Deep Learning Models for ADITYA-U MHD Equilibrium

Technical Summary: Deep Learning Models for ADITYA-U MHD Equilibrium

Problem Statement
Magnetohydrodynamic (MHD) equilibrium reconstruction is fundamental to tokamak research, providing essential quantities such as plasma boundary, magnetic axis position, safety factor ($q$), and current density profiles. Conventional reconstruction relies on iterative free-boundary solvers like EFIT, which are computationally expensive and often unsuitable for real-time plasma control or rapid experimental analysis. While machine learning (ML) surrogates have been explored for other tokamaks (e.g., KSTAR, DIII-D, EAST), there was a lack of comprehensive, data-driven equilibrium modeling specifically tailored to the ADITYA-U medium-sized circular tokamak using a large-scale, experimentally relevant synthetic database.

Methodology

• Dataset Generation:
  The authors generated a synthetic free-boundary equilibrium dataset comprising 100,760 cases using the pyIPREQ Grad-Shafranov solver. The input operating space was derived from 766 ADITYA-U plasma discharges (2021–2025), focusing on the flat-top phase with plasma currents ($I_p$) exceeding 100 kA.

  • Constraints: The dataset was restricted to circular limiter plasmas. To ensure physical realism, the authors applied filtering criteria including limits on magnetic axis displacement ($\Delta R \le 0.04$ m, $Z_{ax} \le 0.02$ m), normalized internal inductance ($\ell_i \in [0.7, 1.7]$), and safety factors ($q_0 \in [0.9, 1.5]$, $q_1 > 2.0$).
  • Input Features: Inputs included toroidal field ($B_\phi$), loop voltage ($V_{loop}$), plasma current ($I_p$), poloidal field coil currents, and poloidal magnetic field measurements ($B_\theta$) from 16 magnetic probes.
  • $\beta_p$ Estimation: Since direct diamagnetic loop measurements were unavailable for all shots, a probabilistic linear regression model was developed to estimate the poloidal beta ($\beta_p$) based on operational parameters, with stochastic noise added to simulate measurement uncertainty.

• Deep Learning Architectures:
  Four classes of models were investigated using TensorFlow/Keras, trained on an 80/10/10 split (based on shot labels to prevent temporal leakage):

  1. Dense Neural Networks (DNN): Used for predicting scalar equilibrium quantities (magnetic axis position $R_{ax}, Z_{ax}$, $\beta_p$, $\ell_i$, edge safety factor $q_1$, and flux values $\psi_{axs}, \psi_{lim}$) and an inverse model for estimating coil currents from desired equilibrium states.
  2. PCA-Based Reduced-Order Models: Applied to 1D safety factor ($q$) profiles and 2D poloidal flux ($\psi$) profiles. The high-dimensional profiles were projected onto a low-dimensional latent space (4 modes for $q$, 13 modes for $\psi$) and reconstructed via DNNs.
  3. Convolutional Neural Networks (CNN):
     • 1D-CNN: Predicted the derivative of the safety factor ($q' = \partial q / \partial \rho$) to enforce monotonicity constraints, subsequently integrating to recover the $q$ profile.
     • 2D-CNN: Modeled the 2D poloidal flux ($\psi$) profile directly as an image generation task using transposed convolutions.
  4. Physics-Informed Neural Networks (PINN): Both PCA and CNN models for $\psi$ were formulated as PINNs. They incorporated the Grad-Shafranov residual ($\delta_{GS}$) as a regularization term in the loss function to penalize solutions inconsistent with the underlying equilibrium physics. The models also predicted the current density profile parameters ($\alpha, \beta, \gamma$) to facilitate this calculation.

Key Contributions and Results

• Scalar Parameter Prediction: Dense neural networks successfully predicted key scalar quantities with high accuracy. For instance, the magnetic axis position ($R_{ax}, Z_{ax}$) was predicted with median absolute errors of $\approx 2.33 \times 10^{-5}$ m and $\approx 6.53 \times 10^{-6}$ m, respectively. The inverse model demonstrated the ability to infer statistically preferred coil current combinations for desired equilibrium states, though the authors note the mapping is not unique.
• Profile Reconstruction:
  • Safety Factor ($q$): Both PCA and 1D-CNN approaches achieved high fidelity. The 1D-CNN approach, which utilized monotonicity constraints, showed slightly improved accuracy near the plasma edge compared to the PCA model. However, both methods exhibited larger relative errors near the magnetic axis ($q_0$), where the profile is sensitive to current density details not fully captured by the inputs.
  • Poloidal Flux ($\psi$): The 2D-CNN model achieved superior overall reconstruction accuracy compared to the PCA model. However, the PCA model produced smoother reconstructions that better preserved the dominant equilibrium structure, whereas the CNN model occasionally exhibited localized spatial distortions.
• Physics-Informed Regularization: The inclusion of the Grad-Shafranov residual loss in the $\psi$ models acted as an effective regularization mechanism, discouraging physically inconsistent predictions and ensuring the reconstructed flux surfaces satisfied the governing equilibrium equation.
• Computational Efficiency: The inference time for the largest model (2D-CNN) was approximately 1 ms per equilibrium instance on a multi-core CPU, indicating suitability for real-time applications.

Significance and Claims
The authors claim this work represents the first comprehensive study on machine learning-based MHD equilibrium modeling for ADITYA-U using a large-scale synthetic database derived from experimentally relevant scenarios. The developed framework provides a computationally efficient alternative to conventional solvers, offering potential utility in:

• Real-time plasma control.
• Rapid equilibrium analysis for experimental discharge interpretation.
• Actuator planning and experimental planning.

The study highlights that the constrained operational space of ADITYA-U admits effective reduced-order representations. While the models are not presented as direct replacements for solvers that operate without magnetic probe inputs, they establish a foundation for future physics-guided real-time control and rapid analysis in ADITYA-U operations.

Limitations Noted
The paper modestly acknowledges several limitations:

• The dataset is restricted to circular limiter plasmas and relies on a specific current density profile model.
• No formal linearized MHD stability analysis was performed.
• The $\beta_p$ estimation relies on a simplified probabilistic model due to limited experimental data.
• The models require reliable magnetic probe and loop voltage measurements as inputs, limiting their applicability in scenarios where these diagnostics are unavailable.
• Prediction of profile shape parameters ($\alpha, \gamma$) and quantities sensitive to them ($\ell_i, q_0$) remains challenging, suggesting a need for additional diagnostics in future work.