A recent study established that for two-dimensional bluff-body wakes (circular cylinder at Re=300-800, square cylinder at Re=100), 95-98% of the long-horizon rollout error of a convolutional-autoencoder/LSTM reduced-order model is accumulated phase error: the model reproduces the vortex-shedding limit-cycle amplitude almost exactly but traverses it at a slightly wrong rate, the drift is linear in time, and a post-hoc one-parameter-per-coordinate correction fitted on a short calibration window removes most of the error, with an R-squared diagnostic predicting when it applies. The open problem is to establish whether this phase-drift mechanism and its correction survive outside the narrow setting in which they were demonstrated: genuinely three-dimensional wakes with spanwise instabilities, broadband/quasi-periodic/turbulent dynamics lacking a dominant frequency, non-stationary shedding requiring a time-varying correction, non-LSTM autoregressive architectures, causal/streaming deployment, and a non-dimensional form of the applicability diagnostic.

Background

A bluff body placed in a stream sheds vortices alternately from its sides once the Reynolds number Re=U∞D/ν\mathrm{Re}=U_\infty D/\nu (free-stream speed times body size over kinematic viscosity) exceeds a critical value, producing the periodic von Karman vortex street — a limit cycle, i.e. a closed periodic orbit to which the flow settles. Numerically resolving such wakes is costly, so much effort has gone into reduced-order models (ROMs): compress each flow snapshot to a low-dimensional latent vector and advance that vector in time instead of the full field. Classical versions use proper orthogonal decomposition (POD), a linear projection onto the most energetic empirical modes, combined with Galerkin dynamics (the governing equations projected onto those modes, yielding a small system of ordinary differential equations); modern versions replace the linear projection with a convolutional autoencoder (a neural network trained to compress and reconstruct fields through a bottleneck) and advance the latent state with a recurrent network such as an LSTM (long short-term memory network). These models are trained on one-step-ahead prediction with true inputs (teacher forcing) but are used autoregressively at inference, feeding their own predictions back as inputs; this train-inference mismatch makes errors compound over long rollouts, a failure usually described as unstructured noise growth. The source study (Samanta and Behera, 2026, arXiv:2608.07189) showed that for two-dimensional circular- and square-cylinder wakes this compounding error is in fact highly structured. Writing each latent coordinate's true and predicted trajectory as an analytic signal (the signal plus ii times its Hilbert transform, an operation that shifts every frequency component by a quarter period; the resulting modulus and unwrapped argument define an instantaneous amplitude and phase), the error decomposes into an amplitude part and a phase part: 95-98% of the rollout-error variance is pure phase error, the error spectrum peaks sharply at the vortex-shedding frequency, and the phase error accumulates linearly in time (limit-cycle amplitudes match to within 0.15%). Because the drift is linear, a post-hoc correction — one constant drift rate per latent coordinate, fitted on about 8 shedding cycles of ground truth — removes 74-80% of the rollout error on the stationary-wake runs its applicability diagnostic accepts (41-71% averaged over all training runs) and about half on a wake with a slowly varying inflow, without retraining. The coefficient of determination R2R^2 of the linear phase fit (applicability threshold R2>0.3R^2>0.3) predicts correction success with r=0.85r=0.85 across 50 trained networks. The mechanism is confined to an intermediate compression regime of the latent dimension, which the source locates without ground truth through the correction-efficacy ratio ρ=εunc2−εCAE2 / εCAE\rho=\sqrt{\varepsilon_{\mathrm{unc}}^{2}-\varepsilon_{\mathrm{CAE}}^{2}}\,/\,\varepsilon_{\mathrm{CAE}}, where εunc\varepsilon_{\mathrm{unc}} is the field-space RMSE of the uncorrected rollout and εCAE\varepsilon_{\mathrm{CAE}} is the autoencoder's reconstruction floor (the RMSE of encoding and immediately decoding the true snapshots). Holding Re=800\mathrm{Re}=800 fixed and varying only the latent dimension, the correction removes 2.1% of the latent error at nz=4n_z=4 and 47.6% at nz=8n_z=8; at Re=400\mathrm{Re}=400 with nz=16n_z=16 the rollout sits essentially on the reconstruction floor (ρ=0.09\rho=0.09), the phase structure disappears, and no coordinate carries a coherent linear drift (R2=0.01R^2=0.01 averaged over the sixteen coordinates, none exceeding 0.03). With too few coordinates there is no clean phase to correct, and at the reconstruction floor there is no drift to correct, so the phase-drift account is a property of the compression-limited intermediate regime that a well-chosen ROM occupies. Qualitatively consistent observations (dominance of progressive phase shifts in autoregressive latent rollouts of wake flows) appear independently in neural-operator (networks that learn solution operators mapping entire function-valued fields to fields) and controlled-wake ROM studies (Viknesh and Arzani 2025; Solera-Rico et al. 2026), but the quantitative decomposition, the correction, and the diagnostic are due to the source study. The demonstration is confined to a deliberately idealized setting. First, it requires a flow with a single dominant shedding frequency, so each latent coordinate is narrowband (its spectrum concentrated near that frequency) and the analytic-signal representation is well posed; for broadband, quasi-periodic, or turbulent dynamics (already at Re=1000 the two-dimensional circular-cylinder wake is temporally broadband) the decomposition returns variance shares exceeding 100%, an unambiguous breakdown. Second, all simulations are two-dimensional, whereas physical cylinder wakes undergo a spanwise (three-dimensional) instability above Re≈190\mathrm{Re}\approx190 for the circular cylinder (the mode A/B wake transition, the two characteristic spanwise-periodic instability patterns that succeed the two-dimensional wake; Williamson 1996) and Re≈150\mathrm{Re}\approx150-200200 for the square cylinder (Robichaux et al. 1999), so the higher-Reynolds-number cases studied are not physically realizable wakes. Third, only an LSTM temporal model was tested, the correction uses a constant drift rate, and it is applied offline with a two-sided Hilbert transform, so it is not causal. Whether the mechanism is a property of the flow physics and the autoregressive rollout structure, or an artifact of this narrow setting, is what the source study itself identifies as the question it leaves open.

Problem Statement

Establish the domain of validity of the phase-drift account of autoregressive rollout error in latent-space reduced-order models of bluff-body wakes — the finding that the compounding long-horizon error is overwhelmingly linearly accumulating phase error at the shedding frequency, correctable post hoc by one constant drift rate per latent coordinate with an R2R^2 applicability diagnostic — beyond the two-dimensional, narrowband, stationary, LSTM setting in which it was demonstrated. The six axes enumerated below jointly define the task and each is required; resolving the three-dimensional axis alone — listed first and the pivotal case — is not a resolution. For each axis, determine whether the phase-drift mechanism and its correction survive beyond the demonstrated setting:

  1. Three-dimensional wakes (primary): for genuinely three-dimensional bluff-body wakes in which spanwise instabilities are present (circular cylinder above Re≈190\mathrm{Re}\approx190, square cylinder above Re≈150\mathrm{Re}\approx150-200200; e.g. the mode A/B regimes), apply the quantitative amplitude-phase decomposition to the autoregressive rollout error of latent-space ROMs of these wakes. Is the error still predominantly linearly drifting phase, and does the constant-rate correction with its diagnostic still apply? Two conditions belong to the question itself. First, the models concerned are ROMs in the intermediate compression regime in which the source demonstrates the mechanism (see Background), so an answer identifies the regime it probes rather than resting on a single unconstrained choice of latent dimension, since a latent space at the reconstruction floor exhibits no coherent drift on any flow. Second, because mode A/B wakes are quasi-periodic rather than strictly periodic, the question includes what well-posed phase description applies there, given that the analytic-signal representation is established only for narrowband coordinates (cf. item 2).
  2. Flows without a dominant frequency: for broadband, quasi-periodic, or turbulent dynamics, where the analytic-signal representation currently breaks down, determine whether a meaningful amplitude-phase (or suitably generalized time-frequency) decomposition of rollout error exists, and whether the error structure remains phase-dominant.
  3. Non-stationary shedding: determine whether the phase-drift account and its correction extend to a wake whose shedding frequency drifts in time, and if so what form the correction takes, quantifying the improvement over the roughly half of the error that the constant-rate correction removes there.
  4. Architecture dependence: test whether the same phase-dominant error structure arises in transformer-based (attention-based sequence models) or neural-operator (learned solution-operator) surrogates rolled out autoregressively on the same flows.
  5. Causal deployment: determine whether the correction survives causal, streaming deployment, where the analytic-signal estimate may use no future data (one-sided or sliding-window), and characterize the accuracy-latency trade-off of such an implementation.
  6. Diagnostic formalization: derive a non-dimensional form of the applicability threshold that replaces the sampling- and window-specific R2>0.3R^2>0.3.

References

  1. Suvam Samanta and Sachidananda Behera, "Autoregressive rollout error in latent-space reduced-order models of bluff-body wakes is accumulated phase drift" (2026), arXiv:2608.07189, doi:10.48550/arxiv.2608.07189, https://arxiv.org/abs/2608.07189
  2. Siva Viknesh and Amirhossein Arzani, "Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling" (2025), arXiv:2510.00233, doi:10.48550/arxiv.2510.00233, https://arxiv.org/abs/2510.00233
  3. Alberto Solera-Rico, Patricia García-Caspueñas, Carlos Sanmiguel Vila, and Stefano Discetti, "The balance between compactness and forecast accuracy of data-driven latent-space reduced-order models in controlled wake flows" (2026), arXiv:2607.24569, doi:10.48550/arxiv.2607.24569, https://arxiv.org/abs/2607.24569
  4. Jiaquan Zhang, Shuxu Chen, Haifan Meng, Yi Lu, Zhihan Lyu, Fan Mo, Wei Dong, Yang Yang, and Chaoning Zhang, "Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction" (2026), arXiv:2608.11237, doi:10.48550/arxiv.2608.11237, https://arxiv.org/abs/2608.11237
  5. C. H. K. Williamson, "Three-dimensional wake transition", Journal of Fluid Mechanics 328, 345-407 (1996), doi:10.1017/S0022112096008750, https://doi.org/10.1017/S0022112096008750
  6. J. Robichaux, S. Balachandar, and S. P. Vanka, "Three-dimensional Floquet instability of the wake of square cylinder", Physics of Fluids 11, 560-578 (1999), doi:10.1063/1.869930, https://doi.org/10.1063/1.869930
  7. Bernd R. Noack, Konstantin Afanasiev, Marek Morzynski, Gilead Tadmor, and Frank Thiele, "A hierarchy of low-dimensional models for the transient and post-transient cylinder wake", Journal of Fluid Mechanics 497, 335-363 (2003), doi:10.1017/S0022112003006694, https://doi.org/10.1017/S0022112003006694
  8. Kookjin Lee and Kevin T. Carlberg, "Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders", Journal of Computational Physics 404, 108973 (2020), doi:10.1016/j.jcp.2019.108973, https://doi.org/10.1016/j.jcp.2019.108973