Reconstruction of Multiscale Plasma Dynamics Across Operating Regimes

• Sparse diagnostics and costly simulations leave a gap between accessible measurements and the high-dimensional plasma fields needed to understand dynamics and develop predictive models and control.Source: §1 · p. 1
• Temporal measurement histories add information because accessible signals reflect modes, instabilities, and transport processes governing unobserved plasma.Source: §1 · p. 2
• Prior plasma reconstructions could favor large-scale morphology over dynamically important smaller-scale fluctuations, with sharper transients and high-frequency details harder to recover [18].Source: §1 · pp. 2–3
• SHRED encodes sparse measurement histories recurrently but maps its latent state through a shallow fully connected decoder without explicit multiscale spatial structure [14].Source: §1 · pp. 2–3
• ReMAIN retains recurrent temporal encoding and uses the recurrent representation as a dense prior and FiLM conditioning throughout a multiresolution U-Net [21] [22].U-Net skip connections preserve fine- and intermediate-scale information while deeper levels capture larger-scale structure.Source: §1 · p. 3
• The parametric extension jointly embeds operating condition and measurement history to adapt reconstruction across regimes, and is evaluated alongside a capacity-matched SHRED comparison on six nonlinear PDE systems [4].Source: §1 · p. 3

Recurrent temporal representation and dense prior

• The measurement history is processed by a stacked LSTM whose final hidden state summarizes recent temporal evolution and conditions the full-state reconstruction.That representation also supplies the dense temporal prior used by the reconstruction network.Source: §2 · p. 4

Dense prior and measurement embedding

• The reconstruction input combines the dense temporal prior, current sparse observations, their sampling mask, and fixed positional-encoding channels.The mask distinguishes unobserved locations from physical measurements equal to zero.Source: §2 · p. 4

Multiresolution convolutional U-Net representation

• A three-level encoder and bottleneck build coarse-scale features, while U-Net skip connections retain fine- and intermediate-resolution features for corresponding decoder levels.Each downsampling step reduces feature resolution and increases channels, enlarging the effective receptive field of deeper representations.Source: §2 · p. 5

FiLM conditioning by the recurrent state

• The recurrent state conditions feature maps through learned channel-wise FiLM transformations at each encoder level, the bottleneck, and each decoder level.Each resolution has its own learned scale and shift parameters, so the shared temporal context can influence feature channels differently across scales.Source: §2 · p. 6

Relation to baseline Shallow Recurrent Decoder (SHRED)

• Unlike SHRED’s shallow fully connected decoder, ReMAIN uses a multiresolution convolutional decoder to convert the recurrent representation into the reconstructed state.The recurrent history encoder is retained; the principal change is how its representation is converted to the state.Source: §2 · pp. 6–7

Parametric ReMAIN

• The parametric extension embeds operating parameters separately, concatenates that embedding with the final LSTM state, and uses the joint representation for the dense prior and U-Net FiLM conditioning.This conditions reconstruction on both observed temporal evolution and the prescribed operating parameter.Source: §2 · pp. 7–8

3 Comparison of ReMAIN and SHRED using numerical PDE benchmark problems

• The six benchmarks are one-dimensional nonlinear PDE systems, with 300 trajectories per system split by trajectory so test trajectories are unseen during training.Inputs use histories from 10 sensors, except compressible Navier–Stokes, which uses 15; each target is the full 128-point state.Source: §3 · pp. 8, 10

3.1 Model settings and training procedure

• History length is selected from autocorrelation timescales, and held-out trajectory-level relative L2 errors quantify generalization to unseen dynamical realizations.The history span uses the larger of the 1/e decay time and integrated autocorrelation time, with a 20% margin.Source: §3 · p. 10, §3.1 · p. 11
• Both models use a two-layer, 64-unit LSTM, while ReMAIN replaces SHRED’s fully connected decoder with a FiLM-conditioned U-Net and a dense temporal prior.Their trainable parameter counts are closely matched, supporting a decoder-focused comparison.Source: §3.1 · pp. 10–11

3.2 Results

• ReMAIN consistently reduces reconstruction error across all six PDE benchmarks, with improvements spanning held-out trajectory error distributions.The comparison uses the same recurrent encoding principle and closely matched parameter counts.Source: §3.2 · p. 11, Figure 3 · p. 12
• For compressible Navier–Stokes and FitzHugh–Nagumo, ReMAIN better preserves sharp-front position and steepness than SHRED.For KdV–KS, sine–Gordon, and CGL, it better captures localized peaks, troughs, and finer spatial variations; the forced Burgers difference is less pronounced.Source: §3.2 · p. 12, Figure 6 · p. 13, Figure 7 · p. 13

4 Application of parametric ReMAIN to a plasma problem

• The test uses PIC-generated collisionless E × B plasma data, with electric-field conditions withheld between training regimes and 50-snapshot histories from ten fixed sensors.The held-out conditions lie within the range spanned by training parameters.Source: §4 · p. 17

4.1 Model settings and training procedure

• Training combines an 80-dimensional history–parameter representation with an auxiliary multiscale loss on signed azimuthal increments at four grid separations.The auxiliary loss has weight 0.02 and is ramped up after the first 50 epochs.Source: §4.1 · p. 18

4.2 Results

• Relative L2 error measures pointwise mismatch, while spectral and second-order structure-function errors assess spatial-scale distributions and real-space fluctuation organization.The structure function is evaluated at multiple grid separations; logarithmic spectral error gives sensitivity across the resolved spectral range.Source: §4.2 · pp. 18–19
• Across held-out operating conditions, spectral and structure-function errors indicate preserved fluctuation scales and organization, with no systematic degradation across the tested parameter range.The relative L2 error remains large; Figure 13 reports mean errors and snapshot-to-snapshot standard deviations.Source: §4.2 · pp. 19–21, Figure 13 · p. 20
• Mean spectra and sampled fields show parameter-dependent spatial scales and dominant morphology, with residual local amplitude, position, or phase offsets.Temporal signals also retain local evolution over the test intervals at sampled locations.Source: §4.2 · p. 20, Figure 14 · p. 21

5 Conclusion

• ReMAIN uses recurrent temporal representations as both a dense state prior and scale-specific conditioning throughout a FiLM-conditioned multiresolution U-Net.Its parametric extension jointly embeds operating condition and measurement history.Source: §5 · p. 25
• Across six nonlinear PDE problems, ReMAIN reduced mean relative L2 reconstruction error on test trajectories by approximately 25–80%, depending on the system.It was especially more faithful on sharp fronts, localized extrema, and finer-scale variations.Source: §5 · p. 25
• The parameter-conditioned model reconstructed E × B plasma states while preserving characteristic dynamics, spectral content, and scale-dependent organization across operating conditions withheld from training.Source: §5 · p. 25
• The framework could augment limited plasma diagnostics with inexpensive state estimates, but practical deployment requires informative feasible measurements and uncertainty or out-of-distribution checks.High-fidelity simulations remain essential for training and validation.Source: §5 · p. 25