Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch

In high-confinement-mode (H-mode) tokamaks, the steep-gradient pedestal region is critical for plasma confinement. Understanding ion-temperature-gradient (ITG) drift-wave instabilities requires solving for both a complex eigenfrequency and a two-dimensional complex-valued mode field. Traditional numerical methods are computationally expensive and struggle when only sparse observational data is available. This paper asks whether a physics-informed neural network (PINN) can accurately perform this joint inverse problem despite the challenges of high-frequency oscillations and strong nonlinear coupling between the field and the eigenfrequency.

The authors propose the CEI-PINN framework, which integrates three key innovations to handle the complexity of the ITG inverse problem:

• Fourier Feature Encoding: To overcome the spectral bias of standard neural networks, the model maps spatial coordinates into a higher-dimensional Fourier space, allowing the network to capture rapid, localized oscillations in the mode field.
• Complex-Valued Feature Propagation: The network architecture is designed to process complex numbers directly, ensuring that the real and imaginary components of the mode field are coupled throughout the learning process.
• Three-Stage Training Strategy: To prevent the model from converging to trivial or compensatory solutions, the training is split into three phases: initial field fitting, eigenfrequency identification (with the field frozen), and final field refinement (with the eigenfrequency frozen). This staged approach stabilizes the joint optimization of the field and the physical parameters.

The CEI-PINN framework successfully recovers the target complex eigenfrequency and reconstructs the 2D complex-valued mode field using only 6% of the full-domain data as sparse supervision. By separating the optimization of the field and the eigenfrequency, the model avoids the instability typically associated with standard PINNs in complex-valued eigenvalue problems. The results demonstrate that the framework is a viable, efficient alternative to traditional numerical eigensolvers for analyzing drift-wave instabilities in fusion plasmas.