{"id":"032b383f-bebe-4d7b-9085-b1663e2c4fe1","arxiv_id":"2606.28569","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"NSDMD performs data-driven resolvent analysis on turbulent flow snapshots by explicitly subtracting nonlinear terms to recover the mean-flow linear operator.","lead":"The paper introduces nonlinearity-subtracted DMD (NSDMD), a data-driven method that subtracts nonlinear forcing terms from flow snapshots to recover a low-rank approximation of the mean-flow linearized Navier-Stokes operator. This lets researchers run resolvent-style linear analysis on turbulent flow data from any standard CFD code without specialized linear solvers.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Subtracting nonlinear term snapshots may not isolate a faithful low-rank approximation to mean-flow-linearized dynamics without residual effects or modeling","rationale":"The reader's weakest assumption directly identifies the same load-bearing step (the subtraction yielding a faithful linear operator). Because the full text was not examined for derivation details or validation metrics, the concern remains untested and the UNVERDICTED verdict is unchanged.","tokens_in":1751,"tokens_out":354,"duration_ms":41877,"concrete_test":"From the method section, recompute the NSDMD operator on the minimal-channel DNS snapshots both with and without the nonlinear-term subtraction; compare the leading resolvent gains and modes against a reference mean-flow linearization computed directly from the same mean profile. If the subtracted version deviates by more than the non-subtracted version in the dominant singular values, the isolation is incomplete.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires that NSDMD produces an operator that is a low-rank approximation of the true mean-flow-linearized NS operator. This hinges on the subtraction step exactly removing the nonlinear forcing contribution from the data so that the DMD operator matches the linearization around the (statistically steady) mean flow. In discrete snapshot data, however, the mean is obtained by time-averaging, the perturbation equations contain Reynolds-stress effects, and the nonlinear terms are evaluated at discrete times and locations; any mismatch in how these are subtracted (e.g., due to interpolation, filtering, or the precise form of the convective term) leaves a residual forcing that the learned operator must absorb. If that residual is not negligible, the resulting operator deviates from the intended linearized dynamics and the post-processing claim does not hold.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces nonlinearity-subtracted DMD (NSDMD), a data-driven method that subtracts snapshots of the nonlinear terms from the perturbation equations to recover a low-rank operator approximating the mean-flow-linearized Navier-Stokes dynamics. This enables data-driven resolvent analysis as a post-processing step on DNS or LES data from arbitrary high-fidelity CFD codes. The approach is demonstrated on a minimal channel flow and scaled to the flow over a full aircraft model.","tokens_in":1945,"tokens_out":618,"duration_ms":23011,"significance":"If the central claim holds, NSDMD would allow mean-flow-based linear analyses (resolvent, etc.) to be performed on existing simulation datasets without specialized linearization codes or access to the underlying solver, extending such tools to complex geometries. The manuscript provides a concrete implementation path that leverages the structure of the NS equations while remaining compatible with black-box CFD output.","major_comments":[{"comment":"§3.1–3.2 (method derivation): The subtraction of nonlinear-term snapshots is presented as yielding an operator that is a faithful low-rank approximation to the true mean-flow-linearized dynamics, yet no derivation or error bound is given showing that the discrete-time, discrete-space subtraction exactly cancels the nonlinear forcing contribution (including any residual Reynolds-stress effects arising from the time-averaged mean). Without this, it is unclear whether the learned DMD operator matches the classical resolvent operator or absorbs discretization artifacts.","section":"§3.1–3.2"},{"comment":"§4.1 (minimal channel results): The paper reports that NSDMD recovers structures consistent with classical resolvent analysis, but provides no quantitative metric (e.g., operator-norm difference, gain-curve L2 error, or singular-value spectrum comparison) against a reference linearization performed on the same mean flow. This leaves the central claim that the method produces “the underlying mean-flow-linearized dynamics” without direct verification.","section":"§4.1"},{"comment":"§4.2–4.3 (scaling to aircraft): While the aircraft demonstration shows practical applicability, the absence of an ablation on the precise form of the convective-term evaluation or interpolation used when subtracting nonlinear snapshots means any residual forcing is unquantified; this directly affects whether the post-processing claim holds for general CFD codes.","section":"§4.2–4.3"}],"minor_comments":[{"comment":"Notation for the perturbation velocity and nonlinear term is introduced without an explicit table relating symbols to the continuous NS equations; a short appendix table would improve readability.","section":null},{"comment":"Figure 3 (channel flow modes): axis labels and color-bar scaling are inconsistent with the corresponding resolvent gain plots in Figure 4; this makes direct visual comparison harder than necessary.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their constructive feedback and for recognizing the potential impact of NSDMD. We provide point-by-point responses to the major comments and indicate where revisions will be made to address the concerns.","responses":[{"response":"We clarify that the method is derived by subtracting the nonlinear term snapshots from the perturbation snapshots, which in the continuous setting exactly isolates the linear operator. In discrete settings, the approximation holds provided the mean flow is accurately computed and discretization is consistent. We will revise §3 to include a more detailed discussion of the assumptions, potential residuals from time-averaging, and discretization effects to address this concern.","revision_made":"yes","referee_comment":"[§3.1–3.2] §3.1–3.2 (method derivation): The subtraction of nonlinear-term snapshots is presented as yielding an operator that is a faithful low-rank approximation to the true mean-flow-linearized dynamics, yet no derivation or error bound is given showing that the discrete-time, discrete-space subtraction exactly cancels the nonlinear forcing contribution (including any residual Reynolds-stress effects arising from the time-averaged mean). Without this, it is unclear whether the learned DMD operator matches the classical resolvent operator or absorbs discretization artifacts."},{"response":"We agree that direct quantitative comparison would provide stronger evidence. Since the minimal channel allows for a reference linearization, we will add in §4.1 comparisons of the resolvent gain curves and singular value spectra between NSDMD and the classical mean-flow linearization to verify the match.","revision_made":"yes","referee_comment":"[§4.1] §4.1 (minimal channel results): The paper reports that NSDMD recovers structures consistent with classical resolvent analysis, but provides no quantitative metric (e.g., operator-norm difference, gain-curve L2 error, or singular-value spectrum comparison) against a reference linearization performed on the same mean flow. This leaves the central claim that the method produces “the underlying mean-flow-linearized dynamics” without direct verification."},{"response":"For the aircraft flow, the nonlinear snapshots are evaluated using the identical discretization scheme as the original simulation to minimize residuals. We will expand the discussion in §4.2–4.3 to include sensitivity analysis to interpolation and note that consistent evaluation is key for the method's applicability to arbitrary CFD codes. A full ablation may require additional simulations but we will quantify the effect where possible.","revision_made":"partial","referee_comment":"[§4.2–4.3] §4.2–4.3 (scaling to aircraft): While the aircraft demonstration shows practical applicability, the absence of an ablation on the precise form of the convective-term evaluation or interpolation used when subtracting nonlinear snapshots means any residual forcing is unquantified; this directly affects whether the post-processing claim holds for general CFD codes."}],"tokens_in":1518,"tokens_out":618,"duration_ms":44841,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The key takeaway is that NSDMD provides a data-driven route to mean-flow linear analysis of turbulent flows by subtracting snapshots of the nonlinear terms, allowing post-processing of standard CFD data.\n\nWhat the paper does well is introduce this construction and demonstrate it on progressively more complex cases: starting with a minimal channel flow from DNS, then moving to LES of the same, and finally applying it to flow over a full aircraft model. This shows the method can handle realistic geometries where traditional linear codes struggle. The use of the Navier-Stokes structure to define the subtraction is a smart way to embed physics into the DMD framework, and it directly addresses the limitation of standard DMD not capturing the linear mechanisms.\n\nThe soft spots are around validation and potential residuals. The stress-test concern is worth taking seriously: in snapshot data, time-averaging for the mean, the discrete evaluation of nonlinear terms, and any interpolation or filtering could leave a residual forcing that gets folded into the learned operator. The abstract claims it yields a low-rank approximation of the linearized dynamics, but without detailed error analysis or side-by-side comparisons to classical resolvent results on the same datasets, it's difficult to gauge how much bias or deviation is introduced. If the paper has those comparisons in the full text, they would strengthen the case considerably.\n\nThis work is aimed at researchers in turbulence who use high-fidelity simulations and want access to linear analysis tools without rewriting their codes. Someone working on flow control or reduced-order modeling in complex geometries would get practical value from the examples and the method.\n\nOverall, the paper shows clear thinking in adapting DMD with NS knowledge, so it deserves to go to peer review for a full assessment of the technical details and results.","headline":"NSDMD is a practical extension of DMD that subtracts nonlinear terms to recover mean-flow linear operators from CFD data, though its fidelity to classical resolvent analysis requires closer scrutiny in the results.","tokens_in":2444,"tokens_out":430,"would_cite":false,"duration_ms":38705,"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":"Nonlinearity-subtracted DMD allows linear analysis of turbulent flows from any CFD simulation data.","keywords":["dynamic mode decomposition","resolvent analysis","turbulent flows","data-driven analysis","Navier-Stokes equations","mean flow","computational fluid dynamics","nonlinear forcing"],"falsifier":"Direct computation of the resolvent operator from the linearized equations around the known mean flow on the same dataset, followed by comparison of the resulting modes and gains to those obtained via NSDMD; significant differences would indicate the approximation does not hold.","tokens_in":2657,"feed_emoji":"📊","tokens_out":439,"duration_ms":53001,"temperature":0.7,"pith_summary":"The paper develops nonlinearity-subtracted dynamic mode decomposition to enable linear analyses such as resolvent analysis on turbulent flows. It does so by using snapshots of the nonlinear terms to remove their effect, so the learned operator approximates the mean-flow linearized Navier-Stokes dynamics. This turns the method into a post-processing tool that works with data from any high-fidelity CFD simulation code. Demonstrations include a minimal channel flow and the flow over a full aircraft model using DNS and LES data. The approach matters because it removes the need for specialized codes when studying complex geometries.","feed_headline":"Subtracting nonlinear terms lets DMD recover linearized flow dynamics","feed_subtitle":"The NSDMD method approximates the mean-flow linearized operator from standard simulation snapshots for use on channel and aircraft flows.","key_machinery":"nonlinearity-subtracted DMD (NSDMD), which explicitly accounts for nonlinear forcing by subtracting its snapshots to isolate the linear operator.","core_discovery":"NSDMD modifies the standard DMD procedure by subtracting the contribution of the nonlinear terms in the perturbation equations from the data snapshots, yielding an operator that is a low-rank approximation of the underlying mean-flow-linearized dynamics and thereby allowing data-driven performance of resolvent analysis.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["NSDMD subtracts nonlinear terms for data-driven linear flow analysis","Data snapshots minus nonlinear terms recover linearized turbulent dynamics","NSDMD approximates mean-flow linear operator via nonlinearity subtraction","Nonlinearity-subtracted DMD for resolvent analysis of aircraft flow data"],"cache_read_input_tokens":64,"weakest_assumption_plain":"Subtracting snapshots of the nonlinear terms from the data produces an operator that faithfully approximates the true mean-flow-linearized dynamics without any extra modeling.","fun_headline_variants_meta":{"raw":{"variants":["NSDMD subtracts nonlinear terms for data-driven linear flow analysis","Data snapshots minus nonlinear terms recover linearized turbulent dynamics","NSDMD approximates mean-flow linear operator via nonlinearity subtraction","Nonlinearity-subtracted DMD for resolvent analysis of aircraft flow data"]},"model":"grok-4.3","cost_usd":0.009764,"raw_usage":{"total_tokens":4348,"prompt_tokens":670,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":97637000,"prompt_tokens_details":{"text_tokens":670,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3611,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":670,"tokens_out":67,"duration_ms":40859,"temperature":1.0,"reasoning_tokens":3611,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T00:45:34.178055+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Direct computation of the resolvent operator from the linearized equations around the known mean flow on the same dataset, followed by comparison of the resulting modes and gains to those obtained via NSDMD; significant differences would indicate the approximation does not hold.","supporting_citations":[],"review_version":1}