{"id":"f988ecd1-1dc2-4493-a0fe-54cc1a77ac76","arxiv_id":"2604.14350","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Weak-DMD applies a Galerkin weak form to Dynamic Mode Decomposition to eliminate timestep constraints and filter noise in modal analysis.","lead":"The paper presents weak-DMD, a Galerkin weak-formulation of Dynamic Mode Decomposition that avoids strict timestep requirements and reduces sensitivity to measurement noise. If effective, this could make DMD more practical for real-world engineering data in nuclear systems and fluid flows.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Reformulating DMD via Galerkin projection does not automatically guarantee noise filtering or removal of timestep restrictions","rationale":"The reader's weakest assumption matches the load-bearing point exactly. Because the full manuscript supplies only application-specific results without the requested controlled verification, the UNVERDICTED status is unchanged.","tokens_in":1608,"tokens_out":305,"duration_ms":33015,"concrete_test":"Generate a linear system with known exact eigenvalues; create snapshot matrices at both uniform and randomly perturbed time steps, add Gaussian noise at 0%, 2%, and 5% levels; compute dominant eigenvalues with both standard DMD and weak-DMD using the paper's test-function choice; if weak-DMD eigenvalue error stays below 5% for irregular/noisy cases while standard DMD exceeds 20%, the claim holds.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests on the assertion that the weak form (via Galerkin projection of the DMD residual) inherently filters measurement noise through averaging and permits arbitrary snapshot spacing. This requires that the chosen test functions suppress the noise component in the inner products while the time integrals remain well-defined and accurate for non-uniform sampling. If the test space is not sufficiently rich or the quadrature for irregular times introduces bias, the eigenvalue and mode approximations can still degrade. The provided results on nuclear applications and cylinder flow offer empirical comparisons but do not include controlled sweeps over noise amplitude or sampling irregularity, leaving the assumption unverified.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1730,"tokens_out":379,"duration_ms":25958,"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":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Results"}],"minor_comments":[{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"partial","referee_comment":"[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."},{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1228,"tokens_out":363,"duration_ms":29020,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that the authors take standard DMD and rewrite it as a weak form using Galerkin projection, arguing this removes the usual equal-timestep requirement and reduces sensitivity to measurement noise through the inner-product averaging. They test the idea on two nuclear engineering datasets and the cylinder wake flow, showing comparisons against a current DMD implementation. That practical focus on engineering data is the clearest strength; the applications are relevant and the results are presented as direct improvements. The formulation itself is new in this exact combination, even if other noise-robust DMD variants exist. The soft spot is the central assumption that the weak form will inherently filter noise and permit arbitrary snapshot spacing without further conditions. The abstract gives no equations, no error bounds, and no controlled sweeps over noise levels or sampling irregularity, so it is not clear whether the chosen test functions actually suppress the noise term or whether quadrature on irregular times introduces its own bias. If the full paper contains the derivations and shows that the projection preserves the eigenvalues and modes under those conditions, the claim holds; otherwise the empirical wins could be case-specific. This is aimed at engineers doing modal analysis on noisy or unevenly sampled data in fluids or nuclear systems. A reader already working with DMD variants would get value from the concrete examples and the new angle. It deserves peer review because the problem is real and the approach is straightforward to check once the math and validation details are on the table.","headline":"Weak-DMD recasts DMD via Galerkin projection to claim noise filtering and flexible timing, but the supporting analysis looks thin on verification.","tokens_in":2186,"tokens_out":356,"would_cite":false,"duration_ms":15193,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Weak-DMD reformulates dynamic mode decomposition with Galerkin projection to filter noise and remove timestep requirements.","keywords":["Dynamic Mode Decomposition","Galerkin projection","weak formulation","noise filtering","data-driven modeling","fluid dynamics","nuclear engineering"],"falsifier":"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.","tokens_in":2523,"feed_emoji":"📊","tokens_out":587,"duration_ms":29444,"temperature":0.7,"pith_summary":"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.","feed_headline":"Galerkin DMD filters noise and drops snapshot timing rules","feed_subtitle":"Weak formulation projects data onto test functions to average errors while keeping mode accuracy in fluid and reactor tests.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Galerkin DMD filters noise without snapshot timing","Weak DMD filters noise via Galerkin no timing rules","DMD noise filtered by weak Galerkin no timing constraints","Weak Galerkin makes DMD noise robust without timing"],"cache_read_input_tokens":64,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Galerkin DMD filters noise without snapshot timing","Weak DMD filters noise via Galerkin no timing rules","DMD noise filtered by weak Galerkin no timing constraints","Weak Galerkin makes DMD noise robust without timing"]},"model":"grok-4.3","cost_usd":0.011274,"raw_usage":{"total_tokens":4821,"prompt_tokens":569,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":112740500,"prompt_tokens_details":{"text_tokens":569,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4192,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":569,"tokens_out":60,"duration_ms":33284,"temperature":1.0,"reasoning_tokens":4192,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-10T11:42:40.238602+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}