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This paper claims that matching scale-resolved graph energy, defined by graph-Laplacian frequency bands, is a practical mechanism for improving autoregressive mesh-based forecasting of chaotic flows, and demonstrates this with three scalabl

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2026-08-02 09:03 UTC pith:SWJOGJTK

load-bearing objection Useful and mostly sound extension of BSP to graph Laplacians, but the reported gains rest on test-set hyperparameter selection and thin evaluation; deserves peer review with a demand for a proper validation split. the 5 major comments →

arxiv 2607.19387 v1 pith:SWJOGJTK submitted 2026-07-02 cs.LG cs.AIphysics.comp-phphysics.flu-dyn

Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses

Kanad Sen , Romit Maulik This is my paper
classification cs.LG cs.AIphysics.comp-phphysics.flu-dyn
keywords graph Laplacianspectral losschaotic dynamicsautoregressive forecastingunstructured meshturbulent flowChebyshev filterGLEAM
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the binned spectral power loss, originally defined on regular Fourier grids, to unstructured and irregular meshes by using the graph Laplacian as a generalized frequency operator. It proposes three variants with increasing scalability: exact Graph BSP, Chebyshev BSP, and GLEAM. The argument is that pointwise losses alone cannot prevent spectral drift during long autoregressive rollouts, so explicitly matching predicted and target energy within graph-frequency bands should reduce that drift. Across three turbulent-flow benchmarks, the spectral variants improve long-horizon rollout error, spectral-energy agreement, and preservation of coherent vortical structure relative to a pointwise baseline.

Core claim

On the paper's own terms, the central discovery is that replacing Fourier band-power matching with graph-Laplacian band-power matching is an effective training signal for surrogate forecasting of chaotic dynamics on unstructured meshes. Exact Graph BSP uses the full eigenspace as a reference, Chebyshev BSP substitutes sparse polynomial filters to avoid eigendecomposition, and GLEAM applies low-rank retained-subspace supervision across multilevel graph hierarchies. The experiments show that these losses reduce long-horizon autoregressive drift and better preserve distributional and spectral content of pressure and velocity fields than deterministic pointwise training.

What carries the argument

The central object is the symmetric normalized graph Laplacian of the mesh, whose eigenvectors serve as generalized Fourier modes and whose eigenvalues order graph frequencies by Dirichlet energy. Bands of eigenvalues define projectors that measure how much field energy sits at each spatial scale. The paper compares predicted and target band energies with a relative ratio loss, and approximates the expensive exact projectors with Chebyshev polynomial graph filters; GLEAM instead builds a low-rank spectral embedding (regularized by eigenvalue weighting, so pairwise distances approximate effective resistance) and adds Fiedler-guided pairwise contrast terms across coarse hierarchy levels.

Load-bearing premise

The load-bearing premise is that graph-Laplacian eigenvalues on an unstructured CFD mesh order physically meaningful spatial scales, so that penalizing band-energy mismatch targets the scale content that drives chaotic rollout drift rather than arbitrary graph harmonics.

What would settle it

Take a mesh on which the graph-Laplacian eigenvectors are known not to correspond to physical spatial scales—say a highly anisotropic or weakly connected mesh—and show that adding Chebyshev BSP or GLEAM fails to reduce or even worsens rollout spectral error compared with the pointwise baseline. A cleaner version: on a structured grid encoded as a graph, compare graph-band and Fourier-band energy matching; if the two lead to different training behavior, the physical interpretation of the bands is suspect.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the central claim is right, spectral losses can be added as a modular auxiliary objective to existing graph-based simulators without changing their architecture.
  • Long-horizon autoregressive forecasting of chaotic CFD on unstructured meshes can be stabilized by scale-aware supervision, not just by better pointwise training or rollout exposure.
  • The cost–fidelity hierarchy offers a practical recipe: exact BSP as a reference diagnostic, Chebyshev BSP for fine-level band control, GLEAM for cheap multilevel supervision on coarse graphs.
  • The results suggest that spectral bias in graph neural networks (the tendency to smooth over large scales) can be counteracted by explicit band-energy penalties.
  • Methods that rely on Fourier or structured-grid assumptions can now be ported to irregular geometries via graph spectral theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • One implicit extension is to combine graph-spectral energy losses with rollout-curriculum training: the paper's BFS comparison suggests pushforward exposure and spectral matching may address different failure modes, and the combination could outperform either alone.
  • The graph-band interpretation could be tested more directly on meshes where a physical wavenumber exists (e.g., a structured grid encoded as a graph); if graph bands systematically misalign with Fourier bands, the loss's effectiveness should degrade, which would be a sharp diagnostic.
  • The same band-energy mechanism could apply to other graph-mediated PDE surrogates, such as unstructured ocean or climate models, where long-horizon drift of invariants is a known bottleneck.
  • The Fiedler-guided pairwise contrast term resembles graph-embedding regularizers; a testable hypothesis is that it mainly helps when error is spatially reorganized (recirculation shifting) rather than merely spread across scales.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper proposes a family of graph-spectral losses for autoregressive forecasting of chaotic fluid flows on unstructured meshes. The authors replace the Fourier-band BSP loss of Chakraborty et al. with graph-Laplacian frequency bands, defining an exact Graph BSP loss (Eq. 16), a scalable Chebyshev-filter approximation (Eq. 27), and GLEAM, a multilevel retained-subspace variant with a Fiedler-guided pairwise contrast term (Eqs. 32, 35, 36). They evaluate on EAGLE, a 2D backward-facing step, and the pOnWing benchmark, reporting that the spectral variants improve long-horizon RMSE, spectral-energy metrics, vorticity/pressure PDFs, and pressure-force errors relative to pointwise-trained baselines. The paper also compares against PyNUFFT and Pushforward baselines on BFS. The central claim, stated in §6, is that matching scale-resolved graph energy is a practical mechanism for improving autoregressive mesh-based forecasting on the studied benchmarks.

Significance. If validated, the proposed modular losses would be a genuinely useful contribution: they provide a principled framework for scale-resolved supervision without a Euclidean Fourier basis, and the Chebyshev/GLEAM hierarchy addresses a real computational bottleneck of exact Graph BSP. The manuscript is unusually transparent about implementation details, reports exact eigensystem diagnostics alongside approximate training losses, and makes code available. The mathematical formulation is mostly coherent, and the physical-space improvements (EAGLE RMSE, BFS global/reverse-flow/high-gradient RMSE, pOnWing force error) are evidence that the method is not merely optimizing its own evaluation metric. However, the empirical claims are weakened by test-set hyperparameter selection, single-sequence BFS evaluation without error bars, and the partial circularity of the spectral-error metrics. As it stands, the evidence supports the claim as a proof-of-concept on the specific configurations tested, but not yet as a general practical mechanism with statistically quantified gains.

major comments (5)
  1. [§4.1, Tables 1–2] The representative EAGLE Chebyshev setting (K=16, M=32, Q=256) is selected because it gives the strongest late-horizon spectral preservation on the 118 test simulations, and the best short-horizon RMSE and best long-horizon RMSE rows are also read off the same tables. No independent validation split is used for this selection. Consequently, the reported improvements over Base could reflect tuning to the test set. Please perform hyperparameter selection on a separate validation split (or via nested cross-validation) and report test metrics only for the selected configuration. This is load-bearing for the central claim because the conclusion explicitly relies on the selected K, M, Q.
  2. [§4.2, §4.3, Appendix B.3] The BFS spectral weights are chosen by test performance: λCheb=0.1 is described as giving 'the best long-horizon rollout accuracy' and λGLEAM=10^-4 as the strongest in 'the saved hyperparameter sweep'; the pOnWing weights are similarly selected from VGAE/FMGN sweeps. With no separate validation set, the risk of selection-induced gains is substantial, especially since many of the reported curves use min–max envelopes rather than error bars. I request a validation/test split protocol for all hyperparameter choices and, at minimum, multiple seeds or independent test sequences for BFS and pOnWing to quantify variance.
  3. [§4.4, Tables 2, 8, Appendix B.2.3, Fig. 10(a)] Several headline metrics measure exactly the band-energy mismatch quantities that the proposed losses are trained to minimize: state spectral-energy relative L1 (Table 2), exact normalized-Laplacian spectral-energy relative L1 (Table 8, Fig. 10(a)), and low-band Uy RMSE (Table 3). Improvements on these metrics are partly expected by construction and should not be presented as independent validation of spectral-drift reduction. The paper does include genuinely independent physical-space metrics — EAGLE RMSE, BFS global/reverse/high-gradient RMSE, pOnWing pressure-force error — and these should be made the primary evidence for the empirical claim. Please either demote the spectral metrics to sanity checks or add independent physical diagnostics (e.g., spatial power spectra computed from physical coordinates, vorticity statistics, structure functions) that do not coincide with the training
  4. [§2.1–2.3, Eq. (11), Appendix E.2, Eq. (60)] The load-bearing premise is that graph-Laplacian eigenvalues on an unstructured CFD mesh order physically meaningful spatial scales, so that penalizing band-energy mismatch penalizes the scale content that drives chaotic rollout drift. This is an assumption: the paper does not demonstrate that the graph-spectral bands correlate with physical wavenumbers or with the scales that actually dominate EAGLE/BFS/pOnWing rollout error. The GLEAM design further assumes the coarsened hierarchy preserves the low/mid-frequency subspaces that matter (Eq. 60). Please provide a concrete validation on at least one benchmark, e.g., by comparing graph-band energies of ground-truth fields against physical-space power spectra, or by ablating the loss with eigenvalue-binning schemes that do not respect smoothness ordering. Without such evidence, the 'scale-aware' interpretation remains plausible but unverifie
  5. [§4.2, Table 3, Appendix C, Table 9] BFS is evaluated on a single held-out test sequence. Table 3 is computed from 'fresh autoregressive rollouts' of one test segment, and Appendix C explicitly says the BFS comparison uses 'one held-out sequence' with no error bars. The strong BFS improvements (Base global RMSE 11.36 vs. GLEAM 3.10; low-band RMSE 6.62 vs. 0.65) could be driven by a favorable draw of this particular sequence. Please report results over multiple independent sequences or multiple training seeds, with summary statistics over those runs.
minor comments (5)
  1. [§4.1 and Appendix B.1] The text states the EAGLE dataset has 948 training, 118 validation, and 118 test simulations; Tables 1–2 and the ablation narrative use the 118 test simulations for selection. Clarify whether the 'validation' split was ever used during the study, and if so, where.
  2. [Fig. 1, Fig. 3, Figs. 6–7] Shaded envelopes are described as min–max ranges; these are not error bars and do not convey statistical uncertainty. Add standard errors or confidence intervals over simulations/seeds, or explicitly relabel the envelopes.
  3. [§3.1, Eq. (19)] The Bartlett windows φm(λ) as defined do not form a partition of unity and the band normalizer dcheb_m is proposed in two variants (trace-normalized or replaced by N). State which normalizer was used in each experiment; otherwise the reported band energies are not comparable across rows of Tables 1–2.
  4. [§4.5, Table 5 and Appendix F] The cost table for GLEAM includes 'PℓC' for pairwise terms, but the text says Pℓ is the number of sampled pairs; clarify whether this is per-channel or per-pair cost and define Pℓ consistently.
  5. [Throughout] Several appendix references in the main text are imprecise (e.g., 'Appendix Fig. 18', 'Appendix B.2.3 support the same interpretation'), and the data/code availability statements could include a version/DOI for the code repository. These are minor presentation issues.

Circularity Check

2 steps flagged

Spectral-drift improvements are partly by construction because the reported band-energy metric is essentially the quantity minimized by the loss, and headline hyperparameters are selected on the same test rollouts used for final reporting.

specific steps
  1. self definitional [§4.1, Tables 1–2; Eq. (27), §4.4]
    "L^{cheb} = 1/(MC) Σ_m Σ_c (1 − (E^{cheb}_u(m,c)+ε_s)/(E^{cheb}_v(m,c)+ε_s))^2 ... The state spectral-energy relative L1 metric measures the relative discrepancy between predicted and target spectral-energy distributions over the state channels, so lower values indicate better preservation of scale-resolved energy."

    The training loss is a relative mismatch of Chebyshev band energies; the headline 'spectral-energy relative L1' diagnostic is the same relative mismatch of the same band-energy quantities. A model explicitly trained to equalize those band energies will, by construction, tend to improve that exact metric, so those tables cannot independently evidence 'spectral drift' reduction. The physical-space RMSE and pressure-force metrics are separate and provide independent support.

  2. fitted input called prediction [§4.1, §4.2; Tables 1–2; Fig. 3]
    "We therefore use K=16, M=32, and Q=256 as the representative EAGLE Chebyshev BSP setting in the following diagnostics, because it gives the strongest late-horizon spectral preservation... among the tested weights, λCheb=0.1 gives the best long-horizon rollout accuracy... In the saved hyperparameter sweep, the strongest GLEAM rollout uses λGLEAM=10−4."

    These hyperparameters and spectral weights are chosen by inspecting the same test rollouts that are later reported as the method's gains. The paper does not state that a separate validation split was used for this selection; EAGLE reports training/validation/test splits, but the ablation tables are computed on the 118 test simulations. Reporting the best member of a test-set sweep as a 'prediction' is a selection artifact rather than independent evidence. This weakens, though does not by itself invalidate, the empirical central claim.

full rationale

The graph-spectral construction itself is not circular: the paper defines band-energy losses from the mesh Laplacian, supplies an eigendecomposition-free Chebyshev approximation, and a multilevel GLEAM variant, with ordinary mathematical assumptions (normalized Laplacian spectrum in [0,2], Parseval, low-rank effective-resistance embeddings). No uniqueness theorem or self-citation is used to force the choice of loss; the prior BSP paper is used only as the structured-grid ancestor, and the contribution is tested against Base, Pushforward, and PyNUFFT baselines. The circularity is confined to evaluation. The 'spectral drift' metric reported as the main scale-resolved evidence is essentially the quantity minimized by the loss, so improvement on it is expected by construction. In addition, the representative Chebyshev and GLEAM hyperparameters are selected using the same test rollouts that produce the reported curves and tables, with only min–max envelopes rather than error bars. That is a test-set selection issue that makes the size of the independent RMSE/force gains uncertain. Physical-space RMSE, pressure-force error, PDFs, and the Pushforward/PyNUFFT comparisons are not derivatives of the loss and give genuine, non-circular evidence, so the paper is not fully circular; but its central empirical claim is partly self-confirming as presented.

Axiom & Free-Parameter Ledger

8 free parameters · 6 axioms · 1 invented entities

The method rests on standard spectral graph theory plus a domain assumption that graph-Laplacian bands correspond to physical scales. The main free choices are spectral weights and approximation hyperparameters selected by ablation rather than derived. No new physical entities are introduced; GLEAM is an algorithmic construct whose external validity is not yet established.

free parameters (8)
  • Spectral weight lambda_Cheb = 0.01 (EAGLE), 0.1 (BFS), 1e-2 (VGAE pOnWing), 1e-4 (FMGN pOnWing)
    Chosen by hyperparameter sweeps to optimize rollout metrics; the paper reports that larger weights over-regularize (BFS §4.2). Not derived from physics.
  • Spectral weight lambda_GLEAM = 1e-4 (BFS), 1e-3 (VGAE pOnWing), 1e-4 (FMGN pOnWing)
    Selected by sweep; report says larger GLEAM weights are worse (§4.2).
  • Band count M = 32 (EAGLE), 24 (BFS), 32 (pOnWing)
    Ablation parameter; Table 2 shows M=64 degrades late-horizon spectral error on EAGLE due to window sharpness.
  • Chebyshev polynomial order K = 16 (EAGLE), 12 (BFS), 16 (pOnWing)
    Ablation parameter; chosen as a balance between approximation quality and cost.
  • Quadrature count Q = 256 (EAGLE), 128 (BFS), 256 (pOnWing)
    Ablation parameter for coefficient computation in Eq. (22).
  • GLEAM retained rank r = 32 (BFS), 16 (VGAE pOnWing), 64 (FMGN pOnWing)
    Ablation parameter; higher rank improves late RMSE in BFS but does not always improve exact spectral L1 (Tables 7–8).
  • Pairwise contrast weight rho = 0.2
    Chosen in GLEAM; ablation shows ρ=0.1 and ρ=0.5 are worse in BFS mid-rollout.
  • Stabilization constants epsilon_s, tau, beta = epsilon_s=1e-6; tau=1e-6 or as specified; beta=0.15
    Small constants preventing division by zero and controlling neighbor-mean smoothing; arbitrary but necessary for numerical stability.
axioms (6)
  • standard math Normalized graph Laplacian has spectrum in [0,2] and an orthonormal eigendecomposition L=Phi Lambda Phi^T.
    Used in §2.1 and Appendix A to define graph frequencies and justify the fixed lambda_max=2 in the Chebyshev rescaling.
  • domain assumption Graph Laplacian eigenvectors are a meaningful analogue of Fourier modes on irregular meshes, and eigenvalue order corresponds to physical scale separation.
    Central premise of §§2.1–2.3. If this fails, band-energy matching constrains arbitrary graph harmonics rather than physical flow scales.
  • ad hoc to paper Linear eigenvalue binning (Eq. 11) is a valid default binning for physical scale separation.
    The authors state linear bins are the default operator-defined choice; no comparison to alternative binning or evidence that they match physical scales.
  • domain assumption The graph hierarchy used by GLEAM retains the low/mid-frequency subspaces that dominate rollout error.
    Used in §3.3 and Appendix E.2, Eq. (60). If coarse levels collapse these subspaces, GLEAM supervision misses the physically important scales.
  • domain assumption Relative band-energy mismatch with stabilization constant epsilon_s is a good proxy for physically damaging spectral drift.
    This is the loss in Eq. (16) and the basis of all spectral evaluation; no physical derivation links this scalar metric to rollout error growth.
  • domain assumption Chebyshev polynomial filters H_m approximate exact band projectors sufficiently well at the chosen K and Q.
    Equations (20)–(27). The approximation has finite accuracy; the paper's own M=64 ablation (Table 2) shows sensitivity to window sharpness and leakage.
invented entities (1)
  • GLEAM (Graph Laplacian Energy Alignment for Meshes) no independent evidence
    purpose: Hierarchy-aware multilevel spectral supervision using retained low-rank embeddings and Fiedler-guided pairwise contrast.
    It is an algorithmic construct introduced by this paper; its benefits are measured only on this paper's benchmarks and no external validation or formal general guarantee is provided.

pith-pipeline@v1.3.0-alltime-deepseek · 33642 in / 14596 out tokens · 137844 ms · 2026-08-02T09:03:00.521011+00:00 · methodology

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read the original abstract

Surrogate modeling for high-dimensional nonlinear dynamical systems that exhibit chaos requires mechanisms that preserve not only pointwise accuracy but also the scale-dependent structure of physical fields. Bandwise spectral power losses, such as the binned spectral loss function, provide such supervision on structured grids, where Fourier modes define a standard frequency decomposition. On irregular meshes, however, no canonical Fourier basis exists, and spectral representations must be constructed from graph operators induced by mesh connectivity and geometry. In this study, we extend the binned spectral power loss for application to unstructured-mesh surrogate modeling of nonlinear dynamical systems. This is obtained by replacing Fourier bands with graph-Laplacian frequency bands, and we provide scalable Chebyshev and multilevel approximations for improving long-horizon rollout fidelity. In its full-spectrum form, our approach uses graph Laplacian eigenspaces to provide a graph analogue of Fourier band-power matching, but incurs the high cost of spectral decomposition. As a scalable approximation, we replace exact band projectors with sparse Chebyshev polynomial graph filters, avoiding explicit eigendecomposition. When utilizing multilevel graph architectures, we introduce Graph Laplacian Energy Alignment for Meshes (GLEAM), which applies retained-subspace scale-aware supervision across graph hierarchies so that coarse and fine representations are regularized during autoregressive rollout. Our results show that the proposed spectral losses improve long-horizon rollout fidelity and preserve statistical invariants for the forecasting of turbulent flows on unstructured meshes, compared to deterministic baselines.

Figures

Figures reproduced from arXiv: 2607.19387 by Kanad Sen, Romit Maulik.

Figure 1
Figure 1. Figure 1: Vorticity rollout diagnostics for the Base model and Chebyshev BSP with [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Qualitative EAGLE velocity-magnitude comparison at [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Autoregressive RMSE on the two-dimensional backward-facing step dataset for (a) Chebyshev final-level [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Backward-facing step rollout fields at t = 1, 5, 10, 20, and 50. Each block shows predicted velocity fields and absolute-error maps for the Base, Chebyshev BSP, and GLEAM models. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: pOnWing surface-centered pressure comparison for simulation 5. Columns are rollout steps t = 1, 5, 10, 50; rows show (a) centered ground truth, (b) Base prediction, (c) Base absolute error, (d) Chebyshev BSP prediction, (e) Chebyshev BSP absolute error, (f) GLEAM prediction, and (g) GLEAM absolute error. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: pOnWing pressure-force error over a 1000-step autoregressive rollout for the trained Base, GLEAM, and Chebyshev BSP models. Curves show the mean over simulations 0–15, and shaded regions show the corresponding min–max range. Panel (a) reports eF = ∥Fbp(t) − Fp(t)∥2/ max(∥Fp(t)∥2, 10−12); panel (b) reports e˜F = ∥Fbp(t) − Fp(t)∥2/ max(P i |pi(t)|Ai , 10−12) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: pOnWing FMGN aggregate rollout diagnostics over 100 autoregressive steps. Curves show the mean over 16 simulations, and shaded regions show the min–max range. Panel (a) reports physical-space RMSE, and panel (b) reports the non-log exact normalized-Laplacian linear BSP spectral error. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Uy graph-band energy mismatch distributions on the backward-facing step rollout at t = 30 and t = 82. Each curve is a smoothed KDE density over log10 |Em(Uˆ y) − Em(Uy)| across 24 graph-Laplacian bands, plotted against the original mismatch values on a logarithmic axis. This diagnostic is complementary to pointwise RMSE because it asks a different question: whether the prediction places the correct amount … view at source ↗
Figure 9
Figure 9. Figure 9: gives the full Uy band-energy RMSE trajectory over the BFS rollout together with the spatial Uy fields and absolute-error maps corresponding to the graph-band mismatch diagnostics in Section 4.4. These snapshots are kept in the appendix because they support the interpretation of the spectral rollout diagnostics but are not needed for the main quantitative argument. (a) Band-energy RMSE. Ground truth Base C… view at source ↗
Figure 10
Figure 10. Figure 10: pOnWing VGAE diagnostics. Panel (a) reports the exact normalized-Laplacian spectral-error diagnostic over a representative 100-step autoregressive rollout, including the GLEAM scalar and no-corrector variants. Panel (b) compares full-field pressure predictions at t = 50 for simulations 5 and 11, with shared field and absolute-error colorbars. evaluation protocol fixed and separates the role of the auxilia… view at source ↗
Figure 11
Figure 11. Figure 11: pOnWing surface-centered pressure comparison for simulation 11. Columns are rollout steps t = 1, 5, 10, 50; rows show (a) centered ground truth, (b) Base prediction, (c) Base absolute error, (d) Chebyshev BSP prediction, (e) Chebyshev BSP absolute error, (f) GLEAM prediction, and (g) GLEAM absolute error. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: pOnWing surface-centered pressure comparison for simulation 14. Columns are rollout steps t = 1, 5, 10, 50; rows show (a) centered ground truth, (b) Base prediction, (c) Base absolute error, (d) Chebyshev BSP prediction, (e) Chebyshev BSP absolute error, (f) GLEAM prediction, and (g) GLEAM absolute error. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: pOnWing pressure-PDF diagnostics at representative autoregressive rollout horizons. The panels compare the pressure distribution of the ground truth, Base model, GLEAM model, and Chebyshev BSP model at t = 10, 100, 350, and 500 using different test simulations. The spectral variants better preserve the broad pressure distribution, while the Base model develops sharper displaced peaks. (a) Velocity RMSE. (… view at source ↗
Figure 14
Figure 14. Figure 14: BFS comparison with related trained baselines. Panel (a) reports autoregressive velocity RMSE, while panel [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: BFS velocity-magnitude probability-density comparison for the Base, Pushforward, Chebyshev BSP, and [PITH_FULL_IMAGE:figures/full_fig_p031_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: BFS absolute velocity-magnitude error maps at autoregressive horizons [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Late-rollout EAGLE visual comparison between Chebyshev BSP and Pushforward training for samples [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Additional EAGLE velocity-magnitude snapshots for sample 50 at [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗

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