Quantitative modeling of inward particle transport in linear plasma device using machine learning techniques Open Access

The global energy crisis has driven the pursuit for sustainable alternatives to fossil fuels, with nuclear fusion emerging as a promising solution offering virtually limitless energy with minimal environmental impact.¹ In fusion devices, such as tokamak, the primary region of fusion reaction is concentrated in the core of the plasma. Thus, continuous and stable operation requires constant particle fueling to maintain plasma core density. Currently, the fueling position of pellet injection and supersonic molecular beam injection is at the edge of the plasma, where the temperature and density are relatively low. However, fuel particles cannot effectively enter the core, which will affect the steady-state operation. The critical challenge lies in controlling how these fuel particles have been transported from the edge to the core plasma region, as this directly impacts the fusion reaction efficiency.² As a result, inward particle transport is critical in achieving magnetic confinement, contributing to peaked density profiles and improving fusion rates in steady-state discharge operations.^3,4 On the other hand, effective helium ash exhaust is also key to achieving steady-state operation. A comprehensive understanding and active controlling the particle transport are crucial for achieving long-pulse, high-parameter, steady-state operation in future fusion devices.⁵ 

Inward particle transport has been observed and studied in various magnetic confinement devices such as stellarator, tokamak, and other linear devices, but the formation mechanism has not been fully understood and remains a subject of ongoing debate.^3,6–16 The complexity of the inward particle transport mechanism arises from multiple interacting mechanisms. Quasi-linear theory of ITG/TEM turbulence, turbulence equipartition principle, and $E×B$ flow shear effect are currently the primary theoretical explanation of inward particle transport.^17–19 

Traditional approaches to studying this phenomenon have relied heavily on experimental observations and theoretical modeling. While these methods have provided valuable insight, they are limited in their ability to capturing the full complexity of plasma behavior, particularly in identifying subtle patterns and correlations in transport mechanisms.³ Machine learning offers a novel approach to this challenge, demonstrating remarkable success in analyzing complex plasma behaviors.

Early applications of machine learning in fusion research mainly focused around prediction of disruption in fusion devices.²⁰ Recent applications of machine learning in tokamak devices, such as EAST, DIII-D, JET, and HL-2A, have significantly advanced our understanding of disruption mechanisms and led to improved predictive capabilities.^5,21–25 Machine learning approach has also been used in simulation to reconstruct poloidal magnetic field profile in FRC configuration fusion devices using LITP diagnostics.²⁶ A recent fusion performance breakthrough in tokamak devices was achieved by using machine learning and real-time adaptive optimization of 3D magnetic fields to achieve the highest fusion performance while successfully suppressing harmful edge instabilities across both DIII-D and KSTAR tokamaks, marking a significant advance for ITER-relevant plasma control.²⁷ These achievements highlight the potential of AI-driven approaches to unravel the complexities of particle transport.

By leveraging artificial intelligence to analyze complex plasma data patterns, we propose a novel approach for investigating correlation between plasma properties and inward particle transport mechanisms. Compared to large magnetic confinement devices, linear devices offer flexible adjustment of experimental parameters and enable high temporal and spatial diagnostics of plasma parameters, making them well suited for studying transport phenomenon. A significant amount of research on plasma transport has been conducted on the linear Peking University Plasma Test (PPT) device.^28,29 Our approach combines neural network modeling with experimental data from the PPT device to identify and quantify the key drivers of inward transport. This study presents a machine learning framework that processes experimental plasma parameters to investigate, predict, and analyze correlations between plasma properties and transport behavior. Our results demonstrate the potential of AI-driven analysis in revealing previously unidentified patterns and correlation in particle transport dynamics, offering new perspectives on this critical aspect of plasma physics.

The dataset used in this analysis comprised of helicon discharge experiment measurements conducted on the PPT device. Detailed description about the device could be found in previous works.³⁰ Critical experimental parameters that could be controlled, including the magnetic field, neutral gas pressure, and power of radio antenna source, were pre-configured for each discharge. Langmuir probes were used to directly measure plasma density and floating potential at various radial positions within the PPT device. In total, 288 discharges worth of data have been collected as our analysis dataset.

Key variables of plasma properties were derived from these Langmuir probe measurements. The selected variables utilized in this analysis were chosen for their significance in determining the property of the confined plasma. The selected variables include plasma density, plasma density gradient, floating potential, vorticity, vorticity gradient, radial electric field, RF, RS, and most importantly, and particle flux. The detailed characteristics of these variables can be seen in Table I, and a general feature overview graph can be seen in Fig. 1.

In our problem setting, we are primarily focused on plasma behavior along the 60 mm long radial axis of the PPT device acquired from Langmuir probes. Our neural network addresses one overarching predictions task: a regression task to predict particle flux. This prediction task would be broken down into two portions. The first prediction task is to predict particle flux given a set of plasma features shown in Table I. The second prediction task is to predict particle flux given a set of plasma properties, which includes a modified plasma density gradient, and subsequently calculated plasma density, to simulate a change in the particle transport barrier.

The key input features listed in Table I were chosen based on their relevance to particle transport dynamics. The dataset was divided into training, validation, and test sets for model evaluation. The training and validation sets include all discharges, or shots, with magnetic field of 660, 920, 1200, 1450, and 1700 G, with 80% of this dataset used as the training set and 20% used as the validation set, chosen uniformly at random with a fixed seed. The test set is comprised of shots with magnetic field of 530, 790, 1050, 1310, 1580, and 1840 G, used to assess model generalization and predictive accuracy under novel conditions.

The input layer received 11 features, as described in Table I, and the output layer produced a single continuous value representing the predicted particle flux. The model was trained for 140 epochs, and early stopping was implemented based on the validation loss to prevent overfitting. The structure of the NN is illustrated in Fig. 2.

Once the neural network has been trained, we examined the model's capabilities in determining feature importance in the region of inward particle transport and compared the findings with auto-power spectra analysis from the Langmuir probe data. A plasma density modulation analysis was then performed, and $∇n$ and n of different shots were modified, amplifying the signal with different intensity, and feed this modified feature along other features into the NN and evaluate the resulting particle flux behaviors, as shown in Fig. 2.

The goal of our deep learning network is to predict particle flux from a set of plasma behavior inputs across different magnetic fields. The model shows high fidelity in predictions most magnetic fields, as demonstrated in Fig. 3. A robust model performance in this operational range is indicated by the close alignment of the predicted particle flux with the experiment values under the majority of field regimes.

Figure 3 also indicates some accuracy limitations at B = 790 to 1310 G. At B = 790 G, the model shows slight inaccuracy in predicting the position of maximum inward flux. While the model can identify the occurrence and dissipation points of inward particle flux at B = 1310 G, it struggles to accurately predict the precise behavior within the inward flux region.

The neural network's successful prediction of particle flux behavior provides significant insight into the mechanisms of inward particle transport in linear plasma devices. To gain a deeper insight into the model's decision-making process and how the neural network determines the feature interactions under different magnetic field conditions, we conducted a feature importance analysis using SHapley Additive exPlanations (SHAP). SHAP quantifies the contribution of each input feature to the predictions of the model, offering a more interpretable understanding of how individual features influence the outcome of the overall predicted particle flux. Given the high-dimensional nature of our data, we took precautions to manage computational load and memory usage. We reduced the size of background dataset for the SHAP analysis by applying a sampling method, retaining a representative subset of 100 samples from our training dataset. This reduction ensures that our SHAP computations remain computationally feasible without sacrificing significant information regarding the underlying data distribution. The SHAP analysis reveals a hierarchical importance of plasma parameters that both confirm the existing theoretical frameworks and suggest new perspectives on transport mechanisms.

Figure 4(a) presents the SHAP value bee swarm graph, illustrating the overall feature impact on the model across all shots. Features are vertically ordered by their overall importance to the model's predictions. Each feature has their values color-coded, with red indicating higher values and blue indicating lower values. The horizontal axis represents SHAP values, showing both magnitude and direction of each feature's impact on the predicted particle flux. Positive SHAP values indicate that the feature is pushing the prediction above the base prediction value in the positive direction, while negative values show that the feature is pushing the prediction below the base prediction value in the negative direction. This visualization allows for a comprehensive understanding of how each feature contributes to the model's output, considering both the feature's value and its impact on predictions.

Key findings from our SHAP analysis of the prediction along the full particle flux include the following: (1) Magnet field emerged as the most influential feature in predicting particle flux. This aligns with plasma confinement theory, where stronger magnetic fields typically lead to improved particle containment. (2) Plasma density is ranked as the second most important feature. (3) Vorticity was ranked as the third most influential feature for prediction across all shots. However, The SHAP analysis reveals that plasma density and vorticity exhibits a complex, bidirectional relationship, where both high and low vorticity values can contribute either positively or negatively to predictions, without an overall trend.

Radial electric field, $PN$⁠, and $PRF$⁠, on the other hand, exhibits distinct bidirectional relationships, with neutral gas pressure and $PRF$'s effect being the most apparent, where larger $PRF$ will lead to increased inward flux contribution. This observation is validated in Sec. III D.

Particular attention was paid to the point of maximum inward particle transport, corresponding to the lowest flux point on the curve. The SHAP values at the lowest flux point of the particle flux curve from shot 23 767 were then computed, where magnetic field is 1450 G, and the particle behavior is relatively stable as seen from the prediction in Fig. 3. The SHAP waterfall plot in Fig. 4(b) reveals that the NN model determines that vorticity has the most significant positive impact on the prediction, pushing it below the base value toward the prediction, followed by plasma density and RS. On the other hand, $PRF$ and vorticity gradients are the only values at this point, which pushes the prediction in the outward direction for this specific shot.

The SHAP analysis was then extended across the full particle flux curve of the same discharge $#$ 23 767 to gain a comprehensive understanding of the factors influencing particle transport. This approach allows us to observe how feature contributions evolve along the radial distance according to our deep learning model, providing potential insight into the changing dynamics of particle transport.

As shown in Fig. 5, under the same magnetic field of 1450 G, the contribution of inward particle flux at 35 mm $≤$ r $≤$ 50 mm is primarily given by vorticity and plasma density, with the plasma density gradient, RS, and magnetic field to follow. However, the behavior of SHAP feature contribution at the minimum flux point is different across magnetic field. Subsequently, we analyzed the SHAP value of each feature at the lowest flux point across the full range of experiment magnetic confinement from 530 to 1840 G to study the change in feature contribution for inward particle flux in Fig. 6. It can be seen that the model determines that the contributor of the formation of the highest inward particle flux point has a clear transition with the increase in the magnetic field strength.

The SHAP analysis was extended to examine the feature contribution of the lowest flux point across all magnetic fields, shown in Fig. 6. The analysis reveals distinct patterns of feature importance across different magnetic confinement regimes, demonstrating a complex evolution of particle transport mechanism. Three magnetic field regimes were proposed to categorize the plasma behaviors in Fig. 6(b), with corresponding auto-power spectra feature demonstrations in Fig. 6(a). A much more detailed view of the transition between auto-power spectra graphs across different magnetic field strength observed on the PPT device can be found in the  Appendix.

Low field regime (530–790 G): Magnetic field was the primary contributor to inward particle transport, followed by plasma density. As the magnetic field increases, its influence on inward particle flux gradually diminishes, while the impact of density remains largely unchanged. Referencing the auto power spectrum graph for plasma density in Fig. 6, there is around 5 kHz coherent mode with long radial coherent length and a series of discrete modes around r = 50 mm.

Intermediate field regime (920–1200 G): Vorticity transitioned with magnetic field and plasma density to become the primary contributor to inward particle transport, driving the prediction value from the base prediction average to the lowest particle flux point. In this region, we can see that the magnetic field, plasma density, and vorticity gradient all exhibit a decrease in their contribution toward inward particle transport. From the autospectrum in the region colored in green, the discrete modes gradually transition into a continuous spectrum. Theoretical studies have found that the formation of coherent modes is related to the mean vorticity gradient.³¹ Therefore, it could be considered that vorticity enhanced inward particle flux through its impact on the coherent modes. The recently proposed topological pinch theory suggests that inward particle flux can be driven by vorticity flux.³² The impact of vorticity on inward particle flux has attracted significant attention, warranting further investigation to understand the underlying physical mechanisms driving particle flux formation.

High field regime (1310–1840 G): Seen in region colored in blue in Fig. 6(b), the third distinct feature contribution pattern would be from 1310 to 1840 G, where a sharp decrease in vorticity's contribution toward inward particle transport, with plasma density's inward contribution steadily decreasing. Correspondingly, the coherent modes in this region also weaken. The SHAP analysis results in this region suggest a complex interplay between these features, highlighting the need for further investigation to disentangle and better understand their contributions to inward particle transport.

These regime-dependent variations in feature importance reveal a more nuanced picture of plasma transport than previously understood. The transitions between different dominate modes suggest that particle transport mechanisms evolve systematically with magnetic field strength, rather than following a simple linear relationship. This understanding provides new insight for optimizing plasma confinement strategies and validates the effectiveness of machine learning approaches in identifying complex plasma physics phenomena.

Inward particle transport is associated with the formation of peaked density profiles, which is frequently observed during L–H transitions in toroidal devices. This establishes a physical link between inward particle transport and density gradients. We leverage this characteristic to validate our model. An increase in RF power would enhance plasma density gradient shown in Fig. 7(a), which has also been seen in other linear devices like PISCES-RF and LEAD.^33,34 The selection of RF power as a control parameter from its direct influence on plasma density and density gradient.

The experimental results in the PPT device shown in Figs. 7(a) and 7(c) demonstrate a positive correlation between enhanced density gradients and amplified inward particle flux, consistent with theoretical prediction. Using this correlation, a plasma density modulation analysis was conducted to validate our model. $∇n$ profile is artificially modulated [Fig. 7(b)], and the corresponding n profile is calculated. Then, the modulated density and density gradient profile are input to our model with other profiles unchanged. The predicted particle flux from the modulated density is shown in Fig. 7(d). In the radial region of 40–50 mm, which corresponds to the peak of inward particle transport, we found that a positive correlation between density gradient $∇n$ and inward flux. Outward transport remained largely unchanged under these changes.

The consistent patterns observed in feature importance across different magnetic confinement regimes suggest a universal mechanism underlying transport barrier formation. This challenges the conventional view that transport mechanisms fundamentally change across different confinement regimes. Instead, our results indicate a continuous evolution of transport dynamics, with the relative importance of different parameters shifting gradually over experimental conditions rather than changing abruptly.

Our model's ability to reproduce experimental results through plasma density gradient modifications demonstrates its practical utility and physical accuracy. The successful replication of transport barrier enhancement through parameter adjustment validates both our machine learning approach and the underlying physical assumptions. This validation is particularly significant as it bridges the gap between computational predictions and experimental observations.

While our machine learning approach successfully captured key aspects of particle transport behavior, several limitations in the current implementation present opportunities for future improvements. The simplicity of our neural network architecture, utilizing only linear layers, potentially limits the model's ability to capture more complex, non-linear plasma interactions. While this approach provided clear interpretability and computational efficiency and served sufficiently as a proof-of-concept model, more sophisticated architecture could potentially reveal additional plasma dynamics patterns with even larger datasets.

Several promising directions for future work could address these limitations. The integration of more sophisticated neural network architecture, such as convolutional neural networks, could lead us to capture spatial correlations of plasma features, and the implementation of recurrent neural networks would allow for temporal plasma behavior analysis. The development of hybrid models combining the two neural network architectures could simultaneously capture both special and temporal dynamics, providing a more complete picture of inward particle transport phenomena.

Expanding our experimental dataset would strengthen the model's predictive capabilities. With our work demonstrating the significance of vorticity's role in our model, we plan to modify the potential via biased endplates in order to actively control vorticity profiles. The introduction of these data to the model will allow for a more systematic examination of vorticity's impact on the underlying transport mechanisms. The incorporation of larger experimental datasets, such as the extension of the model to include broader operational parameter ranges, introducing temperatures as new variables within the model, and the consideration active control methods such as impurity injection, would provide more avenues to understand the established features and their effects on the mechanism of inward particle transport.

Novel analysis methods, such as the implementation of graph neural networks, could significantly expand our analytical capabilities by processing auto power spectrum data to capture frequency-domain dynamics, analyzing high-speed camera footage to incorporate visual plasma behavior patterns, and identifying complex interactions between multiple plasma parameters simultaneously.

These improvements would not only address current limitations but potentially reveal new insight into plasma transport mechanisms that are currently beyond our model's reach. The integration of more sophisticated neural network architectures with expanded datasets could provide a more comprehensive understanding of particle transport phenomena in linear plasma devices.

In this study, we proposed a novel deep learning approach to investigate the correlation between plasma properties and the inward particle transport mechanism in the PPT device. We constructed a deep learning network that successfully models particle flux within the PPT device given key plasma parameters. The network identified features importance patterns across different magnetic field regimes, as seen in Fig. 6, suggesting a continuous evolution of transport dynamics. The magnetic field, plasma density, and vorticity were the primary contributors to the inward particle flux.

The model has suggested three stages at three different magnetic field regimes, each with unique transport characteristics. In the low field regime, magnetic field and plasma density drive inward particle flux. In the intermediate field regime, the influence of vorticity on inward particle flux is significantly enhanced. In the high field regime, vorticity and plasma density continue their drive of inward particle flux at a decreasing rate, while magnetic field become a driver of outward particle flux. The model was validated through plasma density modulation analysis, where the simulation in Fig. 7 also reflected the observation in the discharge experiment that increasing $PRF$ will lead to an increase in inward particle flux, given optimal and stable discharge conditions.

This study demonstrates the potential of machine learning as a powerful tool for plasma physics research, offering both practical insight for experimental operations and theoretical understanding of complex plasma phenomena. The combination of data-driven analysis with physical validation provides a robust framework for future investigations of plasma transport mechanisms.

This work was carried out at the State Key Laboratory of Nuclear Physics and Technology at Peking University. The authors acknowledge the support and facilities provided by the Department of Physics, Peking University. The authors would also like to acknowledge the support of National Key Research and Development Program of China (No. 2022YFA1604600) and National Natural Science Foundation of China Youth Fund Project (No. 12405250).

The authors have no conflicts to disclose.

Y. Zhou: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Z. Zhang: Data curation (equal); Investigation (equal); Methodology (equal); Resources (equal); Software (equal); Writing – review & editing (equal). T. Xu: Conceptualization (supporting); Data curation (equal); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Writing – original draft (supporting); Writing – review & editing (lead). R. Yuan: Data curation (supporting); Resources (supporting); Writing – review & editing (supporting). G. Chen: Data curation (supporting); Resources (supporting); Writing – review & editing (supporting). C. Xiao: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Supervision (equal); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Figure 8 illustrates the distinct features of the auto-power spectrum from 400 to 2000 G. The pattern at the low field regime (400 to 790 G) can be seen in reference to the first general mode in Fig. 6.

The pattern of intermediate field regime (920 to 1200 G) could be seen as the transition to a continuous spectrum at r = 40 mm.

In the high field regime starting at 1310 to 2000 G, it can be seen that the new vertical feature is found at and around r = 40 mm, where it is also different in behavior from the 1700 G graph, which indicates another potential mode shift.