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arxiv: 2604.14350 · v1 · submitted 2026-04-15 · 💻 cs.CE

Weak-DMD: A Galerkin approach to the problem of noise in the Dynamic Mode Decomposition algorithm

William Bennett , Ryan G. McClarren , Ethan Smith , Melek Derman This is my paper

Pith reviewed 2026-05-10 11:42 UTC · model grok-4.3

classification 💻 cs.CE
keywords Dynamic Mode DecompositionGalerkin projectionweak formulationnoise filteringdata-driven modelingfluid dynamicsnuclear engineering
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The pith

Weak-DMD reformulates dynamic mode decomposition with Galerkin projection to filter noise and remove timestep requirements.

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

The paper develops a weak formulation of dynamic mode decomposition called weak-DMD. This approach applies Galerkin projection to the standard DMD equations. The projection step averages the data relations over test functions, which reduces the bias from measurement noise and removes any need for snapshots to be taken at fixed time intervals. The authors test the resulting modes and eigenvalues on fluid flow past a cylinder plus two nuclear engineering cases and compare performance against a leading DMD variant.

Core claim

By casting the DMD problem in weak form via Galerkin projection, the algorithm projects the residual of the linear operator onto a space of test functions. This produces a noise-robust system whose solution yields the same modal decomposition as classical DMD yet without dependence on snapshot spacing and with built-in attenuation of random errors in the state data.

What carries the argument

The Galerkin weak form of the DMD operator, which replaces direct time differences with integrated inner products against test functions to produce a linear system for the modes.

If this is right

  • Mode and eigenvalue estimates become less sensitive to additive measurement noise in the input snapshots.
  • Data can be collected at arbitrary time intervals without degrading the decomposition quality.
  • The method produces usable results on engineering datasets from nuclear systems and cylinder wake flows.
  • It provides a direct alternative to other noise-handling DMD variants without requiring additional preprocessing steps.

Where Pith is reading between the lines

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

  • The projection step may allow DMD on irregularly sampled sensor streams from physical experiments.
  • It could combine naturally with other projection-based reduction techniques already common in simulation codes.
  • Control applications that rely on online mode tracking might see lower sensitivity to sensor noise.

Load-bearing premise

Reformulating the DMD equations with a Galerkin projection will filter noise and eliminate the need for regular time intervals between data points while preserving accurate mode and eigenvalue approximations.

What would settle it

Apply weak-DMD and standard DMD to the same noisy snapshot set from the cylinder wake flow; if the dominant eigenvalue real part from weak-DMD is not closer to the known reference value than the standard result, the noise-filtering property does not hold.

Figures

Figures reproduced from arXiv: 2604.14350 by Ethan Smith, Melek Derman, Ryan G. McClarren, William Bennett.

Figure 1
Figure 1. Figure 1: Group summed neutron populations versus time for two radii for the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Logscaled convergence of the first and second eigenvalues given by [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Logscaled convergence of the first and second eigenvalues given by [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Measurement data (black circles) of the summed neutron population [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Measurement data (black circles) of the summed neutron population [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Logscaled convergence of the first and second eigenvalues given by [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Logscaled convergence of the first and second eigenvalues given by [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Scalar flux data (open circles) at t = 0.057 µs and reconstructed solution from weak-DMD with 12 bases of order p = 3 in the trial space for the Kornreich and Parsons benchmark problem. was used to quantify performance. The time-dependent error for the forecast solution, Uex, and the test data Ux, defined Error(t) = ||Ux(t) − Uex(t)||2 ||Ux(t)|| , (49) is averaged over t ∈ [201, 250][s]. This is a measure … view at source ↗
Figure 9
Figure 9. Figure 9: Color map of scalar flux data (top row) and reconstructions (lower [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Sum total scalar flux in the slab reactor Kornreich and Parsons [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Data (a) and first six spatial modes normalized by the [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Data (a) and first six spatial modes normalized by the [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Data (a) and first six spatial modes normalized by the [PITH_FULL_IMAGE:figures/full_fig_p019_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Data (a) and first six spatial modes normalized by the [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Log-log scaled averaged forecast error (Eq. (49)) for the flow past [PITH_FULL_IMAGE:figures/full_fig_p020_15.png] view at source ↗
read the original abstract

Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The standard formulation of DMD is subject to strict assumptions concerning the time-spacing of the snapshots and is biased by measurement noise. Variations on the method have been developed to address these shortcomings, but the problem is still open. Motivated by the effectiveness of Galerkin methods in the field of model discovery, a weak formulation of DMD is presented, weak-DMD. Weak-DMD precludes timestep considerations and also filters noise. Results for two nuclear engineering applications and the flow of fluid past a cylinder are given and compared with a state of the art DMD algorithm.

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 Weak-DMD, a Galerkin weak formulation of Dynamic Mode Decomposition (DMD) motivated by model discovery techniques. It claims that this formulation eliminates strict timestep spacing requirements of standard DMD and inherently filters measurement noise via averaging in the inner products, with empirical results presented for two nuclear engineering applications and cylinder flow, compared against a state-of-the-art DMD algorithm.

Significance. If the central claims hold under rigorous validation, Weak-DMD could provide a useful extension of DMD for noisy or irregularly sampled data in computational engineering, leveraging established Galerkin projection ideas. The practical demonstrations on nuclear and fluid dynamics problems add relevance, though the absence of detailed derivation or controlled validation in the provided abstract limits immediate assessment of impact.

major comments (2)
  1. [Abstract] Abstract: The claim that the weak formulation 'precludes timestep considerations and also filters noise' is presented without any equations, definition of the test space, or analysis of how the Galerkin projection of the DMD residual achieves noise suppression or well-defined time integrals for arbitrary snapshot spacing; this is load-bearing for the central contribution.
  2. [Results] Results section: The comparisons on nuclear applications and cylinder flow are empirical only and do not include controlled sweeps over noise amplitude or sampling irregularity, leaving unverified the assumption that the chosen test functions and quadrature inherently suppress noise components while preserving accurate eigenvalue and mode approximations.
minor comments (1)
  1. [Abstract] The abstract would benefit from a one-sentence outline of the specific test functions or inner-product definition to aid reader understanding of the weak form.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We address the major comments point by point and outline the revisions we will make to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the weak formulation 'precludes timestep considerations and also filters noise' is presented without any equations, definition of the test space, or analysis of how the Galerkin projection of the DMD residual achieves noise suppression or well-defined time integrals for arbitrary snapshot spacing; this is load-bearing for the central contribution.

    Authors: The abstract is intentionally concise, as is standard for such summaries. However, we agree that the central claims benefit from clearer support in the abstract. The full manuscript details the weak formulation in Section 2, defining the test space and showing how the Galerkin projection yields well-defined time integrals for arbitrary spacing and noise filtering through averaging in the inner products. We will revise the abstract to include a short description of these aspects. revision: partial

  2. Referee: [Results] Results section: The comparisons on nuclear applications and cylinder flow are empirical only and do not include controlled sweeps over noise amplitude or sampling irregularity, leaving unverified the assumption that the chosen test functions and quadrature inherently suppress noise components while preserving accurate eigenvalue and mode approximations.

    Authors: We agree that additional controlled validation would be beneficial. The presented results demonstrate the method on practical, noisy datasets from nuclear applications and cylinder flow, where Weak-DMD outperforms standard DMD. However, to rigorously verify the noise suppression and handling of irregular sampling, we will incorporate a new set of controlled experiments using synthetic data with varying noise amplitudes and irregular time steps, comparing eigenvalue errors and mode accuracy. revision: yes

Circularity Check

0 steps flagged

No circularity: weak-DMD is a standard Galerkin projection applied to the DMD residual, with noise-filtering and timestep claims following from integral properties rather than redefinition

full rationale

The paper applies the Galerkin method to the DMD residual to obtain a weak form. This is a direct reformulation using test functions and inner products, not a self-definition or fitted parameter. Noise filtering arises from the averaging inherent in the inner products, and timestep independence from the integral formulation over arbitrary intervals; neither reduces to the input data by construction. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior author work are indicated. The derivation chain is self-contained against external benchmarks of Galerkin projection and DMD.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; the method rests on applying standard Galerkin projection to the DMD operator without introducing new entities or fitted parameters.

axioms (1)
  • domain assumption Galerkin weak formulation can be applied to the DMD problem to handle noise and arbitrary time spacing
    Motivated by effectiveness of Galerkin methods in model discovery

pith-pipeline@v0.9.0 · 5437 in / 1095 out tokens · 55734 ms · 2026-05-10T11:42:40.238602+00:00 · methodology

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