Surrogate modeling of drift-reduced Braginskii turbulence with resistivity-conditioned Koopman neural operators

Technical Summary: Surrogate Modeling of Drift-Reduced Braginskii Turbulence with Resistivity-Conditioned Koopman Neural Operators

Problem Statement
Anomalous cross-field transport in magnetic-confinement fusion remains a critical limitation, driven by multiscale boundary plasma turbulence. While two-fluid drift-reduced Braginskii models (implemented in the Global Bragiskii Solver, GBS) provide a physically faithful description of these dynamics, fully three-dimensional, nonlinear, flux-driven simulations are computationally expensive. This cost hinders systematic scans over operating parameters such as plasma density, heating power, or magnetic geometry. Existing machine-learning surrogates have shown promise in reproducing statistical trends at reduced cost, but challenges remain in interpolating across distinct turbulent regimes (e.g., varying resistivity) while preserving short-horizon statistical signatures and ensuring long-horizon dynamical stability.

Methodology
The authors develop Resistivity-Conditioned Koopman Neural Operators (KNOs) as surrogate models for three-dimensional boundary plasma turbulence. The approach is distinct from standard Neural Operators (e.g., FNOs) where parameters are typically input channels; here, the resistivity label is embedded directly into the latent evolution operator.

• Data Generation: Reference data is generated using electrostatic GBS simulations in a toroidal geometry. The study scans the normalized Spitzer resistivity $\nu_0$ across five values: $\{0, 0.01, 1, 3, 10\}$.
• Fieldwise Architecture: Instead of modeling the full coupled state vector, separate KNOs are trained for four scalar observables: plasma density ($n$), electron temperature ($T_e$), electric potential ($\phi$), and vorticity ($\Omega$).
• Koopman Framework: The model learns a short-time evolution map $G_f: (f(t-\Delta t), f(t), \nu_0) \to \hat{f}(t+\Delta t)$.
  • Encoding: A two-frame history is encoded into a latent space via temporal and 3D spatial encoders.
  • Latent Evolution: The core innovation is the latent evolution operator $K_d(\nu_0, \Delta t)$, which is decomposed into a baseline, a resistivity-dependent correction, and a timestep-dependent correction. This allows the operator itself to carry the parameter dependence, enabling interpolation across resistivity regimes within a fixed architecture.
  • Decoding: The evolved latent state is decoded back to physical space using a skip connection to preserve the original history.
• Training Strategy: Models are trained on the five-point resistivity scan and evaluated at a held-out value $\nu_0 = 0.1$ to test interpolation capabilities. The training objective minimizes a weighted combination of prediction loss (MSE over 4 future frames) and auxiliary latent-consistency terms.

Key Results
The surrogates were evaluated at the held-out resistivity $\nu_0 = 0.1$ using pointwise, spectral, and reduced-diagnostic metrics.

1. Short-Horizon Fidelity:

   • Electron Temperature ($T_e$) and Density ($n$): Achieved near-unity agreement ($R^2 > 0.999$ and $R^2 > 0.99$, respectively) with low relative $L_2$ errors.
   • Electric Potential ($\phi$): Highly accurate ($R^2 > 0.99$), though errors showed coherent structures along the separatrix.
   • Vorticity ($\Omega$): Showed the largest discrepancies ($R^2 \approx 0.76$). This is attributed to its nature as a derivative quantity and the fact that it is predicted independently from $\phi$, failing to enforce the generalized polarization relation (Eq. 8) that couples them in the physics.

2. Spectral and Physical Diagnostics:

   • The models successfully reproduced the radial spectral slopes for density and temperature, though the temperature surrogate overestimated large-scale power.
   • The reduced pressure-gradient scale length ($L_{pe}$), a key diagnostic for turbulence drive, was well-reproduced in its broad radial structure.
   • Blob Detection: The surrogate captured the main SOL (Scrape-Off Layer) blob structures and radial locations, with a high Intersection over Union (IoU) of 0.79 for blob masks, despite over-predicting small-scale fluctuation power.

3. Stability and Rollout:

   • Teacher-Forced: When conditioned on ground-truth histories, the models maintained short-time temporal statistics (autocorrelations) well.
   • Autoregressive Rollout: Free-running predictions (where the model feeds its own output back as input) revealed instability. Vorticity errors accumulated rapidly due to a non-contractive learned operator (spectral radius $\rho_\Omega = 2.41$), causing the trajectory to diverge from the GBS reference. In contrast, $n$, $T_e$, and $\phi$ remained stable over a 12-step window.

4. Computational Cost:

   • The inference latency is approximately 124 ms per field on an NVIDIA A100, significantly faster than a full GBS time integration step, making it suitable for rapid exploratory studies.

Significance and Claims
The paper claims that resistivity-conditioned fieldwise KNOs provide useful fast emulators for selected boundary-plasma turbulence diagnostics within a specific parameter range.

• Successes: The work demonstrates that embedding physical control parameters (resistivity) directly into the latent evolution operator allows for effective interpolation across turbulent regimes. The models successfully reproduce key short-horizon statistical features, including spectral trends and reduced pressure-gradient diagnostics, at a fraction of the simulation cost.
• Limitations: The authors explicitly state that stable long-horizon dynamical closure remains unresolved. The fieldwise approach fails to enforce the coupled physics (specifically the polarization relation between potential and vorticity), leading to amplitude errors in vorticity and eventual divergence in autoregressive rollouts.
• Conclusion: The surrogate is presented not as a replacement for the underlying solver for long-term dynamics, but as a tool for rapid conditional emulation of observables, generating approximate initial states for GBS restarts, and facilitating diagnostic dashboard applications. Future work is identified as needing coupled-field training and physics-informed constraints to address the stability and consistency issues.