{"id":"6570fb5f-28b4-4233-a80e-df44f2d9994b","arxiv_id":"2507.20463","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A greedy quadratic-manifold training procedure that incorporates operator-inference prediction error into the basis selection yields substantially more accurate reduced models than reconstruction-only training.","lead":"Researchers propose a new way to train quadratic manifolds for nonintrusive model reduction that accounts for both reconstruction error and the prediction error of the downstream reduced model. On transport and turbulent flow tests, the approach improves reduced-model accuracy by up to two orders of magnitude compared to manifolds trained on reconstruction error alone.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No ablation isolates the OI-aware criterion from the new isotropic scaling, so the headline accuracy gain may be due to scaling alone.","rationale":"The reader's weakest_assumption focused on the training-rollout proxy and the interleaved turbine split, but their rationale also lists 'the absence of an ablation separating the two new components' as a reservation. I agree that the generalization concern is real, but the missing ablation is the more load-bearing issue for the central claim: the abstract explicitly attributes the improvements to the greedy procedure that 'takes into account both the reconstruction error ... and the prediction error,' yet the experimental comparison cannot distinguish the new OI-aware selection criterion from the simultaneously introduced isotropic scaling. Because isotropic scaling plausibly improves the operator-inference fit on its own, the causal attribution in the headline is not established by the current evidence. This is an internal-validity problem that a single ablation experiment would resolve, so the conditional verdict remains appropriate rather than a rejection. The interleaved test split in the turbulent-flow experiment is a related but secondary concern; the transport experiment has a clean temporal split and still shows the advertised trend, although it too is confounded by the missing ablation. Therefore I keep the reader's conditional verdict and request the ablation as the decisive check.","tokens_in":14738,"tokens_out":10288,"duration_ms":98250,"concrete_test":"Re-run the transport experiment of Section 4.1 with three configurations: (a) unscaled greedy [11] (the existing baseline), (b) isotropic scaling of Section 3.1 but with γ_op = 0 in Eq. (10), so the operator-inference prediction-error term is disabled and selection uses reconstruction error only, and (c) the full OI-aware method with γ_op = 1.0. Compare operator-inference test errors (Eq. 18) at r = 20 and γ = 10^-4, the setting where the paper reports OI-aware error near 6e-4 versus baseline near 1. If configuration (b) matches (c) within a factor of 2, the accuracy gain is attributable to isotropic scaling, not to the OI-aware criterion. If (c) clearly outperforms (b), the criterion's causal contribution is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that including operator-inference prediction error in the greedy selection (the OI-aware criterion, Eq. 10) is what produces the large accuracy gains. However, the proposed method differs from the compared baseline 'QM, greedy [11]' in two ways at once: the isotropic scaling of Section 3.1 and the new selection criterion. All experiments in Figures 2-6 compare the combined method against the unscaled, reconstruction-only baseline; no configuration with isotropic scaling but the prediction-error term disabled (γ_op = 0) is reported. Isotropic scaling changes the magnitudes of reduced coordinates and can substantially improve the conditioning and fit of the operator-inference regression independently of any effect of the OI-aware criterion. Indeed, the baseline's documented failure mode (oscillatory coordinates, Figure 1) could be mitigated by scaling alone, since well-scaled coordinates make the least-squares problem in Eq. (8) better behaved even if the coordinate trajectories remain non-smooth. Consequently, the experiments as presented do not establish that the OI-aware criterion, rather than the scaling, causes the observed up-to-two-orders-of-magnitude improvements. The interleaved train/test split in the turbine experiment (even vs. odd snapshots) further weakens the generalization claim, but the ablation gap is the primary internal-validity threat to the central causal assertion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a greedy training procedure for quadratic manifolds in nonintrusive model reduction. The key idea is to augment the reconstruction-error-based greedy selection of [11] with a term that measures the rollout error of an operator-inference model fit to the reduced trajectory, and to use isotropically scaled reduced coordinates. Numerical experiments on a linear transport problem and a large-eddy simulation of a wind-turbine flow demonstrate that the proposed approach achieves lower operator-inference test errors than the compared baselines, in some cases by up to two orders of magnitude.","tokens_in":15031,"tokens_out":11002,"duration_ms":107805,"significance":"If the central causal claim were established, this would be a useful advance: the greedy selection objective explicitly accounts for downstream model prediction error, which is a natural and potentially important idea for nonlinear nonintrusive model reduction. The paper is clearly written, discusses the computational complexity of the greedy objective, and includes a realistic wind-turbine test case. However, the current experimental evidence does not isolate the proposed selection criterion from the isotropic scaling, and the turbine experiment's train/test split is not a genuine temporal holdout. The ideas are promising and the issues appear addressable within the manuscript's scope.","major_comments":[{"comment":"The definition of the prediction-error term is inconsistent. The text states that the operator-inference model is trained once per greedy iteration on the reduced trajectory eS_{i-1}=f_{V^{σ^{-1}}_{i-1}}(S), which has dimension i−1, and that \\hat{eS} is the corresponding prediction. Equation (10) then defines eS = f_{[V^{σ^{-1}},v]}(S), which has dimension i, so the Frobenius norm ||eS − \\hat{eS}||_F is not well-defined. Equation (15) and the cost discussion in Section 3.3 imply that an operator-inference model is refit for the candidate-augmented trajectory, but this is not what Section 3.2 describes. This is central to the method and must be clarified.","section":"Section 3.2, Eq. (10)"},{"comment":"The proposed method differs from the 'QM, greedy [11]' baseline in two respects: it uses the isotropic scaling of Section 3.1 and the operator-inference-aware criterion of Eq. (10). No experiment is reported with isotropic scaling but with the prediction-error term disabled (γ_op=0), so the observed gains cannot be attributed to the new criterion; they may be due to scaling alone, which changes the conditioning of the operator-inference least-squares problem in Eq. (8). Please add an ablation with γ_op=0 and isotropic scaling to isolate the effect of the OI-aware criterion.","section":"Section 4, Figures 2–6"},{"comment":"The training and test sets for the turbine experiment are defined as S_train = [s(0),s(2),...,s(998)] and S_test = [s(1),s(3),...,s(999)], i.e., even-odd interleaving of the same trajectory with a 0.1 s time step. The test snapshots are thus one time step from training states and do not represent an unseen trajectory or a temporally disjoint test condition. A hold-out interval (e.g., early vs. late time) or a different inflow condition should be used to support the generalization claim.","section":"Section 4.2.1"},{"comment":"The text says that for the operator-inference models 'we show results for the regularization parameters γA and γH with the lowest error.' If this selection is performed on the test set, the reported model prediction errors are optimistically biased. Please specify how these parameters were selected (e.g., using a separate validation set) and apply the same rule to all methods. In addition, γop is set to 1.0 (transport) and 10^2 (turbine) without a stated validation procedure; provide a selection rule or sensitivity analysis for these values.","section":"Sections 4.1.2 and 4.2.2"}],"minor_comments":[{"comment":"The argument \\tilde{v} in J_rec(\\tilde{v}, V^{σ^{-1}}) is not defined; it should presumably be the candidate vector v.","section":"Eq. (10)"},{"comment":"Equation (14) defines W' ∈ R^{n×r(r+1)/2}, while Algorithm 1 and the surrounding text use W' ∈ R^{(k-i)×p}; the dimensions should be made consistent.","section":"Eq. (14) and Algorithm 1"},{"comment":"The caption uses 'relative model prediction error' without defining the normalization; please state the precise formula.","section":"Figure 1"},{"comment":"The grid search for the reconstruction regularization parameter γ is described, but it is not stated whether γ is chosen on a training or validation set; this should be stated explicitly to avoid ambiguity.","section":"Section 4.1.1"},{"comment":"The complexity statement says the model prediction error costs O(r^3 k) per candidate; multiplying by q candidates and r iterations gives O(q r^4 k), which could be significant, and a brief discussion of this total cost would be helpful.","section":"Section 3.3"}],"recommendation":"major_revision","confidential_remarks":"The manuscript does not currently provide code or data availability statements; given the strong experimental emphasis, the editor may wish to request them. The novelty relative to [11] is an incremental but potentially important extension, and the missing ablation between scaling and the OI-aware criterion is the main internal-validity threat. The paper fits the journal's scope, but the experimental evidence needs to be strengthened before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper before you spend a full afternoon on it: the core idea is good, the transport experiment is genuinely encouraging, but the paper does not actually demonstrate that the new operator-inference-aware selection criterion is what causes the big accuracy gains. The method changes two things at once relative to the prior greedy approach [11] — it adds isotropic coordinate scaling and replaces the selection objective with one that includes a rollout error. No configuration with scaling but the OI-aware term disabled (γ_op = 0) is shown, so the observed order-of-magnitude improvement could plausibly come from the scaling alone. That is the main soft spot, and it is fixable.\n\nWhat is genuinely new and useful: the greedy selection criterion in Eq. (10) that fits an operator-inference model to the current reduced trajectory and uses its rollout error when picking the next basis vector is, to my knowledge, new. The isotropic scaling of reduced coordinates is a sensible addition, and the authors give a convincing explanation of why unscaled greedy bases produce oscillatory coordinates that are hard to fit. The fast evaluation trick that reuses the SVD to avoid solving the full reconstruction least-squares problem is a practical contribution that makes the method affordable.\n\nThe transport results are the best evidence: the combined method beats the reconstruction-only greedy baseline by more than an order of magnitude in model prediction error across several reduced dimensions and regularization settings. The turbine experiment is weaker because the test snapshots are interleaved with training snapshots (even vs. odd time indices), so the test set is essentially one-step-ahead interpolation from the training trajectory. That does not support a claim about generalization to unseen operating conditions. In addition, the values of q and the selection/tuning of γ_op are not reported with any detail, and no code or data are released. None of these are fatal, but they matter for a methods paper.\n\nThe authors are clearly competent and the paper is honest about what it builds on; self-citation is transparent and appropriate. I would send this to peer review, but I would ask the authors to add an ablation separating the isotropic scaling from the OI-aware criterion, and to redo the turbine experiment with a held-out time block rather than interleaved snapshots. With those changes the empirical claim would be much more solid. As it stands, treat the abstract's causal claim as plausible but unproven.","headline":"Promising extension of greedy quadratic manifolds, but the headline accuracy gain is not causally isolated from the new isotropic scaling, and the turbine test is interleaved.","tokens_in":15536,"tokens_out":2038,"would_cite":true,"duration_ms":25092,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","65L10","65L12","65L20","65L70"],"pacs":[],"model":"deepseek-v4-flash","headline":"Training quadratic manifolds with a greedy criterion that includes the reduced model's prediction error, not just reconstruction error, produces nonintrusive reduced models that are up to two orders of magnitude more accurate.","keywords":["scientific machine learning","operator inference","model reduction","quadratic manifolds","greedy basis selection","isotropic reduced coordinates","prediction error"],"falsifier":"A decisive check is to evaluate the method on a test set that is temporally separated from the training set by more than one time step (e.g., a later, non-interleaved time window), at matched reduced dimension, and see whether the operator-inference-aware greedy manifolds still outperform reconstruction-only greedy manifolds by the claimed margin; if the advantage shrinks or disappears, the training-rollout proxy is the weak link.","tokens_in":14550,"feed_emoji":"📉","tokens_out":15534,"duration_ms":128075,"temperature":0.7,"pith_summary":"The paper claims that quadratic manifolds—reduced-state spaces formed by a linear subspace plus a quadratic correction term—should be trained not only to reproduce snapshot data, but to make the downstream reduced model accurate. It proposes a greedy column-selection procedure whose objective combines the manifold's reconstruction error with the rollout prediction error of an operator-inference model, a low-dimensional quadratic dynamical system fitted to the encoded training trajectory. On a linear transport problem and a large-eddy simulation of flow around a wind turbine, the resulting reduced models achieve up to two orders of magnitude lower prediction error than manifolds trained on reconstruction error alone. The accompanying isotropic scaling of the basis, dividing each selected singular vector by its singular value, keeps reduced coordinates on a common scale and thereby avoids the oscillatory embeddings that reconstruction-only greedy training produces.","feed_headline":"Training on prediction error beats reconstruction-only reduced models","feed_subtitle":"Greedy manifold training that includes model rollout error cuts test prediction error by up to two orders of magnitude","key_machinery":"The central mechanism is the operator-inference-aware greedy selection criterion used together with isotropic reduced coordinates. At each greedy step the next column of the reduced basis is chosen as the scaled left-singular vector $\\phi^{(j)}_{\\sigma^{-1}} = \\phi^{(j)}/\\sigma_j$ that minimizes $J'_{\\mathrm{tot}}$, the manifold reconstruction error plus $\\gamma_{\\mathrm{op}}\\|\\Psi_I^\\top - \\hat{e}S\\|_F^2$, the rollout error of an operator-inference model trained on the encoded trajectory. Division by $\\sigma_j$ gives every reduced coordinate unit $\\ell^2$ norm over the training snapshots, so no coordinate is dominated by scale, and the trajectory-level error then guides selection toward embeddings that remain stable under the fitted quadratic model. This combination is what prevents the oscillatory, non-smooth reduced coordinates that make reconstruction-only greedy manifolds hard to model.","core_discovery":"The central discovery is that the way a quadratic manifold embeds snapshot data determines how well a reduced dynamical model can be fitted to it, and this dependence can be exploited in training. The paper defines a greedy selection criterion $J'_{\\mathrm{tot}}$ that scores each candidate singular vector by the sum of the manifold reconstruction error and a penalty proportional to the squared Frobenius distance between the encoded training trajectory and the trajectory predicted by an operator-inference model—a quadratic low-dimensional dynamical system—fitted to that encoding. Because the criterion penalizes trajectory-level rollout error rather than the one-step least-squares residual, it prefers embeddings that a first-order quadratic model can track over many time steps. Scaling the basis columns by the inverse singular values $\\sigma_j^{-1}$ makes the reduced coordinates isotropic, so coordinates from later singular vectors are not tiny and fast-oscillating; this avoids the non-smooth embeddings that reconstruction-only greedy training can produce. On the transport and turbulent-flow test problems, this changes the reduced-model prediction error by up to two orders of magnitude.","pith_inferences":["The same trajectory-rollout criterion could in principle be applied to other learned encoders and decoders, such as polynomial manifolds or autoencoders, wherever a downstream reduced dynamical model is fitted to the latent trajectory; the paper's argument is not tied to the specific quadratic feature map.","The turbulent-flow experiment interleaves test snapshots with training snapshots (odd versus even time indices), so the claimed gain on unseen data should be re-tested on a temporally disjoint window before generalizing it to longer forecasting horizons.","The isotropic-scaling step is separable from the greedy selection and could improve operator inference even with the leading-singular-vector basis whenever the singular values decay steeply.","The scalar $\\gamma_{\\mathrm{op}}$ controls the trade-off between reconstruction fidelity and model learnability; the experiments use fixed values (1.0 and $10^2$), so tuning it per problem is a natural next step."],"forward_implications":["On the transport and turbulent-flow test sets, operator-inference models built on the proposed greedy quadratic manifolds achieve up to two orders of magnitude lower prediction error than manifolds trained on reconstruction error alone.","The greedy procedure avoids oscillatory reduced coordinates, so the fitted operator-inference models remain stable over the test horizon rather than producing divergent or artifact-ridden predictions.","The reconstruction error of the proposed manifolds stays comparable to reconstruction-only greedy manifolds, so the accuracy gain in predictions does not come at the cost of data fidelity.","Because the selection criterion is evaluated without differentiating through the time stepper, the greedy search has per-evaluation cost $O(r^3 k)$ and can reuse a precomputed singular value decomposition of the snapshot matrix."],"supporting_citations":[{"why":"Provides the iterative greedy column-selection algorithm for quadratic manifolds that the new operator-inference-aware criterion modifies.","marker":"[11]"},{"why":"Defines operator inference, the downstream model-fitting procedure whose prediction error is folded into the greedy objective.","marker":"[12]"},{"why":"Establishes operator inference with quadratic manifolds and the baseline of using leading singular vectors as the reduced basis.","marker":"[46]"},{"why":"Justifies using trajectory rollout error rather than the one-step residual when judging reduced-model quality in the selection criterion.","marker":"[37]"},{"why":"Introduces the quadratic manifold decoder form and the quadratic feature map used throughout the paper.","marker":"[48]"}],"fun_headline_variants":["Greedy manifold training that includes rollout error cuts error up to 100x","Rollout-aware manifold training beats reconstruction-only by up to 100x","Reduced models improve up to 100x when manifold training includes prediction error","Quadratic manifold training with rollout error yields 100x better reduced models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method depends on the assumption that the operator-inference rollout error on the training trajectory is a faithful stand-in for prediction error on unseen data, an assumption that the turbulent-flow test (Section 4.2.1) weakens by placing test snapshots only one time step away from training snapshots.","fun_headline_variants_meta":{"raw":{"variants":["Greedy manifold training that includes rollout error cuts error up to 100x","Rollout-aware manifold training beats reconstruction-only by up to 100x","Reduced models improve up to 100x when manifold training includes prediction error","Quadratic manifold training with rollout error yields 100x better reduced models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001182,"raw_usage":{"total_tokens":4850,"prompt_tokens":884,"completion_tokens":3966,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":3884}},"tokens_in":500,"tokens_out":3966,"duration_ms":28073,"temperature":1.0,"reasoning_tokens":3884,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:44:01.258872+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to evaluate the method on a test set that is temporally separated from the training set by more than one time step (e.g., a later, non-interleaved time window), at matched reduced dimension, and see whether the operator-inference-aware greedy manifolds still outperform reconstruction-only greedy manifolds by the claimed margin; if the advantage shrinks or disappears, the training-rollout proxy is the weak link.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines operator inference, the downstream model-fitting procedure whose prediction error is folded into the greedy objective."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes operator inference with quadratic manifolds and the baseline of using leading singular vectors as the reduced basis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Justifies using trajectory rollout error rather than the one-step residual when judging reduced-model quality in the selection criterion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the quadratic manifold decoder form and the quadratic feature map used throughout the paper."}],"review_version":2}