TCV plasma emulator: what the present evidence supports

01Dataset, cadence and toroidal modes

The study uses two simulations in one operating regime. Shot 85604 supplies training and same-shot validation windows; shot 85606 supplies a held-out temporal test. The stored frame interval is 3.1319 μs. This supports within-regime short-horizon tests, but it does not establish robustness across equilibria, forcing conditions or discharge regimes.

Toroidal coordinates

The simulated interval is 2π/5 of the full torus and is repeated five times. Therefore only physical toroidal modes n=0,5,10,… occur:

For a field's wedge Fourier coefficient F_k(t,x,y), lag-1 complex coherence measures persistence of complex phase and amplitude direction between adjacent stored frames. The temporal mean is removed before the toroidal FFT:

The non-axisymmetric power share p_k is the energy in one wedge harmonic divided by the energy in all positive harmonics:

Coherence and power answer different questions: a small mode can remain predictable, while an energetic mode can lose exact phase.

The practical target suggested by these truth statistics is phase-sensitive prediction through approximately k=7–8 (n≈35–40), together with statistically faithful treatment of the low-coherence tail beginning around k=16 (n≈80). That tail cannot simply be discarded: vorticity retains 12.3% of its non-axisymmetric power there.

02Emulator pipeline and representation choices

LOLA is a latent diffusion emulator. An autoencoder (AE) compresses a five-field 3D state into a smaller latent grid. A masked diffusion transformer samples future latent frames conditional on one or more known context frames. The decoder returns the sampled latent trajectory to field space. Independent Gaussian diffusion noise produces different ensemble members; in that precise sense, the model is still “rolling dice.”

The fields are electron density Ne, electron temperature Te, ion temperature Ti, electrostatic potential φ and parallel ion velocity Vi. f8, z22 and z44 are experiment labels for codecs with 11, 22 and 44 toroidal latent cells. Their direct real-FFT Nyquist ceilings are k=5, 11 and 22, or physical n=25, 55 and 110.

AE reconstruction VRMSE is computed in physical field units for each field and frame, then averaged:

The “dominant reconstructed Ne edge mode” is the largest positive non-axisymmetric toroidal component in reconstructed edge density. It is an empirical content check, distinct from the Nyquist ceiling, which is only a mathematical capacity limit.

03Two LOLA training procedures

Procedure A: masked-denoising continuation

Jobs 6791660 (z22) and 6791711 (f8 equal-budget control) continue training from an existing best checkpoint. “Continuation” means the denoiser weights begin from a pretrained run rather than random initialization; the script creates a new optimizer and learning-rate schedule. Each example contains five consecutive frames. A contiguous prefix of one to four frames is marked as context, and the model learns to denoise the remaining latent cells at a random diffusion time.

With clean latent trajectory x, Gaussian noise z, diffusion time t uniformly sampled on [0,1], and schedule coefficients α_t and σ_t, the corrupted input and denoising objective are

Here m is the context mask, μ_θ is the denoiser mean prediction, v_t,i is the fixed schedule variance used as a weight, and stopgrad means that weight receives no gradient. Context cells are pinned to √(α_t²+σ_t²)x. These runs use no CRPS or spectral term. Both use 40 continuation epochs, 1,024 training examples per epoch, batch 8, AdamW at 5×10⁻⁵ with cosine decay, and gradient clipping at 1.

Procedure B: sampled rollout-CRPS fine-tuning

Job 6791994 initializes the denoiser from the pretrained f8 state_best, freezes the autoencoder and updates the entire denoiser at a lower learning rate of 10⁻⁵. This is a fine-tune: it asks whether changing the training objective can reshape an existing sampler while holding the representation fixed. A from-scratch retrain would also change initialization and representation learning, making that narrow causal comparison harder.

Each training example samples M=4 diffusion members. Every member uses two composed five-frame windows with one-frame overlap: four new frames per block and eight predicted frames total. Each block uses 16 reverse-diffusion steps. Gradients pass through every reverse step and through the carried state between blocks. The 40 epochs contain 128 examples each, with batch 2 and four-step gradient accumulation for an effective batch of 8; toroidal-roll augmentation is enabled.

For predictions x^(m), truth y, ensemble size M and α=0.95, define ε=(1−α)/M=0.0125. The index i below runs over scored latent cells:

A rewards accuracy; D rewards ensemble separation; fair CRPS uses ε=0. bandCRPS applies a toroidal FFT, filters a band, divides by that truth band's detached RMS, evaluates afCRPS and averages bands. The requested edges [1,6,16] meet an f8 real-FFT grid containing only k=0,…,5, so the effective non-axisymmetric band is k∈[1,6). This objective cannot directly score k=7 (n=35) because that component is absent from the f8 latent.

04Held-out rollout accuracy and calibration

All rollout errors below are measured in standardized model space: Ne is log-transformed and then standardized; Te, Ti, φ and Vi are standardized directly. Each standardized field and grid cell receives equal weight. These values are dimensionless and should not be read as physical-unit errors.

For an M-member ensemble at forecast frame t, the evaluator uses ensemble-mean error E_t, ensemble spread S_t, spread-to-error ratio SSR and anchoring gain ρ:

SSR=1 means spread and ensemble-mean error have the same average magnitude under this estimator; SSR<1 is underdispersion. ρ>1 means anchoring lowers mean error. Figure 5 uses a different aggregate validation SSR estimator, so its ≈0.7 values are not a before/after pair with the ≈0.3–0.4 held-out framewise estimator.

The epoch-4 result is modest but real under this gate: all three starts contribute to the means, and both free and anchored error improve relative to the parent. It is not the checkpoint chosen by minimum validation fair CRPS, and all evaluated checkpoints remain underdispersed. The training objective decreases from 0.2416 to 0.2246, while fixed-slice validation fair CRPS and validation SSR do not improve over the parent. The evidence therefore supports a useful early fine-tune checkpoint, but not a general calibration solution.

05ETKF conditioning and sensor experiments

The ensemble transform Kalman filter (ETKF) is a data-assimilation method. At each analysis time it compares synthetic observations from each forecast member with an observed synthetic value. Ensemble anomalies estimate the covariance between measured channels and every latent-state direction. The observation innovation is then mapped into a state update. Multiplicative inflation expands forecast anomalies before the analysis; M is ensemble size and H is rollout horizon in frames.

In compact form, the ensemble-mean update has the Kalman structure

Superscripts f and a denote forecast and analysis, h is the synthetic observation operator, y is the observation and K is estimated from the ensemble covariance and assumed observation error. Narrow or misoriented ensemble covariance can therefore produce an early update that raises global error even when later updates help. This behavior diagnoses covariance, localization, observation-error or forward-model mismatch; it does not by itself imply that early sensor samples should be removed.

Figure 9 is a baseline model-conditional information ranking. The potential φ is gauge-dependent: replacing φ(x) with φ(x)+C leaves the electric field and E×B velocity unchanged because their dynamics depend on spatial gradients. A physical follow-up should use a declared gauge, assimilate φ anomalies or gradients, or implement a floating-potential forward model with realistic transfer and noise. The localization settings k_max=2 and 5 correspond to physical n≤10 and n≤25; a test that includes n=35 requires k_max≥7 and a representation capable of k=7.

06Physics validity beyond aggregate error

Low standardized RMSE is useful but insufficient for the research goal. A trustworthy edge/SOL emulator should also preserve the relationships that drive transport: spectral power in the coherent band, density–potential phase, radial E×B particle flux, intermittency and the direction and speed of localized structures.

07Evidence table and proposed sequence

1. Complete the missing comparison. Evaluate z22 job 6791660 on shot 85606 with the same three starts, M=64, H=48 and stated ETKF protocol.
2. Adopt an explicit codec gate. Require edge/SOL reconstruction and phase fidelity through k=7 (n=35), alongside physical-unit reconstruction error.
3. Build a middle, band-aware representation. Test 16–22 toroidal latent cells or an explicit spectral branch, with reconstruction weight through k=7–8; use z44 as an upper-capacity reference.
4. Separate predictable and stochastic targets. Use phase-sensitive rollout losses through n≈35–40, and distributional scores for the n≥80 tail, including vorticity statistics.
5. Align selection with deployment. Evaluate checkpoints on absolute held-out free and anchored errors, calibration and physics gates; do not select on gain alone.
6. Upgrade sensor realism. Declare observation operators, gauge handling, transfer functions, spatial averaging, correlations and noise for GPI, reflectometry and target probes.
7. Expand the data. Add shots or operating points before interpreting architecture or diagnostic rankings as robust rather than within-regime.

08Definitions and reading guide

09Provenance, limitations and reproducibility

Primary method references

1. BOUT++ input options: ZPERIOD defines the number of identical toroidal periods.
2. LOLA: Lost in Latent Space (NeurIPS 2025): latent diffusion for probabilistic physical-system emulation.
3. Probabilistic Retrofitting of Learned Simulators (2026): proper-scoring-rule retrofits for deterministic simulators; this motivates the separate comparison branch in Figure 2.