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REVIEW 4 major objections 6 minor 53 references

Reduced-order control using low-rank Dynamic Mode Decomposition

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proposes lrDMD, a rank-constrained linear model with distinct input and output subspaces, and shows it builds full-state LQR controllers that stabilize the Ginzburg-Landau system at lower rank and suppress flat-plate vortex…

desk verdict lrDMD is a solid application of a known rank-constrained idea; the GL results are clean, but the flat-plate 'cost lower' claim is not backed by any computed cost. read the letter →

arxiv 1908.02249 v1 pith:ZW4ISXBI submitted 2019-08-06 physics.flu-dyn

Palash Sashittal , Daniel J. Bodony This is my paper · ORCID
classification physics.flu-dyn
keywords flowcontrolDynamicModeDecompositionlow-rankapproximationmodelreductionLQRfeedbackadjointmodesGinzburg-Landauequationvortexshedding
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that a reduced-order model for unsteady fluid flows should not be forced to use the same low-dimensional subspace for the input and the output of its linear map. It proposes low-rank Dynamic Mode Decomposition (lrDMD), which solves $\min_{L,D,R}\|Y-LDR^{T}X\|_F^2$ over separate orthonormal left and right subspaces $L$ and $R$. On the linearized Ginzburg-Landau equation in the globally unstable regime, lrDMD stabilizes the system with a rank-5 full-state LQR controller, whereas DMD and OMD leave an unstable eigenvalue at that rank. On the unsteady flow over a flat plate inclined at 35 degrees, lrDMD suppresses vortex shedding with lower actuation cost than DMD and matches or exceeds OMD, especially when the controller is switched on after nonlinearities have grown. The paper's point is that the separate input and output subspaces, not merely the rank constraint, are what make the reduced-order controllers better.

What carries the argument

The load-bearing object is the rank-constrained least-squares surrogate with the closed-form optimal $D^{*}$ and the observation that the objective depends only on the subspaces spanned by $L$ and $R$. The paper solves it with two methods: a fast subspace-projection iteration that alternates between an optimal $L$ for a fixed projected basis and an Orthogonal Procrustes update for $R$, initialized from the DMD/POD subspace, and a Riemannian trust-region gradient method on the product Grassmann manifold initialized by the subspace projection. In the controller construction, the LQR Riccati equation is solved by Galerkin projection onto the row-space basis of the rank-$r$ model, which for lrDMD is the right subspace $R$; the same projection choice is made for DMD and OMD, so the comparison isolates the learned model rather than the controller projection.

What would settle it

On a small linear system where the global rank-$r$ optimum can be found by exhaustive search, run the lrDMD solvers from many random initial subspaces instead of the DMD subspace and compare reconstruction error and the smallest rank of a stabilizing LQR controller. If a substantial fraction of random starts land in local minima with errors no better than DMD or OMD, or fail to stabilize at the rank the paper reports, the central improvement is initialization-dependent rather than a property of the rank-constrained formulation.

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Extended reading notes

Core claim

The paper's central claim is that the best rank-$r$ linear model of a dynamical system from snapshot data is not generally obtained by projecting onto one subspace, as DMD and OMD do, but by solving the rank-constrained least-squares problem $\min_{L,D,R}\|Y-LDR^{T}X\|_F^2$ over orthonormal $L,R\in\mathbb{R}^{m\times r}$. For fixed $L$ and $R$ the optimal middle factor is $D^{*}=(L^{T}YX^{T}R)(R^{T}XX^{T}R)^{-1}$, and the remaining problem depends only on the subspaces spanned by $L$ and $R$, so it lives on a product of Grassmann manifolds. The freedom to choose the input subspace $R$ and output subspace $L$ separately is what the paper identifies as the source of improvement: it fits the data more closely, represents the adjoint mode of the unstable Ginzburg-Landau eigenmode more faithfully, and yields full-state LQR controllers that stabilize that system at rank 5, suppress vortex shedding behind a 35-degree flat plate from later nonlinear initial conditions, and recover the true optimal actuator location at lower rank than DMD or OMD.

Load-bearing premise

The load-bearing premise is that the numerical solvers for the non-convex problem find a good enough local minimum; the paper states that the gradient method has no global convergence guarantee and that the subspace-projection method is a heuristic initialized from the DMD subspace. If those solvers often land in poor local minima on other systems, the claimed advantages over DMD and OMD would not generalize.

Editorial extensions

If this is right

  • At a fixed rank, lrDMD achieves lower snapshot reconstruction error than DMD and OMD on the flat-plate data, and in several cases the fast subspace-projection solver is both more accurate and cheaper than the gradient solver.
  • lrDMD stabilizes the Ginzburg-Landau system with a rank-5 LQR controller, while DMD and OMD leave an unstable eigenvalue at rank 5; at rank 9 all three approaches approach the full-order optimal controlled spectrum.
  • lrDMD recovers the true optimal actuator location for the Ginzburg-Landau system at rank 5, where DMD selects the wrong location until rank 9.
  • For flow over the inclined flat plate, lrDMD returns the flow to the steady state from later, more nonlinear initial conditions and does so with lower actuation cost than DMD, while also giving a lower projection error for the unstable adjoint mode at a given rank.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the separate input and output subspaces are the active ingredient, lrDMD-type models should also improve other data-driven tasks in which the row space matters, such as non-intrusive receptivity and sensitivity analysis of nonnormal flows.
  • The comparison inherits whatever local minima the non-convex solvers find; a natural stress test is to initialize the Grassmannian optimization from many random subspaces and measure how often reconstruction error and stabilizing controller rank stay better than OMD.
  • The rank-constrained model is not tied to LQR, so the same lrDMD representation could be plugged into other controller designs, such as output-feedback or $\mathcal{H}_\infty$ synthesis, for problems where full-state measurements are unavailable.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes low-rank Dynamic Mode Decomposition (lrDMD), a reduced-order modeling method that solves the rank-constrained least-squares problem min_{L,D,R} ||Y - LDR^T X||_F^2 with separate input and output subspaces, and compares it with DMD and OMD. Two numerical solvers are presented: a heuristic subspace-projection method and a Riemannian trust-region gradient method. The reduced-order models are used to design full-state LQR feedback controllers, which are tested on the linearized complex Ginzburg-Landau equation in the globally unstable regime and on two-dimensional incompressible flow past an inclined flat plate at Re=100. The paper claims that lrDMD achieves lower reconstruction error, stabilizes the GL system at lower rank, provides improved estimates of adjoint modes, and suppresses flat-plate vortex shedding at lower control cost than DMD or OMD.

Significance. The core idea of using separate input and output subspaces for data-driven reduced-order models is a meaningful extension of DMD and OMD, particularly for non-normal flow systems where adjoint information matters for control. The GL experiments provide a clean, full-order-validated demonstration: lrDMD stabilizes the system at rank 5 where DMD and OMD fail, and the adjoint-mode projection error is reduced. The authors are also honest about the non-convexity of the optimization problem in Remark 2. However, the strongest comparative claims—lower reconstruction error in general and lower control cost on the flat plate—are not yet supported by the evidence as presented. The flat-plate comparison rests on single deterministic runs with no quantified cost, and the reconstruction-error comparison is in-sample. If these gaps are fixed, the paper could make a solid contribution to data-driven flow control.

major comments (4)
  1. [Section 5.2.3, Figs. 7-8, Table 5; abstract] The abstract claims that lrDMD suppresses vortex shedding from the inclined flat plate "at a cost lower than either DMD or OMD," but no quantitative control cost is reported anywhere in Section 5.2. The evidence consists of CD vs. time plots and actuation-strength plots for three activation times (t0 = 170, 190, 210) and a binary pass/fail table (Table 5) for two snapshot spacings. This is insufficient to establish a cost ordering. The authors should compute and report an integrated cost, such as the actual LQR objective J = sum (q^T Q q + u^T S u) evaluated on the closed-loop nonlinear trajectories, or an equivalent metric (e.g., time-integrated deviation of CD from the steady-state value plus control effort), for each method, activation time, and snapshot spacing. Without such a quantitative comparison, the headline "cost lower" claim is not load-bearing evidence and may not generalize.
  2. [Section 3.1, Table 2] The reconstruction errors reported in Table 2 are computed as epsilon = ||Y - A_hat X||_F on the same data matrices X, Y that were used to construct each reduced-order model. This is an in-sample evaluation of a training objective, and lrDMD has more degrees of freedom than DMD or OMD in optimizing that objective, so its lower error is partly expected. This does not demonstrate better predictive accuracy or a generally superior model. The paper should include an out-of-sample test, for example by withholding a portion of the time series or snapshots from a different initial condition, and reporting the prediction error on that held-out data. Without such evidence, the "lower reconstruction error" claim in the abstract is overstated.
  3. [Appendix A.1, Algorithm 1, line 10] The formula for D in Algorithm 1, line 10, includes an extra factor R^T L: D = (L^T Y X^T R)(R^T X X^T R)^{-1}(R^T L). This is inconsistent with Eq. (7), which gives D*(L,R) = (L^T Y X^T R)(R^T X X^T R)^{-1}. Since L and R are generally different orthogonal matrices, R^T L is not the identity, and as printed the algorithm does not compute the least-squares optimal D for the given L and R. Please correct the formula or explicitly clarify whether R^T L is an intentional part of a different definition. If the implementation actually uses Eq. (7), then this is a typo that should be fixed; otherwise the algorithm and the numerical results may be based on a different model than the one analyzed in Section 2.3.
  4. [Remark 2 and Section 3] All claimed advantages of lrDMD over DMD and OMD depend on the non-convex optimization problem (6) being solved to a sufficiently good local minimum. Remark 2 states that the gradient method has no global convergence guarantee, and Algorithm 1 is a heuristic initialized from the DMD subspace. The paper does not report any sensitivity analysis with respect to initialization or convergence diagnostics. For example, one could initialize from the OMD solution, from random orthogonal matrices, or from several perturbed DMD bases and quantify the spread in reconstruction error and closed-loop control performance. Without this, the single-run comparisons in Sections 3.1 and 5 cannot rule out the possibility that the observed lrDMD advantages are artifacts of the particular local minimum found by the proposed initialization. This is a load-bearing concern for the generality of the central claim.
minor comments (6)
  1. [Section 2 heading] The heading "Mathematical F ormulation" contains an erroneous space; it should read "Mathematical Formulation."
  2. [Section 5.1.3] In the paragraph after Eq. (14), the sentence "the eigenvector of X that corresponds to the largest eigenvalue" should refer to the eigenvector of F, not X.
  3. [Section 3.1] The notation Y = U^T Y and X = U^T X reuses symbols X and Y for the projected data matrices; this is confusing and should be clarified, for example with new symbols X_p, Y_p.
  4. [Table 5] The check and cross symbols in Table 5 do not render correctly in the text version (they appear as box glyphs); ensure that the caption and symbols display properly in the final PDF.
  5. [Section 5.2.3] The paper should clarify that T in Eq. (15) is the number of timesteps advanced by the nonlinear solver between control updates, and that Table 5 varies this parameter for the snapshot data matrices; this distinction is implicit but not stated clearly.
  6. [References] Reference [36] is listed as "forthcoming"; if the associated CDC paper has appeared, the complete publication details should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: control claims are tested against independent full-order benchmarks.

full rationale

The paper's central control claims are not circular. lrDMD is a rank-constrained least-squares problem (Eq. 6). The Ginzburg-Landau comparisons in Secs. 5.1.2 and 5.1.3 evaluate the LQR gains on the full-order complex Ginzburg-Landau dynamics and compare optimal actuator locations against the full-order system's Lyapunov cost, not against the reduced model's own reconstruction. Likewise, the flat-plate tests in Sec. 5.2.3 apply the reduced-order gains to the nonlinear immersed-boundary solver and report closed-loop vorticity and drag trajectories; the benchmark is the full-order nonlinear flow, so the lrDMD controller's success is not forced by its training objective. The adjoint-mode comparison in Sec. 5.1.1 uses the true adjoint mode of the discretized system as an external target. The only self-citation, [36], is for implementation details of the Riemannian gradient method and is outlined in Appendix A.2; it is not load-bearing, and the closed-form reduction of the lrDMD problem is credited to the independent work [37]. Remark 2 honestly states that the gradient method has no global convergence guarantee; this is a numerical limitation, not a circularity. The lower reconstruction error of lrDMD relative to DMD/OMD follows from the larger feasible set in (6) over (3), but the paper presents this as a property of the optimization rather than as an independent prediction. I find no prediction or derivation that reduces to its own inputs.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a few user-chosen hyperparameters (rank, snapshot count, sampling interval) and on the assumption that the non-convex lrDMD optimization is solved accurately enough. No new physical entities or forces are introduced. The main domain assumptions are linearity of the underlying dynamics in the data-collection region and the fidelity of the numerical simulations used to generate data.

free parameters (5)
  • Reduced-order model rank r = 5, 9 (Ginzburg-Landau); 20 (flat plate)
    The rank is chosen by hand for each case; the central comparison of methods at the same rank depends on this user choice.
  • Number of snapshots n = 15 (GL), 200 (flat plate)
    The data matrices are built from n snapshot pairs; the comparison and control performance depend on this user choice.
  • Snapshot time separation T = 20, 50, 100 for flat plate
    Controls how far apart the snapshots are in time; this affects the difficulty of identification, as shown in Table 5.
  • GL actuator center xa and width sigma = xa=8, sigma=5
    Hand-chosen for the GL control problem; the actuator location sweep varies xa, but the base controller uses this location.
  • Flat plate actuator constants = Table 4: xc1=0, yc1=1.3423, xc2=0, yc2=0.89, a=20, c=2
    Taken from the cited work [21], not fitted here, but the control results depend on these values.
assumptions (5)
  • domain assumption The flow dynamics of interest are well approximated, in the region of state space used for data collection, by a linear time-invariant discrete-time system x_{k+1} = A x_k + B u_k.
    The entire method builds a linear ROM from snapshots and uses LQR control, which assumes linearity. Section 4 states the LQR is used for stabilization about an unstable base flow.
  • ad hoc to paper The rank-r linear least-squares problem min_{L,D,R} ||Y - LDR^T X||_F^2 has its global minimizer attained and the proposed algorithms find a sufficiently good local minimum.
    Remark 2 admits no global convergence guarantee, and the subspace projection method in Algorithm 1 is heuristic. The reported advantages of lrDMD rely on this.
  • domain assumption The immersed boundary method simulation [50] and the Newton-GMRES steady-state computation are accurate enough to generate trustworthy snapshots.
    Section 5.2.1 uses these to produce data; errors in the simulation propagate to the reduced-order models.
  • domain assumption The unstable eigenmode and adjoint mode of the Ginzburg-Landau system are the relevant modes for control, and the projection errors of these modes measure controller-relevant fidelity.
    Section 5.1.1 evaluates only the single unstable mode and its adjoint; the claim about improved adjoint estimation rests on this choice.
  • standard math Standard results from numerical linear algebra (SVD, least squares, Lyapunov and Riccati solvers) and Riemannian optimization (ManOpt trust-region) are correct.
    These established tools are used throughout the paper without proof.

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Pith. "Pith review of Reduced-order control using low-rank Dynamic Mode Decomposition." pith.science (2026). https://pith.science/paper/ZW4ISXBI

@misc{pith2026190802249,
  author       = {Pith},
  title        = {Pith review of: Reduced-order control using low-rank Dynamic Mode Decomposition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZW4ISXBI}},
  note         = {Machine review of arXiv:1908.02249}
}
read the original abstract

In this work we perform full-state LQR feedback control of fluid flows using non-intrusive data-driven reduced-order models. We propose a model reduction method called low-rank Dynamic Mode Decomposition (lrDMD) that solves for a rank-constrained linear representation of the dynamical system. lrDMD is shown to have lower data reconstruction error compared to standard Optimal Mode Decomposition (OMD) and Dynamic Mode Decomposition (DMD), but with an increased computational cost arising from solving a non-convex matrix optimization problem. We demonstrate model order reduction on the complex linearized Ginzburg-Landau equation in the globally unstable regime and on the unsteady flow over a flat plate at a high angle of attack. In both cases, low-dimensional full-state feedback controller is constructed using reduced-order models constructed using DMD, OMD and lrDMD. It is shown that lrDMD stabilizes the Ginzburg-Landau system with a lower order controller and is able to suppress vortex shedding from an inclined flat plate at a cost lower than either DMD or OMD. It is further shown that lrDMD yields an improved estimate of the adjoint system, for a given rank, relative to DMD and OMD.

Figures

Figures reproduced from arXiv: 1908.02249 by the authors.

Figure 1
Figure 1. (a) Eigenvalues of the discrete GL system matrix (black circles). Unit cir [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The eigenvalues of the uncontrolled (◦) and controlled GL system with reduced-order controllers constructed using DMD, OMD and lrDMD with (a,b) rank 5 and (c) rank 9 approximations. (d) Shows the eigenvalues of uncontrolled and optimally controlled GL system. also fails to stabilize the system with one eigenvalue outside the unit circle. This shows that 5 modes are not sufficient for reduced order control even with … view at source ↗
Figure 3
Figure 3. Supremum of the cost J over all initial conditions for the controlled GL system with DMD (red), OMD (blue) and lrDMD (green) reduced-order controllers for (a) rank 5 approximation and (b) rank 9 approximation. 5.2 Flow past a flat plate In this section we demonstrate the performance of the controllers on stabilizing the two-dimensional uniform flow approaching an inclined flat plate. The freestream flow is at a low … view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) Vorticity contours and velocity streamlines for steady flow over 35 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Vorticity field generated by the actuator placed near the leading edge of a [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: CD vs. time plot of the impulse response of 35◦ inclined flat plate with steady state as the initial condition. Snapshots of this simulation were used in the study [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: CD vs. time for 35◦ inclined flat plate with the actuator activated at (a) t0 = 170 (b) t0 = 190 and (c) t0 = 210 using DMD, OMD and lrDMD based reduced-order controllers. Method t0 = 170 t0 = 190 t0 = 210 T = 50 T = 100 T = 50 T = 100 T = 50 T = 100 DMD ✓ ✓ ✓ ✗ ✗ ✗ OM…
Figure 8
Figure 8. Figure 8: Strength of the actuation when the controller was switched on at (a) [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.