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arxiv: 2606.19731 · v1 · pith:66HDPLWJnew · submitted 2026-06-18 · ⚛️ physics.flu-dyn

Forcing-informed resolvent analysis: Identification of input-output relations in self-sustained flows

Yuta Iwatani , Kunihiko Taira , Soshi Kawai This is my paper

Pith reviewed 2026-06-26 16:01 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords forcing-informed resolvent analysisself-sustained flowsinput-output relationsnonlinear forcingenergy transfer mapcylinder wakeboundary layer transition
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The pith

Forcing-informed resolvent analysis extracts forcing and response modes that match actual self-sustained flow fields by incorporating nonlinear forcing structures from data.

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

The paper develops a forcing-informed resolvent framework that modifies the standard resolvent operator to account for the actual spatiotemporal patterns of nonlinear terms acting as forcing around a mean flow. Bases for the input subspace (from forcing snapshots) and output subspace are estimated directly from simulation data, so the resulting modes and singular values align with the observed flow rather than idealized linear assumptions. This holds at frequencies where nonlinear amplification dominates. The same snapshots can build the linear operator itself, producing a fully data-driven version. The framework also yields a nonlinear energy transfer map that shows where each forcing mode adds or removes fluctuation energy in space.

Core claim

The forcing-informed resolvent operator is built by estimating basis vectors for the input subspace spanned by forcing snapshots and for the output subspace from simulation data. The extracted response modes are linear combinations of the output basis, forcing modes are linear combinations of the input basis, and the singular values equal the actual output amplitudes. These properties make the identified modes and gains consistent with the statistics of the underlying self-sustained flow, as shown on the Stuart-Landau oscillator, two-dimensional cylinder wake, and three-dimensional transitional boundary layer.

What carries the argument

The forcing-informed resolvent operator, constructed from estimated input and output subspace bases derived from nonlinear forcing and response snapshots.

If this is right

  • Extracted FI response and forcing modes remain consistent with the actual self-sustained flow fields.
  • Singular values of the FI resolvent operator equal the measured output amplitudes.
  • Forcing snapshots alone suffice to build the linear operator for a fully data-driven analysis.
  • The nonlinear energy transfer map locates spatial regions where each extracted forcing mode injects or removes fluctuation energy.

Where Pith is reading between the lines

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

  • The energy transfer map could be used to rank spatial regions by their contribution to sustaining the flow at each frequency.
  • The data-driven construction might reduce the need for an explicit mean-flow linearization when only snapshot data are available.
  • The approach could be tested on additional self-sustained flows such as bluff-body wakes or mixing layers to check whether the same consistency holds.
  • If the input subspace basis is truncated too aggressively, the recovered forcing modes may miss localized nonlinear structures that still affect the overall energy balance.

Load-bearing premise

Basis vectors estimated from a finite number of simulation snapshots accurately represent the relevant structures of the nonlinear terms that act as forcing.

What would settle it

If the modes and gains produced by the method on the cylinder wake at a nonlinear frequency fail to reproduce the dominant fluctuation structures seen in the original simulation data, the consistency claim is falsified.

Figures

Figures reproduced from arXiv: 2606.19731 by Kunihiko Taira, Soshi Kawai, Yuta Iwatani.

Figure 1
Figure 1. Figure 1: Schematic of workflow of the present forcing-informed (FI) resolvent analysis, shown for an example of a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Time trajectories of the state vector 𝑞 ′ (blue) and the nonlinear forcing vector 𝑓 ′ NL (orange) of the Stuart–Landau oscillator. (a) The first components 𝑞 ′ 1 and 𝑓 ′ NL,1 from the initial condition to the limit cycle oscillation (LCO) at 0 ≤ 𝑡 ≤ 120. (b) All components at 152 ≤ 𝑡 ≤ 161 in the LCO domain. Solid lines, the first components of 𝑞 ′ and 𝑓 ′ NL; dashed lines, the second components. proposed … view at source ↗
Figure 3
Figure 3. Figure 3: Results for the Stuart-Landau oscillator. ( [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Time evolutions of ⟨𝑞 ′ , 𝑳𝑞 ′ ⟩ (solid blue line), ⟨𝑞 ′ , 𝑓 ′ NL⟩ (solid orange line), and their sum (solid green line), for the Stuart-Landau system. The dashed blue/orange line is the time-averaged energy supply/removal due to the linear/nonlinear mechanisms during the LCO, evaluated using the leading FI response 𝑞ˆ𝐺,1 and forcing modes ˆ𝑓𝐺,1. Finally, to assess the energy balance between the linear and… view at source ↗
Figure 5
Figure 5. Figure 5: Data-driven estimate of linear time-invariant operator [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Leading gains (singular values 𝜎1) for 2D cylinder case, obtained from the fully data-driven 𝑮FDD (orange circle), semi-data-driven 𝑮SDD (blue triangle), and the ordinary resolvent operator 𝑹 (gray plus), along with the secondary leading gains of 𝜎2 (𝑮FDD)(orange cross). The square roots of the leading eigenvalues SPOD gains 𝑺qq(red cross) are also shown. FI resolvent operator, which holds at the harmonics… view at source ↗
Figure 7
Figure 7. Figure 7: Spatial distributions of the leading FI response and forcing modes in 2D cylinder wake extracted by the fully [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Conventional resolvent modes obtained using [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Phase evolutions of the FI (a) response and (b) forcing modes at 𝑆𝑡𝐾 , along with the (normalized) streamwise component of phase velocity of FI (c) response and (d) forcing modes. In (b), the black arrows indicate the direction of the movement of the FI forcing mode structure. Spatial distributions of −∇∠𝑞ˆ𝐺,𝑚𝑥 /∥∇∠𝑞ˆ𝐺,𝑚𝑥 ∥ and −∇∠ ˆ𝑓𝐺,𝑚𝑥 /∥∇∠ ˆ𝑓𝐺,𝑚𝑥 ∥ are shown in (c) and (d), respectively. The gray isoli… view at source ↗
Figure 10
Figure 10. Figure 10: Same as figure 7 but for semi-data-driven approach using 𝑮SDD. X 3.4 4.0 4.6 5.2 5.8 6.4 7.0 Z 3 2 1 0 0 0.05 0.1 Y [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Topview of the instantaneous iso-surfaces of Q criterion [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Late stage of H-type transition, visualized by iso-surfaces of Q criterion ( [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Eigenvalues 𝜆𝑟 + i𝜆𝑖 of the estimated linear operator 𝑳data obtained using 𝑁POD = 10, 90, 170, 250: (a) overall view; (b) enlarged view near unstable eigenvalues. X Z Y m̂ x /∥m̂ x∥∞ m̂ y /∥m̂ y∥∞ m̂ z /∥m̂ z∥∞ −0.38 0.38 −0.07 0.07 −0.02 0.02 −0.07 0.07 0.0 2.0 0.0 2.0 Z Z 0.0 2.0 Z 4.8 4.8 4.8 3.5 3.5 3.5 X X X 0.1 0.1 0.1 Y Y Y [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Isosurfaces of momentum components 𝑚ˆ for the most unstable eigenmodes of the H-type transitional boundary layer (𝜆𝑖/𝜔2𝐷 ≈ 3/2), computed from the data-driven linear operator 𝑳data with 𝑁pod = 90: (left) streamwise 𝑚ˆ 𝑥, (middle) wall-normal 𝑚ˆ 𝑦, (right) spanwise 𝑚ˆ 𝑧 momentum components. For visualization, the iso-surfaces are copied once in the spanwise direction. amplification mechanism is predominant… view at source ↗
Figure 15
Figure 15. Figure 15: Gains, 𝜎𝐺,1 and 𝜎𝐺,2, of the FI resolvent operator 𝑮FDD for the transition boundary layer case, compared with the square root of the leading SPOD gains 𝜆 1/2 1 (𝑺qq). X 3.5 4.0 4.5 4.8 Z 𝜔/𝜔2D = 1/2 q ̂ mx (FI resp.) (FI forc.) (NL. energy trans. map) f ̂ mx 𝓣mx 1 0 1 0 1 0 X ≈ 4.25 X ≈ 4.5 X 3.5 4.0 4.5 4.8 Z 𝜔/𝜔2D = 2 q ̂ mx f ̂ mx 𝓣mx 1 0 1 0 1 0 X ≈ 4.25 X ≈ 4.5 (𝑎) (𝑏) [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 16
Figure 16. Figure 16: Results of FI resolvent analysis of transitional boundary layer for streamwise momentum component at ( [PITH_FULL_IMAGE:figures/full_fig_p022_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Input-output relations at 𝑋 ≈ 4.5 where a perpendicularly standing ring-like vortex (PSRV) is observed. Streamwise momentum component of the FI response and forcing modes at 𝜔/𝜔2𝐷 = 1/2 (left) and 2 (middle). Snapshots of streamwise momentum component of the output 𝑚 ′ 𝑥 and nonlinear forcing 𝑓 ′ 𝑚𝑥 (right). attenuates the energy at this region, whereas that at 𝜔/𝜔2𝐷 = 2 in [PITH_FULL_IMAGE:figures/full_… view at source ↗
Figure 18
Figure 18. Figure 18: Comparison between the eigenmode 𝑚𝑥 (global stability mode) of 𝑳data using 𝑁POD = 90 and nonlinear energy transfer map 𝑻𝑚𝑥 , for streamwise momentum component, at 𝜔/𝜔2𝐷 = 1/2 and 2. Eigenvalues 𝜆𝑟 + i𝜆𝑖 of 𝑳data indicate the stability of the corresponding eigenmodes. 2012], the positive nonlinear energy transfer near the wall suggests a possible connection between nonlinear, intense sweep motions under th… view at source ↗
Figure 19
Figure 19. Figure 19: Comparison between the present nonlinear energy transfer maps computed from the most dominant FI [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
read the original abstract

We present a forcing-informed (FI) resolvent analysis framework to identify input-output relations for statistically stationary self-sustained unsteady flows. The central idea of this method is to inform the resolvent operator about the spatiotemporal structures of the nonlinear terms that act as exogenous forcing with respect to the mean flow. To construct the FI resolvent operator, we estimate the basis vectors for the input subspace spanned by forcing snapshots and, similarly, for the output subspace, from simulation data. The extracted FI response and forcing modes are expressed through the estimated bases of the output and input subspaces, respectively, and the singular values of the FI resolvent operator correspond to the actual output amplitudes. These properties ensure that the extracted modes are consistent with the actual self-sustained flow fields. Additionally, the forcing snapshots can be used to construct the linear operator, enabling a fully data-driven FI resolvent analysis. The proposed framework is validated using the Stuart-Landau oscillator and demonstrated for a two-dimensional cylinder wake and a three-dimensional transitional boundary layer. We successfully identify the gains and the corresponding pairs of forcing and response modes, even at frequencies where the nonlinear amplification mechanism is crucial. Furthermore, leveraging the balance between the time-averaged energy amplification/attenuation by the linear operator and nonlinear forcing, we introduce a nonlinear energy transfer map that identifies the spatial domains where the extracted forcing mode injects or removes fluctuation energy, thereby providing key physical insight into the self-sustaining mechanisms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a forcing-informed (FI) resolvent analysis framework for statistically stationary self-sustained unsteady flows. It estimates bases for the input subspace (spanned by forcing snapshots) and output subspace from simulation data to construct an informed resolvent operator that incorporates the spatiotemporal structure of nonlinear terms acting as exogenous forcing. The extracted FI response and forcing modes are expressed via these bases, with the singular values asserted to equal actual output amplitudes, ensuring consistency with the self-sustained fields. A fully data-driven variant constructs the linear operator from the same snapshots; the framework is validated on the Stuart-Landau oscillator, 2D cylinder wake, and 3D transitional boundary layer, and includes a nonlinear energy transfer map based on time-averaged energy balance.

Significance. If the consistency properties and validation hold under scrutiny, the approach could provide a practical route to extract input-output relations and physical insight into nonlinear self-sustaining mechanisms where standard mean-flow resolvent analysis is insufficient. The multi-case demonstration and the energy-transfer map are potentially useful contributions if the data-dependence issues are resolved.

major comments (2)
  1. [Abstract and method description] The central consistency claim (singular values equal actual output amplitudes; modes consistent with self-sustained fields) rests on the estimated input/output subspace bases from finite snapshots accurately spanning the relevant nonlinear forcing structures. The abstract and method description do not quantify truncation or sampling error in these bases, nor demonstrate that the projected operator preserves the true energy balance between linear amplification and nonlinear forcing.
  2. [Validation cases and fully data-driven variant] In the fully data-driven variant (linear operator also built from the same forcing snapshots), subspace estimation errors propagate simultaneously into the operator and the forcing representation. This compounds the risk that the reported gains and modes are artifacts of the shared data rather than robust identifications; explicit robustness checks (e.g., convergence with snapshot count or cross-validation) are required.
minor comments (1)
  1. Notation for the estimated bases, projection operators, and how the FI resolvent is assembled should be made fully explicit (including any regularization or truncation thresholds) to allow independent reproduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. We address each major comment below, indicating planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and method description] The central consistency claim (singular values equal actual output amplitudes; modes consistent with self-sustained fields) rests on the estimated input/output subspace bases from finite snapshots accurately spanning the relevant nonlinear forcing structures. The abstract and method description do not quantify truncation or sampling error in these bases, nor demonstrate that the projected operator preserves the true energy balance between linear amplification and nonlinear forcing.

    Authors: We agree that the abstract and method description do not explicitly quantify truncation or sampling errors from finite snapshots. In the revised manuscript we will expand the method section to discuss the approximation properties: as the snapshot count increases the estimated bases converge to the true forcing/response subspaces and the singular values recover the actual amplitudes. We will also add a brief demonstration on the Stuart-Landau oscillator showing preservation of the linear-nonlinear energy balance once the snapshot ensemble is sufficiently rich. These additions will be placed after the definition of the FI resolvent operator. revision: yes

  2. Referee: [Validation cases and fully data-driven variant] In the fully data-driven variant (linear operator also built from the same forcing snapshots), subspace estimation errors propagate simultaneously into the operator and the forcing representation. This compounds the risk that the reported gains and modes are artifacts of the shared data rather than robust identifications; explicit robustness checks (e.g., convergence with snapshot count or cross-validation) are required.

    Authors: The concern about compounded data dependence in the fully data-driven variant is valid. Although the Stuart-Landau and cylinder-wake validations already compare against known reference solutions, we will add an explicit robustness subsection. This will include (i) convergence of the leading gains and modes versus snapshot count for the cylinder wake and (ii) a simple cross-validation by partitioning the snapshot set. These checks will be reported for both the standard and fully data-driven FI operators. revision: yes

Circularity Check

1 steps flagged

Subspace bases fitted to simulation snapshots force singular values to match data amplitudes by construction

specific steps
  1. fitted input called prediction [Abstract]
    "To construct the FI resolvent operator, we estimate the basis vectors for the input subspace spanned by forcing snapshots and, similarly, for the output subspace, from simulation data. The extracted FI response and forcing modes are expressed through the estimated bases of the output and input subspaces, respectively, and the singular values of the FI resolvent operator correspond to the actual output amplitudes. These properties ensure that the extracted modes are consistent with the actual self-sustained flow fields."

    The bases are obtained by fitting to the identical simulation snapshots whose amplitudes are later declared to be recovered by the singular values. Because the FI operator is the projection of the (mean-flow or data-derived) linear operator onto these bases, its SVD singular values are algebraically forced to reproduce the projected amplitudes; the claimed 'correspondence' and 'consistency' therefore reduce to the fitting step rather than constituting an independent prediction.

full rationale

The paper estimates input/output subspace bases directly from the same simulation snapshots that supply the nonlinear forcing and fluctuation amplitudes. The FI resolvent is then formed by projection onto these bases, after which the paper states that its singular values equal the actual output amplitudes and that the resulting modes are consistent with the flow fields. This match holds by the linear-algebra construction of the reduced operator rather than by independent verification; the fully data-driven variant (linear operator also built from snapshots) inherits the same dependence. No self-citation chain or external uniqueness theorem is invoked, so the circularity is moderate and localized to the data-driven projection step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on treating nonlinear terms as exogenous forcing to the mean flow and on the assumption that finite snapshots yield representative bases for the relevant subspaces; no free parameters or new entities are mentioned.

axioms (1)
  • domain assumption Nonlinear terms in the Navier-Stokes equations can be treated as exogenous forcing with respect to the mean flow for the purpose of input-output analysis.
    This is the foundational modeling choice that allows the resolvent operator to be informed by forcing structures.

pith-pipeline@v0.9.1-grok · 5804 in / 1211 out tokens · 26114 ms · 2026-06-26T16:01:53.017183+00:00 · methodology

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