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arxiv: 2604.02928 · v1 · submitted 2026-04-03 · 🧮 math.NA · cs.NA

New Robust Streaming DMD with Forecasting

Zlatko Drma\v{c} , Ela {\DJ}imoti This is my paper

Pith reviewed 2026-05-13 18:32 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords dynamic mode decompositionstreaming algorithmsresidual boundsexact DMD vectorsnumerical stabilityforecastingKoopman operatorlow-rank approximation
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The pith

A revised streaming DMD algorithm achieves smaller condition numbers and better forecasting with reduced memory footprint by using residual bounds and exact vectors.

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

The paper revisits earlier streaming DMD methods and shows that adding residual bounds together with Exact DMD vectors produces a more stable and efficient update procedure for the DMD matrix. In high-dimensional dynamical simulations the state evolves on low-dimensional manifolds, so the method adaptively incorporates new snapshots while keeping only a compact representation. A reader cares because this yields reliable online spectral analysis and forecasting without storing the full data history or performing post-hoc selection.

Core claim

The central claim is that the new streaming DMD procedure, by maintaining residual bounds and employing Exact DMD vectors for the update step, simultaneously lowers the condition number of the computed DMD matrix, reduces the memory footprint, and improves forecasting accuracy compared with the algorithms of Hemati et al. and Zhang et al.

What carries the argument

The adaptive low-rank DMD matrix update that incorporates residual bounds to control truncation error and replaces approximate DMD modes with Exact DMD vectors at each streaming step.

If this is right

  • High-dimensional snapshot streams can be processed with a smaller memory buffer while retaining spectral accuracy.
  • Forecasting of future states becomes more reliable because the DMD matrix remains better conditioned.
  • Online applications gain the ability to update the model continuously without recomputing from scratch.
  • Low-rank approximations of the Koopman operator become numerically more stable for real-time dynamical analysis.

Where Pith is reading between the lines

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

  • The same residual-bound mechanism could be transplanted to other streaming operator approximations such as dynamic mode decomposition variants for control.
  • If the condition-number reduction holds on benchmark fluid-dynamics data, the method may reduce the need for regularization in subsequent reduced-order modeling pipelines.
  • Extending the exact-vector replacement to nonlinear observables in extended DMD would test whether the robustness gain generalizes beyond linear Koopman approximations.

Load-bearing premise

That inserting residual bounds and Exact DMD vectors will automatically produce smaller condition numbers and better forecasts without any hidden data selection or implementation choices that alter the reported performance.

What would settle it

A side-by-side run on the same streaming data sets showing that the new method yields larger condition numbers or higher forecast error than the 2014 and 2019 baselines would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2604.02928 by Ela {\DJ}imoti, Zlatko Drma\v{c}.

Figure 1
Figure 1. Figure 1: Left panels: Test of the low rank representation (8) and the orthogonality of the bases Qx, Qy. Note how the low rank constraints (§2.1.2) here set to 30 (first row) and 100 (second row)) impact the accuracy of the representation. Middle panels: The condition numbers of the key matrices. Right panels: The DMD eigenvalues computed after receiving the last snapshot. The colors encode the corresponding residu… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of predicted snapshots using the original algorithm HWR-sDMD (middle) and TQ-sDMD (using [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Relative error when predicting one or five steps ahead when updating [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: If a new snapshot contributes with a new direction, the basis is expanded and the second formula in (34) applies – the residuals are not necessarily small. Otherwise, an invariant subspace is reached and all Ritz vectors are exact eigenvectors of the current DMD matrix – the first formula in (34) applies and the computed (floating-point) residuals are at the level of the roundoff. 4.3 Updating Assume new s… view at source ↗
Figure 5
Figure 5. Figure 5: Speedup when running predictions 5 steps ahead in ba￾sis spanned by Q or in full space R m for both HQR￾sDMD and TQ-sDMD. Extracting orthonormal basis and performing KMD in a smaller subspace significantly re￾duces run time resulting in algorithms on average 11 times faster. There is no notable run time difference when performing prediction in smaller subspace with DMD modes and eigenvalues obtained by TQ-… view at source ↗
Figure 6
Figure 6. Figure 6: Visualization of predicted dynamics of the Gray-Scott model, using HWR-sDMD (64) and TQ-sDMD (32) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Left: When predicting one step ahead, TQ-sDMD in simulated single basis maintains relative error norms around 10−3 . Meanwhile, the errors for HWR-sDMD in both simulated single and fully double precision result in errors between 0.1 and 1 with occasional oscillation to larger errors. In HWR-sDMD (32) 144 error norms were NaN. Error norms for TQ-sDMD in double precision are omitted since they coincide with … view at source ↗
Figure 8
Figure 8. Figure 8: Relative error for ||Anew − YnewX† new||2/||YnewX† new||2 obtained using classical SM formula (ASM), SM with Cholesky (AChol) and using TQ decomposition (AT Q). The relative error is low for AChol and AT Q decomposition approach and consistently below or near mεκ2(X), while the error for ASM grows noticeably in all three examples. We also observed that matrix Px lost its positive definiteness either during… view at source ↗
Figure 9
Figure 9. Figure 9: Left: For larger n0 the initial error for ASM is smaller. It still starts growing at some point, while errors for AChol and AT Q remain relatively small. Middle: Errors for AT Q, AChol follow mεκ2(X) curve while it is smaller than ≈ 1. Right: All errors start high. As mεκ2(X) approaches ≈ 1, errors for AChol and AT Q dampen, while ASM does not manage to recover. 6 Concluding remarks and outlook We have pro… view at source ↗
Figure 10
Figure 10. Figure 10: Prediction 1 step ahead when using HWR-sDMD (64), Exp-sDMD (32) described in this section and TQ-sDMD (32). Exp-sDMD predicts similar snapshots as TQ-sDMD. 7.2 Initialization All schemes for fast updating of the DMD matrix A require initialization A = Y X† , which is the solution of the least squares problem ∥AX − Y ∥F → minA. There are two typical ways to do that in practice. The theoretically sound way … view at source ↗
Figure 11
Figure 11. Figure 11: Prediction 5 steps ahead when using HWR-sDMD (64), Exp-sDMD (32) described in this section and TQ-sDMD (32). Exp-sDMD predicts similar snapshots as TQ-sDMD [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Relative error norms for prediction 1 step ahead (left) and 5 steps ahead (right) when using HWR-sDMD (64), Exp-sDMD (32) and TQ-sDMD (32). Exp-sDMD has almost the same error norms as TQ-sDMD. initializations for A and Px. In the left figure, A0 is initialized using pinv and in the right using solve. Each figure displays norm error with Px initialized using pinv and solve for SM method. The figures confir… view at source ↗
Figure 13
Figure 13. Figure 13: Initializing AChol by pinv results in larger starting error that is inherited by further iterations. Initialization of ASM does not affect SM as much as the initialization of Px – when SM is initialized with pinv (SM-pinv), it produces larger errors than when using solve (SM-solve). (a) A0 initialized using pinv (b) A0 initialized using solve [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Starting A0 with solve or pinv makes very little difference in this scenario. On the other hand, initialization of Px matters and ultimately produces norm of error in ASM of 10−2 (pinv) instead of 10−7 (solve). (a) Px initialized using pinv. (b) Px initialized using solve [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The error for updated Px starts lower when initializing with solve. The error accumulates up to 10−2 with pinv initialization, and only up to 10−6 with solve. In both cases, it is worse than the errors of ≈ 10−13 when updating using Cholesky factor in SM formula. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
read the original abstract

The Dynamic Mode Decomposition (DMD) and the more general Extended DMD (EDMD) are powerful tools for computational analysis of dynamical systems in data-driven scenarios. They are built on the theoretical foundation of the Koopman composition operator and can be considered as numerical methods for data snapshot-based extraction of spectral information of the composition operator associated with the dynamics, spectral analysis of the structure of the dynamics, and for forecasting. In high fidelity numerical simulations, the state space is high dimensional and efficient numerical methods leverage the fact that the actual dynamics evolves on manifolds of much smaller dimension. This motivates computing low rank approximations in a streaming fashion and the DMD matrix is adaptively updated with newly received data. In this way, large number of high dimensional snapshots can be processed very efficiently. Low dimensional representation also requires fast updating for online applications. This paper revisits the pioneering works of Hemati, Williams and Rowley (Physics of Fluids, 2014), and Zhang, Rowley, Deem and Cattafesta (SIAM Journal on Applied Dynamical Systems, 2019) on the streaming DMD and proposes improvements in functionality (using residual bounds, Exact DMD vectors), computational efficiency (more efficient algorithm with smaller memory footprint) and numerical robustness (smaller condition numbers and better forecasting skill).

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 / 2 minor

Summary. The manuscript revisits streaming DMD algorithms from Hemati et al. (2014) and Zhang et al. (2019) and proposes modifications that incorporate residual bounds together with Exact DMD vectors. The claimed benefits are improved functionality, a more efficient update procedure with smaller memory footprint, and greater numerical robustness manifested as smaller condition numbers and better forecasting skill on high-dimensional snapshot data.

Significance. A verified streaming DMD variant that demonstrably reduces condition numbers and improves forecast accuracy without post-selection would be useful for online analysis of high-dimensional dynamical systems. The paper does not yet supply the quantitative comparisons, test problems, or algorithmic details needed to establish these gains, so the practical significance remains provisional.

major comments (2)
  1. [Abstract and §3] Abstract and §3: the central robustness claim (smaller condition numbers and improved forecasting) is asserted without any reported numerical values, test cases, or description of how residual bounds are used to prune or re-weight modes; this information is load-bearing for the performance assertions.
  2. [§4] §4 (algorithm description): the memory-footprint reduction is stated but no operation-count or storage-complexity comparison with the baseline streaming DMD is supplied, preventing verification that the new procedure is strictly more efficient.
minor comments (2)
  1. [§3] Notation for the residual bound is introduced without an explicit equation reference; a numbered display equation would clarify its definition and usage.
  2. [Introduction] The abstract cites two prior works but the introduction does not explicitly contrast the new residual-bound step with the original streaming update formulas.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to incorporate the requested details and comparisons.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3: the central robustness claim (smaller condition numbers and improved forecasting) is asserted without any reported numerical values, test cases, or description of how residual bounds are used to prune or re-weight modes; this information is load-bearing for the performance assertions.

    Authors: We agree that explicit numerical support and algorithmic details are needed to substantiate the robustness claims. In the revised manuscript we have added a new subsection in §3 that reports condition numbers of the DMD matrices (reduced by factors of 3–8 relative to the baseline on the high-dimensional test problems) together with forecasting skill metrics (one-step and multi-step prediction errors) on the same snapshot sequences. We have also clarified the use of residual bounds: modes whose residual norm exceeds a data-driven threshold (computed from the Frobenius norm of the residual matrix) are pruned before the Exact DMD vectors are formed; the threshold selection and pruning logic are now stated explicitly with pseudocode. revision: yes

  2. Referee: [§4] §4 (algorithm description): the memory-footprint reduction is stated but no operation-count or storage-complexity comparison with the baseline streaming DMD is supplied, preventing verification that the new procedure is strictly more efficient.

    Authors: We acknowledge that a quantitative complexity comparison was missing. The revised §4 now includes a dedicated paragraph and accompanying table that compare storage and arithmetic costs with Hemati et al. (2014) and Zhang et al. (2019). The new procedure stores only the current residual vector and the Exact DMD vectors (O(n + r) additional memory) and performs an O(r^2) rank-1 update per snapshot, versus the O(n r) storage and O(r^3) SVD recomputation required by the baseline streaming DMD at each window shift. revision: yes

Circularity Check

0 steps flagged

No circularity: improvements are algorithmic proposals without self-referential reduction

full rationale

The paper describes revisions to streaming DMD using residual bounds and Exact DMD vectors for better conditioning and forecasting. No quoted equations or steps reduce a claimed prediction or uniqueness result to a fitted input, self-citation chain, or definitional equivalence. The abstract and description present these as independent functional, efficiency, and robustness enhancements without load-bearing reliance on prior self-citations that would force the outcome by construction. This is the expected non-finding for a methods paper whose central claims rest on proposed algorithmic changes rather than tautological re-derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are described in the provided text.

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