{"id":"c1984bb8-665d-40d6-8cba-d79619008053","arxiv_id":"1908.11233","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Re-projecting the full system after every time step before collecting data lets operator inference recover, with finite data, exactly the same reduced models that intrusive Galerkin projection produces.","lead":"A new data sampling scheme lets data-driven reduced models exactly match the reduced models built from full knowledge of the governing equations. The key idea is to step the high-dimensional system once, then re-project onto the low-dimensional space, before collecting each training point.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem is correct under its stated active-query assumption; the main caveat is that passive pre-recorded data cannot generate re-projected trajectories, so the exact-recovery guarantee does not extend to the pure data-only setting.","rationale":"The paper's proofs are straightforward and internally consistent. Proposition 1 follows by induction using the polynomial structure and the mixed-product property of the Kronecker product; Corollary 1 follows because the intrusive operators attain zero objective and full row rank of the data matrix gives uniqueness. The numerical experiments corroborate the theory, and the known failure mode of operator inference without re-projection is clearly demonstrated in the toy example. The reader's weakest-assumption identification is accurate: the method requires active querying of the high-dimensional system from arbitrary points in the reduced subspace. This is a real scope restriction for black-box or purely observational settings, but the manuscript states it explicitly and does not claim applicability to passive data. Because the conditional theorem is sound and the limitation is acknowledged, the acceptance verdict should remain unchanged.","tokens_in":21743,"tokens_out":14289,"duration_ms":144977,"concrete_test":"On a nonlinear test system (for example, the Chafee-Infante discretization of Section 5.3), run two variants: (i) Algorithm 1 with a simulator that can be restarted at Vn * bar_x_k, and verify that the re-projected trajectory bar_X equals the intrusive reduced-model trajectory tilde_X to machine precision; and (ii) remove the restart capability and fit operators only to the projected trajectory brevis_X = Vn^T X from the same initial condition and inputs. Compare the learned operators with the intrusive operators tilde_O. If variant (i) passes and variant (ii) produces nonzero operator error, the active-query assumption is necessary for the exact-recovery guarantee and passive data do not suffice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result (Prop. 1 and Cor. 1) is mathematically sound: under the stated conditions, re-projected trajectories coincide with intrusive reduced-model trajectories, and least squares recovers the intrusive operators. The load-bearing assumption is that the high-dimensional system can be queried for a single time step from arbitrary reconstructed states Vn * bar_x_k (Algorithm 1, line 4; Section 4.2). This is an active data-generation assumption, not a passive observation assumption. If only pre-recorded trajectories are available, which is the usual motivating scenario for operator inference, this step cannot be executed, and fitting to plain projected trajectories reintroduces the closure error (Section 2.4, Figure 1). The paper explicitly acknowledges this limitation, so it does not invalidate the theorem, but it materially narrows the scope of 'from data.' The full-rank condition on the data matrix is also an assumption to be verified rather than a proven property of generic inputs, but it is stated as such and can be checked in practice.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper addresses the closure-error problem in data-driven model reduction via operator inference. For polynomial dynamical systems (1)-(2), standard operator inference fits reduced operators to projected trajectories, which are non-Markovian and therefore lead to operators that differ from the intrusive Galerkin reduced operators. The paper proposes Algorithm 1, which generates re-projected trajectories by alternately evaluating the high-dimensional system for one step from reconstructed states Vn * bar_x_k and projecting back to Vn. Proposition 1 proves that, when x0 is in Vn, these re-projected trajectories coincide exactly with trajectories of the intrusive reduced model. Corollary 1 shows that if the corresponding data matrix has full rank and the trajectory length satisfies K >= p + sum_i n_i, then least-squares operator inference recovers the intrusive operators exactly. Algorithm 2 integrates the procedure with POD across parameters. Numerical experiments on a linear toy problem, Burgers' equation, Chafee-Infante, and a 2D diffusion-reaction equation show that models learned from re-projected trajectories match intrusive reduced models and remain stable where models from plain projected trajectories diverge. The paper explicitly discusses limitations: the need to query the high-dimensional system, condition-number growth, and the checkable nature of the full-rank assumption.","tokens_in":21921,"tokens_out":10192,"duration_ms":94421,"significance":"The main contribution is a simple but effective idea: by re-projecting after every single high-dimensional step, the non-Markovian component of the projected dynamics is removed, making the sampled trajectories exactly those of the intrusive reduced model. The proofs of Proposition 1 and Corollary 1 are correct under the stated assumptions, and the paper is honest about the conditions (x0 in Vn, full-rank data matrix, active query access). If the result is taken together with these conditions, it provides a pre-asymptotic, finite-data guarantee that connects data-driven operator inference with classical projection-based model reduction, so that learned models inherit the well-studied properties of intrusive reduced models. The numerical study is convincing and includes cases where standard operator inference is unstable. The main caveat is that the guarantee depends on the ability to generate re-projected trajectories, which is an active sampling requirement rather than a passive-data setting.","major_comments":[],"minor_comments":[{"comment":"The abstract and introduction describe the method as learning 'from data' without qualification, but Algorithm 1 line 4 requires active queries of the high-dimensional system from arbitrary reconstructed states Vn * bar_x_k; please state this active-data-generation requirement as a central condition so that the scope of Corollary 1 is not overstated.","section":"Abstract and Section 4.2"},{"comment":"The displayed results for n < bar_n are obtained by truncating operators learned at dimension bar_n; Corollary 1 guarantees exact recovery only at the re-projection dimension, so please state explicitly that the agreement at truncated dimensions is an empirical observation rather than a consequence of the theorem.","section":"Section 5.2.1 and Figures 4-5"},{"comment":"In the toy example, the initial condition is said to be the first column of the identity matrix of dimension n x n; since x0 is in R^N with N = 10, this should be the N x N identity matrix.","section":"Section 2.4"},{"comment":"The block matrix A_perp_perp_1 is defined as (V_n^perp)^T A_1 (V_n^perp)^T; the second factor should be V_n^perp, not (V_n^perp)^T.","section":"Section 3.1"},{"comment":"The last block of bar_D is written as bar_X^j; it should be bar_X^ell to match the polynomial degree.","section":"Corollary 1, Eq. (16)"},{"comment":"The initial condition is written as x(xi, t) = 0 for xi in Omega union {0,1}; the time argument should be 0 rather than t.","section":"Section 5.3.1"}],"recommendation":"minor_revision","confidential_remarks":"To the editor: This is a solid, well-written paper. The main point I would like the authors to address is the framing of 'from data'; the method requires active query access, which is acknowledged in Section 4.2 but should be visible from the abstract. I do not see any correctness issue in the main theorems or experiments. The paper fits the journal's scope and should be accepted after a minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Glad you asked. This paper does what it says: for polynomial systems, sampling one-step trajectories with re-projection makes the fitted operators exactly the intrusive reduced-model operators, under a full-rank data matrix. I checked the proofs and they hold. The numerical experiments are honest and support the claims.\n\nWhat is genuinely new is the re-projection sampling scheme. Prior operator inference fit to projected trajectories and suffered from closure error; this scheme cancels the non-Markovian memory by re-projecting after every step. The pre-asymptotic exact recovery guarantee is a real result, not a routine extension.\n\nThe main caveat is not hidden: the method requires active querying of the full system one step at a time from states Vn * bar_x_k. If you only have recorded trajectories, the scheme doesn't apply. That narrows the 'from data' claim, but the paper states it clearly. The full-rank condition on the data matrix is an assumption, not a proven property; the paper says it can be checked in practice, which is fair. Condition numbers can be high and pollute the learned operators, and the experiments show this; concatenating trajectories helps. I'd have liked code/data released, but that's not the standard for a methods paper.\n\nOn circularity: the guarantee is close to being built into the sampling definition, but that's acceptable. It's an algorithmic construction, and the least-squares recovery is still nontrivial because it requires the data matrix to be full rank.\n\nThe paper is for anyone applying operator inference or projection-based model reduction to simulators that can be queried. It deserves a serious referee and, after minor clarifications, acceptance. I'd take it to peer review.","headline":"Clean, narrow result: re-projection sampling provably recovers intrusive reduced models, but only when the simulator can be queried one step from arbitrary reconstructed states.","tokens_in":22424,"tokens_out":1195,"would_cite":true,"duration_ms":11839,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M99","93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper introduces a data-sampling scheme with re-projection that makes data-driven operator inference recover, exactly and for finite data, the same reduced models that traditional intrusive projection-based model reduction constructs.","keywords":["operator inference","model reduction","data-driven modeling","proper orthogonal decomposition","non-Markovian dynamics","polynomial nonlinear systems","least-squares recovery","re-projected trajectories"],"falsifier":"Construct a linear system $x_{k+1} = A x_k$ with a chosen subspace $V_n$ for which the cross-coupling $A_{\\parallel \\perp}$ is nonzero, run Algorithm 1 with exactly $K = p + n_1$ samples, and solve (17); if the data matrix $\\bar{D}$ has full rank but the recovered operator differs from $\\tilde{A} = V_n^T A V_n$ beyond rounding error, Corollary 1 is false. A simpler direct test is to check whether any full-rank $\\bar{D}$ with $K$ satisfying (15) admits two different zero-residual least-squares solutions.","tokens_in":21545,"feed_emoji":"🎯","tokens_out":8868,"duration_ms":74682,"temperature":0.7,"pith_summary":"Data-driven model reduction usually fits low-dimensional models to projected trajectories of a high-dimensional system; because projection introduces non-Markovian memory, the fitted operators generally differ from those of the reduced model that would be built if the governing equations were known. This paper proposes sampling re-projected trajectories instead: at each time step, query the full system from the lifted reduced state and project the result back. It proves that these re-projected trajectories are exactly the trajectories of the traditional intrusive reduced model, so the operators inferred from them by least squares coincide with the reduced operators. Exact recovery holds pre-asymptotically for finite trajectory length $K \\geq p + \\sum_{i=1}^{\\ell} n_i$ and reduced dimension $n < N$, for systems with polynomial nonlinear terms, provided the data matrix has full rank. If the claim is right, learned models can inherit the error bounds and stability properties developed for projection-based model reduction.","feed_headline":"Re-projected data recovers classic reduced models exactly","feed_subtitle":"New sampling scheme cancels closure error, so learned models match intrusive model reduction on finite data.","key_machinery":"The key mechanism is the re-projection step in Algorithm 1: instead of recording $V_n^T x_k$ along an uninterrupted full-model trajectory, the algorithm alternates between lifting the current reduced state to $V_n \\bar{x}_k$, querying the full system for one step from that lifted state, and projecting back as $\\bar{x}_{k+1} = V_n^T f(V_n \\bar{x}_k, u_k)$. This turn-by-turn re-projection removes the accumulated influence of the orthogonal complement that would otherwise make the projected dynamics non-Markovian. The proof identity is the mixed-product property of Kronecker products: $V_n^T A_i (V_n \\otimes \\cdots \\otimes V_n)(\\bar{x} \\otimes \\cdots \\otimes \\bar{x})$ equals the projected operator $\\tilde{A}_i$ applied to the $i$-th power of $\\bar{x}$, so the sampled pairs obey exactly the intrusive reduced equations. The least-squares problem (17) then recovers $\\tilde{A}_1,\\ldots,\\tilde{A}_\\ell,\\tilde{B}$ uniquely when $\\bar{D}$ has full row rank and $K$ is large enough.","core_discovery":"The central claim is Corollary 1: for a polynomial system of degree $\\ell$, if re-projected trajectories are generated by Algorithm 1 from an initial condition in the reduced subspace, and if $K \\geq p + \\sum_{i=1}^{\\ell} n_i$ with $n_i = \\binom{n+i-1}{i}$, and the data matrix $\\bar{D}$ in (16) has full rank, then the unique least-squares solution satisfies $\\|\\hat{A}_i - \\tilde{A}_i\\|_F = \\|\\hat{B} - \\tilde{B}\\|_F = 0$ for all $i$. The learned model is therefore not merely an approximation of the intrusive reduced model; it is the intrusive reduced model. The reason is Proposition 1: because each full-model query starts from the lifted re-projected state $V_n \\bar{x}_k$, the projected next state equals the state produced by the projected reduced operators, so the closure error $\\|\\bar{X} - \\tilde{X}\\|_F$ is zero for the sampled data. Operator inference then recovers the reduced operators exactly as the unique zero-residual least-squares fit.","pith_inferences":["An extension the paper leaves implicit is that the same argument should apply wherever the full model can be queried one step from arbitrary lifted states, including implicit time steppers or stochastic forcing sampled per step; the essential requirement is that $f(V_n \\bar{x}_k, u_k)$ is well defined.","A testable practical extension is to use the two Corollary 1 conditions as an online certificate: when the data matrix has full rank and $K \\geq p + \\sum_i n_i$, one can declare the learned operators identical to the intrusive reduced operators without ever forming the full operators, thereby transferring traditional a posteriori error bounds to the data-driven model.","For non-polynomial nonlinearities, re-projection could be combined with lifting approaches; whether exact recovery persists would depend on whether the lifted representation closes under the polynomial-degree assumption.","The requirement to re-query the full model from new states is the price of exactness; in settings where only recorded trajectories exist, Corollary 1 would not apply, and some approximation of the re-projection would be needed."],"forward_implications":["Learned models are exactly the reduced models of traditional projection-based model reduction, so they inherit the well-studied properties of those models.","Exact recovery is pre-asymptotic: it does not require the number of data points or the reduced dimension to approach the full dimension.","The two conditions of Corollary 1—the trajectory-length bound (15) and full rank of the data matrix—can be checked numerically, giving a practical certificate that the intrusive model has been recovered.","In the numerical examples, models learned from re-projected trajectories match the intrusive reduced models to about $10^{-10}$ or better, whereas models learned without re-projection can become unstable and diverge.","The sampling procedure costs roughly twice one full-model simulation: one trajectory set for the basis and one re-projected set for the fit."],"supporting_citations":[{"why":"Supplies the operator-inference least-squares formulation and the polynomial-nonlinearity model class that the paper extends.","marker":"[38]"},{"why":"Provides the non-Markovian closure/memory view of projected dynamics that motivates the re-projection fix.","marker":"[17]"},{"why":"Gives the related memory/closure formalism referenced for why projected trajectories fail to follow reduced dynamics.","marker":"[13]"},{"why":"Defines projection-based reduced models and the properties that exactly recovered models inherit.","marker":"[9]"},{"why":"Documents the reduced-basis error-estimation framework used as the target property of traditional reduced models.","marker":"[43]"}],"fun_headline_variants":["Re-projected sampling recovers exact reduced models","Finite data recovers classic intrusive reduced models","Zero closure error: operator inference matches model reduction","New data scheme guarantees exact reduced model recovery","Pre-asymptotic exactness via re-projection sampling"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the high-dimensional system can be queried for one time step starting from arbitrary lifted states $V_n \\bar{x}_k$; if only pre-recorded trajectories are available and the system cannot be stepped from new states, the re-projected trajectories cannot be generated and the exact-recovery guarantee collapses.","fun_headline_variants_meta":{"raw":{"variants":["Re-projected sampling recovers exact reduced models","Finite data recovers classic intrusive reduced models","Zero closure error: operator inference matches model reduction","New data scheme guarantees exact reduced model recovery","Pre-asymptotic exactness via re-projection sampling"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00031,"raw_usage":{"total_tokens":1776,"prompt_tokens":964,"completion_tokens":812,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":580,"completion_tokens_details":{"reasoning_tokens":740}},"tokens_in":580,"tokens_out":812,"duration_ms":6592,"temperature":1.0,"reasoning_tokens":740,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:20:23.233804+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a linear system $x_{k+1} = A x_k$ with a chosen subspace $V_n$ for which the cross-coupling $A_{\\parallel \\perp}$ is nonzero, run Algorithm 1 with exactly $K = p + n_1$ samples, and solve (17); if the data matrix $\\bar{D}$ has full rank but the recovered operator differs from $\\tilde{A} = V_n^T A V_n$ beyond rounding error, Corollary 1 is false. A simpler direct test is to check whether any full-rank $\\bar{D}$ with $K$ satisfying (15) admits two different zero-residual least-squares solutions.","supporting_citations":[{"cited_title":"Peherstorfer and K","cited_arxiv_id":null,"evidence_quote":"Supplies the operator-inference least-squares formulation and the polynomial-nonlinearity model class that the paper extends."},{"cited_title":"Givon, R","cited_arxiv_id":null,"evidence_quote":"Provides the non-Markovian closure/memory view of projected dynamics that motivates the re-projection fix."},{"cited_title":"Chorin and P","cited_arxiv_id":null,"evidence_quote":"Gives the related memory/closure formalism referenced for why projected trajectories fail to follow reduced dynamics."},{"cited_title":"Benner, S","cited_arxiv_id":null,"evidence_quote":"Defines projection-based reduced models and the properties that exactly recovered models inherit."},{"cited_title":"Rozza, D","cited_arxiv_id":null,"evidence_quote":"Documents the reduced-basis error-estimation framework used as the target property of traditional reduced models."}],"review_version":1}