NN is named as SExFC with the differences between ExFC-NN and SExFC detailed in Table 1. Most importantly, it is critical to select the appropriate plasma parameters for constructing the dataset. Figure 2 shows the process of constructing the database and model in detail. According to the flow chart, further points are added iteratively. Though in the nonlinear steady state, the simulation results still have slight fluctuations in the nonlinear evolution.

Fig. Fig. 2. Schematic diagram for the construction of simulation results in order to obtain a reliable model.

DownLoad: Full-Size Img PowerPoint

Table 1. Comparisons between ExFC-NN and SExFC.

	ExFC-NN	SExFC
Input	R/L_n,R/L_Te,R/L_Ti,	R/L_n,R/L_Te,R/L_Ti,
parameters	T_i0(r),T_e0(r),n₀(r),ε	T_i0(r),T_e0(r),n₀(r),q(r), s,ε
Output 1	Dominant type of turbulence	Dominant type of turbulence
Output 1	(ITG, TEM, ITG & TEM)	(ITG, TEM, ITG & TEM)
Output 2	Radial averaged fluxes	Radial averaged fluxes
Output 2	〈Q_e〉_r, 〈Q_i〉_r, 〈Γ〉_r	〈Q_e〉_r, 〈Q_i〉_r, 〈Γ〉_r
Output 3	Radial profiles of perturbations and fluxes	Radial profiles of perturbations and fluxes
Output 3	${\tilde{T}}_{i} (r), {\tilde{T}}_{e} (r), \tilde{n} (r), Q_{i} (r), Q_{e} (r), Γ (r)$	${\tilde{T}}_{i} (r), {\tilde{T}}_{e} (r), \tilde{n} (r), Q_{i} (r), Q_{e} (r), Γ (r)$
Output 4	Experimental process	Experimental process
Output 5		Real-time evolution of fluxes and perturbations

| Show Table

DownLoad: CSV

As for the description in the introduction, enhancing the understanding of physical mechanisms of anomalous transport, through both ion and electron channels, is of central importance for achievement of burning plasmas. The transport of heat across flux surfaces in magnetic fusion experiments has been observed in previous research. Accordingly, a thorough comprehension of the mechanisms of turbulent transport and improvement of energy confinement is of great significance from both theoretical and experimental points of view. An anomalous pinch (a net inward flow of particle/energy transport channel) was observed in various devices, such as ASDEX-Upgrade,^[30] Tore Supra,^[31] DIII-D,^[32] and JET,^[33] and attracted particular attention. Particle transport has been researched in our previous work, therefore, in the present research, the phenomenon of electron heat transport is described with simulations. The contour plots of radial averaged electron heat transport 〈Q_e〉_r for given R/L_n = 1 and R/L_n = 9 are separately illustrated in Fig. 3. Directly, the density gradient R/L_n may cause more electron heat transport both in an outward and inward direction. In some parameter regions, 〈Q_e〉_r ∼ 0 is observed. As shown in Ref. [23], the inward heat flows in tokamaks were analyzed. For local quasilinear heat fluxes, the expression was separated as follows:^[23,34]

$q_{j} = - n_{j} χ_{j j} \nabla T_{j} + T_{j} χ_{j n} \nabla n_{j} - V_{j} n_{j} T_{j},$

(13)

in which j denotes species.

Fig. Fig. 3. Contour plots of electron heat transport 〈Q_e〉_r in the plane of (R/L_Ti,R/L_Te) with (a) R/L_n = 1 and (b) R/L_n = 9.

DownLoad: Full-Size Img PowerPoint

However, it is not unique because of the coefficients which depend on the profile scale lengths nonlinearly. The diffusivities and electron velocity are illustrated as well. The strength and direction of heat fluxes are determined through the relative strength of plasma temperature, density profiles as well as other associated scale lengths. Namely, the balance between temperature and density gradient driven flows plays an essential role in the direction of the heat flux. Moreover, electron heat pinch is observed in Fig. 4. A similar conclusion can be found in previous research^[23] in which the heat pinch was determined by the ∇n component. Such a finding indicates a mechanism for heat flux pinch. Also, the results may be valuable for exploring the improvement of energy confinement. However, the heat pinch may require specified conditions, thus further underlying investigation will be needed in future work.

Fig. Fig. 4. Time evolution of radial electron heat flux Q_e with R/L_n = 9: (a) R/L_Ti = 1, (b) R/L_Ti = 9.

DownLoad: Full-Size Img PowerPoint

On the other hand, for a credible machine learning model, further comparisons between SExFC and ExFC are required. Figure 5 shows the regression plot for each of the outputs of the NN ensemble. The regression plot is obtained by feeding the validation data set into the SExFC regression and the ExFC models. The distribution of points along the line with slope unity demonstrates that the NN-based model can predict results of the full ExFC calculation with accuracy. The training data of radial averaged electron heat flux (〈Q_e〉_r) is considered with the coefficient of determination metric ℜ² = 0.97. A further comparison of 〈Q_e〉_r between SExFC predictions and ExFC simulations is displayed in Fig. 6. By giving random plasma parameters selected out of the databases, the prediction results of SExFC are approximative to the simulation results of ExFC. The primary characteristics of transport can be captured by the SExFC.

Fig. Fig. 5. Regression histograms of electron heat flux 〈Q_e〉_r comparing between SExFC model and ExFC simulations.

DownLoad: Full-Size Img PowerPoint

Fig. Fig. 6. 〈Q_e〉_r as a function of random parameters selected out of databases comparing between SExFC predictions and ExFC simulations. The dashed line represents the 〈Q_e〉_r ∼ 0.

DownLoad: Full-Size Img PowerPoint

Finally, in this work, a framework for a machine learning-based surrogate model is established and constructed, which is for gyro-fluid electrostatic toroidal plasma simulations with a global effect. The surrogate model, SExFC, is trained to predict the type of dominant turbulence, radially averaged fluxes, and radial profiles of perturbations in the nonlinear phase, including the electron temperature T_e(r), ion temperature T_i(r) and density n(r), as well as the corresponding electron heat flux Q_e(r), ion heat flux Q_i(r) and particle flux Γ(r). Furthermore, the SExFC may also capture features of special inward transport. With the ability for real-time prediction, the possibility of taking physics-based turbulent transport information from massively parallel simulations into plasma control may come true.

Acknowledgment: This work was supported by the National Natural Science Foundation of China (Grant Nos. 12205035, 11925501, 12275071, and U1967206).

References (34)

References

[1]	Humphreys D, Kupresanin A, Boyer M D, Canik J, Chang C S, Cyr E C, Granetz R, Hittinger J, Kolemen E, Lawrence E, Pascucci V, Patra A, Schissel D 2020 J. Fusion Energy 39 123 doi: 10.1007/s10894-020-00258-1 CrossRef Google Scholar
[2]	Degrave J, Felici F, Buchli J, Neunert M, Tracey B, Carpanese F, Ewalds T, Hafner R, Abdolmaleki A, de Casas D L, Donner C, Fritz L, Galperti C, Huber A, Keeling J, Tsimpoukelli M, Kay J, Merle A, Moret J M, Noury S, Pesamosca F, Pfau D, Sauter O, Sommariva C, Coda S, Duval B, Fasoli A, Kohli P, Kavukcuoglu K, Hassabis D, Riedmiller M 2022 Nature 602 414 doi: 10.1038/s41586-021-04301-9 CrossRef Google Scholar
[3]	Kates-Harbck J, Svyatkovskiy A, Tang W 2019 Nature 568 526 doi: 10.1038/s41586-019-1116-4 CrossRef Google Scholar
[4]	Joung S, Kim J, Kwak S, Bak J G, Lee S G, Han H S, Kim H S, Lee G, Kwon D, Ghim Y C 2020 Nucl. Fusion 60 016034 doi: 10.1088/1741-4326/ab555f CrossRef Google Scholar
[5]	Abbate J, Conlin R, Kolemen E 2021 Nucl. Fusion 61 046027 doi: 10.1088/1741-4326/abe08d CrossRef Google Scholar
[6]	Zheng W, Wu Q Q, Zhang M, Chen Z Y, Shang Y X, Fan J N, Pan Y, Team J T 2020 Plasma Phys. Control. Fusion 62 045012 doi: 10.1088/1361-6587/ab6b02 CrossRef Google Scholar
[7]	Böckenhoff D, Blatzheim M, Hölbe H, Niemann H, Pisano F, Labahn R, Pedersen T S, The W7-X Team 2018 Nucl. Fusion 58 056009 doi: 10.1088/1741-4326/aab22d CrossRef Google Scholar
[8]	Schmidhuber J 2015 Neural Netw. 61 85 doi: 10.1016/j.neunet.2014.09.003 CrossRef Google Scholar
[9]	Rea C, Montes K J, Erickson K G, Granetz R S, Tinguely R A 2019 Nucl. Fusion 59 096016 doi: 10.1088/1741-4326/ab28bf CrossRef Google Scholar
[10]	Citrin J, Breton S, Felici F, Imbeaux F, Aniel T, Artaud J F, Baiocchi B, Bourdelle C, Camenen Y, Garcia J 2015 Nucl. Fusion 55 092001 doi: 10.1088/0029-5515/55/9/092001 CrossRef Google Scholar
[11]	Meneghini O, Smith S P, Snyder P B, Staebler G M, Candy J, Belli E, Lao L, Kostuk M, Luce T, Luda T, Park J M, Poli F 2017 Nucl. Fusion 57 086034 doi: 10.1088/1741-4326/aa7776 CrossRef Google Scholar
[12]	Poli F M 2018 Phys. Plasmas 25 055602 doi: 10.1063/1.5021489 CrossRef Google Scholar
[13]	Doyle E J, Houlberg W A, Kamada Y, Mukhovatov V, Osborne T H, Polevoi A, Bateman G, Connor J W, Cordey J G, Fujita T, Garbet X, Hahm T S, Horton L D, Hubbard A E, Imbeaux F, Jenko F, Kinsey J E, Kishimoto Y, Li J, Luce T C, Martin Y, Ossipenko M, Parail V, Peeters A, Rhodes T L, Rice J E, Roach C M, Rozhansky V, Ryter F, Saibene G, Sartori R, Sips A C C, Snipes J A, Sugihara M, Synakowski E J, Takenaga H, Takizuka T, Thomsen K, Wade M R, Wilson H R, ITPA Transport Physics Topical Group, ITPA Confinement Database and Modelling Topical Group, and ITPA Pedestal and Edge Topical Group 2007 Nucl. Fusion 47 S18 doi: 10.1088/0029-5515/47/6/S02 CrossRef Google Scholar
[14]	Horton W, Hu B, Dong J Q, Zhu P 2003 New J. Phys. 5 14 doi: 10.1088/1367-2630/5/1/314 CrossRef Google Scholar
[15]	Allen L, Bishop C M 1992 Plasma Phys. Control. Fusion 34 1291 doi: 10.1088/0741-3335/34/7/008 CrossRef Google Scholar
[16]	Wakasa A, Murakami S, Itagaki M, Oikawa S I 2007 Jpn. J. Appl. Phys. 46 1157 doi: 10.1143/JJAP.46.1157 CrossRef Google Scholar
[17]	Lister J B, Schnurrenberger H 1991 Nucl. Fusion 31 1291 doi: 10.1088/0029-5515/31/7/005 CrossRef Google Scholar
[18]	Clayton D J, Tritz K, Stutman D, Bell R E, Diallo A, LeBlanc B P, Podestá M 2013 Plasma Phys. Control. Fusion 55 095015 doi: 10.1088/0741-3335/55/9/095015 CrossRef Google Scholar
[19]	Svensson J, von Hellermann M, König R W T 1999 Plasma Phys. Control. Fusion 41 315 doi: 10.1088/0741-3335/41/2/016 CrossRef Google Scholar
[20]	Vega J, Murari A, Dormido-Canto S, Moreno R, Pereira A, Acero A, Contributors J E 2014 Nucl. Fusion 54 123001 doi: 10.1088/0029-5515/54/12/123001 CrossRef Google Scholar
[21]	Dong G, Wei X, Bao J, Brochard G, Lin Z, Tang W 2021 Nucl. Fusion 61 126061 doi: 10.1088/1741-4326/ac32f1 CrossRef Google Scholar
[22]	Liu Y Q, Lao L, Li L, Turnbull A D 2020 Plasma Phys. Control. Fusion 62 045001 doi: 10.1088/1361-6587/ab6f56 CrossRef Google Scholar
[23]	Nordman H, Strand P, Weiland J 2001 Plasma Phys. Control. Fusion 43 1765 doi: 10.1088/0741-3335/43/12/310 CrossRef Google Scholar
[24]	Li H, Li J Q, Fu Y L, Wang Z X, Jiang M 2022 Nucl. Fusion 62 036014 doi: 10.1088/1741-4326/ac486b CrossRef Google Scholar
[25]	Beer M A 1995 Ph.D. Thesis Princeton University Google Scholar
[26]	Garbet X, Garzotti L, Mantica P, Nordman H, Valovic M, Weisen H, Angioni C 2003 Phys. Rev. Lett. 91 035001 doi: 10.1103/PhysRevLett.91.035001 CrossRef Google Scholar
[27]	Garbet X, Dubuit N, Asp E, Sarazin Y, Bourdelle C, Ghendrih P, Hoang G T 2005 Phys. Plasmas 12 082511 doi: 10.1063/1.1951667 CrossRef Google Scholar
[28]	Nordman H, Weiland J, Jarmén A 1990 Nucl. Fusion 30 983 doi: 10.1088/0029-5515/30/6/001 CrossRef Google Scholar
[29]	Li H, Fu Y L, Li J Q, Wang Z X 2021 Plasma Sci. Technol. 23 115102 doi: 10.1088/2058-6272/ac15ec CrossRef Google Scholar
[30]	Ryter F, Angioni C, Peeters A G, Leuterer F, Fahrbach H U, Suttrop P 2005 Phys. Rev. Lett. 95 085001 doi: 10.1103/PhysRevLett.95.085001 CrossRef Google Scholar
[31]	Hoang G T, Bourdelle C, Garbet X, Giruzzi G, Aniel T, Ottaviani M, Horton W, Zhu P, Budny R V 2001 Phys. Rev. Lett. 87 125001 doi: 10.1103/PhysRevLett.87.125001 CrossRef Google Scholar
[32]	Baker D R, et al.. 2000 Nucl. Fusion 40 1003 doi: 10.1088/0029-5515/40/5/301 CrossRef Google Scholar
[33]	Garzotti L, Valovič M, Garbet X, Mantica P, Parail V, JET EFDA Contributors 2006 Nucl. Fusion 46 994 doi: 10.1088/0029-5515/46/12/002 CrossRef Google Scholar
[34]	Fröjdh M, Strand P, Weiland J, Christiansen J 1996 Plasma Phys. Control. Fusion 38 325 doi: 10.1088/0741-3335/38/3/008 CrossRef Google Scholar

[1]	ISOBE Mitsutaka, LIU Yi, YANG Jin-Wei, CHEN Wei, JI Xiao-Quan, YUAN Guo-Liang, ZHANG Yi-Po, ZOU Gui-Qing, LI Wei, LEI Guang-Jiu, DING Xuan-Tong, YAN Long-Wen, YANG Qing-Wei, DUAN Xu-Ru, OHDACHI Satoshi, MORITA Shigeru, TOI Kazuo. Beam Ion Transport from Neutron Emissivity Measurement in Deuterium Neutral Beam-Heated Deuterium Plasmas of the HL-2A Tokamak [J]. Chin. Phys. Lett., 2009, 26(10): 105201. doi: 10.1088/0256-307X/26/10/105201
[2]	GAO Hong, YU Qing-Quan. Heat Diffusion across a Strong Stochastic Magnetic Field in Tokamak Plasmas [J]. Chin. Phys. Lett., 2009, 26(6): 065201. doi: 10.1088/0256-307X/26/6/065201
[3]	ZHANG Xing, TAKAHASHI Koji, FUJII Motoo. Charge and Heat Transport in Polycrystalline Metallic Nanostructures [J]. Chin. Phys. Lett., 2008, 25(9): 3360-3363.
[4]	LI Heng-Chao, ZHANG Jia-Shu. Local Prediction of Chaotic Time Series Based on Support Vector Machine [J]. Chin. Phys. Lett., 2005, 22(11): 2776-2779.
[5]	FU Hong-Jun, DING Xuan-Tong, LIU Yi, XU De-Ming. Electron Heat Transport in an HL-1M Tokamak ECRH Plasma [J]. Chin. Phys. Lett., 2003, 20(11): 1973-1976.
[6]	WANG Ai-Ke, QIU Xiao-Ming. Drift Mode Growth Rate and Associated Ion Thermal Transport in Reversed Magnetic Shear Tokamak Plasma [J]. Chin. Phys. Lett., 2001, 18(5): 650-652.
[7]	QIU Xiao-ming, WANG Ai-ke. Pinches-Induced Transport Reduction in Reversed Shear/Optimized Shear Tokamak Plasmas [J]. Chin. Phys. Lett., 1999, 16(9): 662-664.
[8]	QIU Xiao-ming, WANG Ai-ke. Model for Heat Pinch in Reversed Magnetic Shear Tokamak Plasmas [J]. Chin. Phys. Lett., 1998, 15(10): 721-723.
[9]	HUANG Lin, QIN Xiaoming, DING Ning, LONG Yongxing. Heating Finite Beta Tokamak-Plasmas by Alfven Waves [J]. Chin. Phys. Lett., 1991, 8(5): 232-235.
[10]	LIU Min, WANG Long, LIN Ruiyan. SIMULATION OF THE RADIAL TRANSPORT IN CT-6B TOKAMAK DURING GAS PUFFING [J]. Chin. Phys. Lett., 1987, 4(3): 121-124.

Supplements (0)

Cited By

Figures(6) / Tables(1)

Article views (22) PDF downloads (56)

Turn off MathJax

Article Contents

Abstract

Article Text

References

Simulation Prediction of Heat Transport with Machine Learning in Tokamak Plasmas

Abstract

Article Text

References

Related Articles

About This Article

Cite this article:

Catalog

Simulation Prediction of Heat Transport with Machine Learning in Tokamak Plasmas

Abstract

Article Text

References

Related Articles

About This Article

Cite this article:

Catalog

Export File

Citation

Format

Content