{"id":"a7451cf2-7c44-483c-b042-7ae6ea592e92","arxiv_id":"2505.01244","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For 1D linear advection, a discrete-time sparse full-order model inferred by least squares is guaranteed stable only if the training data satisfy Δt/Δx ≤ (m+1)/(3c), a 'sampling CFL' bound.","lead":"This paper analyzes data-driven inference of sparse discrete-time models and derives a 'sampling CFL' bound linking training-data step sizes to stability of the inferred model. The result gives practitioners a concrete rule for choosing snapshot spacings when learning advection-dominated dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sampling CFL bound (39) is derived for a first-order Taylor surrogate of the LS problem, not for the exact LS problem (7); the bridge is unproven.","rationale":"The paper's main contribution is the analytic sampling CFL condition. I re-derived the symmetric-stencil case: with c3=0, (32) gives β_j = 1/(2m+1) + j r/[m(m+1)(2m+1)/3], and substituting into the sufficient l1 condition (12) yields exactly r ≤ (m+1)/3, so the internal algebra is sound for the Taylor surrogate. However, the article identifies (39) as a property of the inferred sFOM from the exact least-squares problem (7), while the derivation applies to the first-order surrogate (26)-(27). The reader's weakest assumption points to the same gap: the uncontrolled Taylor approximation. The numerical validation is qualitative, uses only a smooth initial condition, and does not report errors or exact thresholds. This is a genuine but fixable weakness: a direct comparison of exact LS coefficients to the Taylor coefficients, or a search for counterexamples just below the claimed bound, would settle whether (39) governs the true inference problem. Since the reader already issued a CONDITIONAL verdict and this concern supports that assessment, no change is needed.","tokens_in":14493,"tokens_out":6265,"duration_ms":62949,"concrete_test":"Test the bound on the exact LS problem. Fix c=1, symmetric 3-point stencil (m=1), domain [-1,1], N=500 equally spaced snapshots, and use two initial conditions: u0=cos(πx) and a sharper u0=exp(-x^2/0.02). For Δx=0.02 and Δt = rΔx with r ∈ {0.2, 0.4, 0.6, 0.66, 0.68, 0.75, 1.0}, form D_i and d_i from the exact solution u(x,t)=u0(x+ct), solve (7) exactly, assemble the Toeplitz sFOM operator, and record max |λ| and ∥β∥_1. Also compute the Taylor-predicted β from (32) and report the relative error. If max |λ| > 1 for any r ≤ 2/3, the guarantee fails for the exact inference; if the coefficients differ by more than a few percent, (39) is not established for (7).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central bound (39) is proven only for the first-order Taylor surrogate of the least-squares problem, not for the exact problem (7). Equations (26)-(27) replace the data matrix D_i by a rank-2 approximation and the right-hand side d_i by an explicit-Euler approximation; the solution (32) solves this surrogate, not the actual LS problem that is implemented in Figure 3 and described in the abstract. For symmetric stencils the algebra from (32) to (39) does close: with c3=0, β_j = 1/(2m+1) + j r/[m(m+1)(2m+1)/3], and the l1-norm condition (12) gives exactly r≤(m+1)/3. So the bound is internally correct for the surrogate. But the manuscript never bounds the difference between the surrogate solution (32) and the exact LS solution of (7), nor proves that the spectral radius of the exact inferred operator is ≤1 whenever r≤(m+1)/3. The only evidence is a visual 'closely follow' on a smooth cosine problem with no error bars and no quantitative threshold. If the Taylor higher-order terms are not small, the exact LS coefficients can differ from (32) and the sufficient stability guarantee can fail even inside the claimed region; conversely the bound may be too conservative. The sentence 'This inequality provides a priori bounds ... that guarantee stability' overstates the rigorous support.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates the inference of discrete-time sparse full-order models (sFOMs) from snapshot data by solving a local least-squares problem (7). It reviews the SVD-based closed-form solution, discusses how l2 regularization relates to the Gershgorin sufficient stability condition (12), and analyzes the 1D linear diffusion example analytically. The central contribution is a 'sampling CFL' condition for 1D linear advection: for a symmetric stencil with m neighbors on each side, the training-data discretization should satisfy Δt/Δx ≤ (m+1)/(3c) (Eq. (39)) for the inferred sFOM to be stable. The condition is validated on a smooth cosine advection problem and the approach is demonstrated on 2D Burgers and an oscillating lid-driven cavity. The paper's main claim is that this condition provides an a priori bound on the spatial and temporal sampling of training data that guarantees stability of the inferred model.","tokens_in":14779,"tokens_out":7465,"duration_ms":73371,"significance":"If the sampling CFL condition were rigorously established for the exact least-squares problem, it would be a valuable a priori diagnostic for data-driven PDE inference: it is simple, falsifiable, and directly actionable for choosing snapshot spacing. The paper also usefully connects l2 regularization to the Gershgorin sufficient condition and provides a closed-form expression for a model problem. The theoretical derivation is transparent for the first-order Taylor surrogate, and the numerical test on smooth advection is consistent with the predicted threshold. The main caveat is that the bridge from the surrogate to the exact inference problem is currently missing; resolving this gap would turn a plausible heuristic into a theorem.","major_comments":[{"comment":"The sampling CFL condition is derived after replacing the data matrix D_i by the rank-2 first-order Taylor surrogate (26) and the right-hand side d_i by the explicit-Euler surrogate (27). The coefficients β_ij in (32) solve this surrogate, not the least-squares problem (7) that is actually solved in Figure 3 and described in the abstract. The manuscript never bounds the difference between the surrogate solution and the exact LS solution, nor proves that the spectral radius of the operator inferred from (7) is at most 1 whenever (39) holds. Since (39) is the central advertised result, this gap is load-bearing. Please either prove a rigorous error bound under explicit smoothness assumptions with quantitative constants, or re-state (39) as a heuristic condition for the surrogate and provide numerical evidence that it predicts the exact threshold. The sentence 'This inequality provides a priori bounds ... that guarantee stability' overstates the current support.","section":"Section 3.2, Eqs. (26)-(32) and (39)"},{"comment":"The algebra that turns the sufficient condition (37) into the explicit bound Δt/Δx ≤ (m+1)/(3c) for symmetric stencils is omitted. This is load-bearing because (37) is an l1-condition on the stencil coefficients, and its reduction to a simple ratio requires controlling the signs of the two terms in (35), not just their magnitudes. For m = l, c3 = 0, and the reduction is not automatic from the expression for β_ij. Please present the full computation, including the treatment of the absolute values in (37), so that the reader can verify the bound.","section":"Section 3.2.1, derivation of Eq. (39) from Eq. (37)"},{"comment":"The numerical evidence for (39) is qualitative and limited to a single smooth cosine initial condition. There is no metric for how closely the empirical stability boundary follows the predicted line, no error bars or variation over initial conditions, and no test with non-smooth data, where the first-order Taylor approximations (26)-(27) are questionable. Please quantify the threshold agreement, for example by reporting the empirical critical ratio (Δt/Δx)_crit divided by (m+1)/(3c) for both stencils, and state the smoothness regime in which (39) is intended to apply.","section":"Figure 3 and Section 3.2.1"}],"minor_comments":[{"comment":"The statement that l2 regularization is stability-promoting because of (13) should be softened: a penalty on ||β_i||_2 does not by itself enforce ||β_i||_1 ≤ 1; it biases toward small-norm solutions and hence, heuristically, toward satisfying (12).","section":"Section 2.2"},{"comment":"Calling the condition that the sum of coefficients equals 1 'consistency' is imprecise; it is the condition for reproducing a constant state, which is necessary but not sufficient for consistency of the underlying PDE discretization.","section":"Section 3.1.1, Eq. (24)"},{"comment":"The notation D_a for the augmented data block is inconsistent with d_α; use D_{α_j} for each augmented block.","section":"Equation (9)"},{"comment":"Equation (32) contains a typesetting artifact ('Œœ'), and the norm in (37) should be absolute value, since K1i and K2i are scalars.","section":"Equations (32) and (37)"},{"comment":"Theorem 1 is essentially tautological once the ansatz and support match the data-generation scheme; presenting it as a remark would better reflect its logical status.","section":"Theorem 1"}],"recommendation":"major_revision","confidential_remarks":"The central gap is fixable in revision: either add an error bound between the surrogate and the exact LS solution, or explicitly reframe the sampling CFL result as a heuristic and validate it more thoroughly. No circularity concerns. If the authors cannot close the gap, the advertised guarantee should be downgraded."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe thing to know: this paper gives a clean, explicit \"sampling CFL\" condition for discrete-time sFOM inference on 1D linear advection, Δt/Δx ≤ (m+1)/(3c). That bound is new and practically useful as a heuristic. But the proof as written applies to a first-order Taylor surrogate of the least-squares problem, not to the exact problem (7), and the manuscript never bridges that gap. Read it as a promising insight rather than as a proven theorem.\n\nWhat's actually new: the analytic bound (39), the closed-form SVD solution (16), and the l2 regularization–stability connection in discrete time. The SVD solution is standard linear algebra, but the CFL formula is not in the cited literature; earlier works only hinted at the trade-off. The paper is also honest about the approximate nature of the derivation—it says \"we can derive a Taylor approximation\"—which is more than many data-driven papers do.\n\nSoft spots: first, the gap between the surrogate and the exact LS problem. Equations (26)–(27) replace the data with a rank-2 model and explicit Euler. The algebra from (32) to (39) does close for symmetric stencils (I checked; the stress-test note's reconstruction is right), so the bound is internally consistent for the surrogate. But no error analysis shows that the exact solution of (7) has the same threshold. The numerical validation on a smooth cosine problem shows the instability boundary \"closely follow\" the formula, but without error bars or a quantitative comparison, that's suggestive, not conclusive. Second, the step from (37) to (39) is just asserted; the reader has to take the algebra on faith. That's easy to fix in revision. Third, the nonlinear sections are empirical demonstrations—hand-selected hyperparameters, no error bars—so they read as anecdotes, not validations.\n\nWho this is for: anyone working on data-driven discovery of discrete-time dynamical systems, especially operator inference and stencil learning. The sampling CFL condition gives a concrete rule of thumb for choosing training data discretizations. A serious reviewer should engage with it, because the core idea is plausible and the paper is clearly written.\n\nMy recommendation: send it to peer review, with a request that the authors provide the missing algebra from (37) to (39), an error analysis or numerical demonstration that the exact LS solution obeys the same bound, and ideally code/data. With those, it could be a solid contribution.","headline":"The sampling CFL condition is a genuinely new and useful heuristic, but the proof only covers a Taylor surrogate of the LS problem, not the exact problem.","tokens_in":15315,"tokens_out":2903,"would_cite":true,"duration_ms":26630,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M99","65M06","65M12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper derives a sampling CFL condition, $\\Delta t/\\Delta x \\le (m+1)/(3c)$, that governs whether a sparse full-order model inferred from snapshots of 1D linear advection will be stable.","keywords":["sparse full-order model inference","discrete-time dynamical systems","operator inference","least-squares regression","stability","sampling CFL condition","linear advection","Gershgorin circle theorem"],"falsifier":"Solve the exact least-squares problem for 1D linear advection over a sweep of $\\Delta x$ and $\\Delta t$, first with the smooth cosine initial condition and then with a sharp or non-smooth profile, and compare the empirical instability boundary of the inferred sFOM with $\\Delta t/\\Delta x \\le (m+1)/(3c)$. A systematic discrepancy for non-smooth data, or an exact-solution threshold that differs from the Taylor-based bound even for smooth data, would show that the sampling CFL condition is only an approximation, not the true stability threshold of the inference problem.","tokens_in":14284,"feed_emoji":"🌊","tokens_out":6954,"duration_ms":66356,"temperature":0.7,"pith_summary":"This paper studies sparse full-order model (sFOM) inference, a data-driven approach that learns the local numerical stencil of a discretized PDE from state snapshots by solving a least-squares problem. The authors set out to show that the stability of the inferred discrete-time model is controlled not only by regularization but by how the training data are sampled: for 1D linear advection they derive a 'sampling CFL' condition, $\\Delta t/\\Delta x \\le (m+1)/(3c)$, that bounds the ratio of the time step to the space step in the training data. If that bound holds, the inferred model is stable under a Gershgorin sufficient condition; if it is violated, the model can blow up. A sympathetic reader would care because this gives a priori guidance for collecting training data for operator inference and explains instabilities seen in practice. The argument is carried by a closed-form solution of the least-squares problem and a first-order Taylor approximation of the local data.","feed_headline":"Stable data-driven PDE models obey a sampling CFL bound","feed_subtitle":"For 1D linear advection, too large a time step in the training data can make the inferred model blow up.","key_machinery":"The machinery is the local least-squares inference problem equipped with an adjacency-based sparse stencil, together with the SVD closed-form solution $\\beta_i = \\Phi_i\\Sigma_i^\\dagger\\Psi_i^T\\psi_{S_i}\\sigma_{S_i}\\phi_{S_i}^T$ and its rank-2 Taylor reduction, which replaces each stencil by the pair $(u_i, \\partial u/\\partial x)$ and the time shift by an explicit Euler step. The Gershgorin circle theorem turns the requirement that all eigenvalues of the inferred operator lie in the unit disk into the sufficient row-sum condition $\\|\\beta_i\\|_1 \\le 1$, and applying that condition to the approximated solution produces the sampling CFL bound. The coefficients $c_1, c_2, c_3$ encode the stencil geometry, so widening the stencil changes the allowed sampling ratio.","core_discovery":"The paper's central claim is that the least-squares inference problem has a closed-form solution expressible through the SVD of the snapshot data, and that for fine enough data this solution can be approximated analytically. For 1D linear advection with a symmetric stencil of half-width $m$, substituting that approximation into the Gershgorin sufficient stability condition yields the explicit sampling CFL bound $\\Delta t/\\Delta x \\le (m+1)/(3c)$. The paper also argues that the commonly used $\\ell_2$ regularization is stability-promoting in discrete time, because it shrinks $\\|\\beta_i\\|_2$ and hence the $\\ell_1$ norm that enters the sufficient stability condition $\\|\\beta_i\\|_1 \\le 1$. These results are validated numerically on smooth advection data and used to interpret instabilities in 2D Burgers and oscillating lid-driven cavity sFOMs.","pith_inferences":["If the bound reflects the data's information content rather than the specific Taylor argument, then other discrete-map learners, such as learned stencils or convolutional surrogates, should show a similar $\\Delta t/\\Delta x$ threshold for advection; this is directly testable.","The Taylor derivation assumes smoothness, so for shock-like or under-resolved data the exact least-squares threshold likely differs, and a higher-order Taylor reduction or a data-dependent correction may be needed.","Data augmentation changes the effective stencil width $m$, so the bound should be recomputed for augmented problems; this could explain the diffusion-case instability at very small $\\Delta t$ reported in the paper.","An analogous sampling condition should hold for continuous-time operator inference, with the time-derivative discretization playing the role of $\\Delta t$; the paper hints at this connection."],"forward_implications":["Training data for advection-dominated sFOM inference can be planned: choose $\\Delta x$ and $\\Delta t$ satisfying the bound to guarantee a stable model without extra stabilization.","Wider stencils raise the allowed ratio, so using larger local supports makes discrete-time sFOM inference more forgiving of coarse temporal sampling.","The stabilizing role of $\\ell_2$ regularization in discrete-time inference is quantified: it acts on the $\\ell_1$ row sums that the sufficient stability condition controls.","The same sampling mechanism plausibly explains the Burgers' instabilities at high advection speed, where the effective wave speed $c$ in the bound grows.","Instability of an inferred sFOM can signal an ill-sampled training set rather than a wrong model structure or insufficient regularization."],"supporting_citations":[{"why":"Supplies the Gershgorin circle theorem that converts eigenvalue stability into the sufficient row-sum condition $\\|\\beta_i\\|_1 \\le 1$.","marker":"[12]"},{"why":"Prior work establishing the $\\ell_2$-regularization/stability connection for sFOM and data augmentation; the discrete-time argument in Section 2.2 follows it.","marker":"[13]"},{"why":"Earlier study of discrete operator inference with linear models, including rank deficiency of advection data, which the paper extends.","marker":"[33]"},{"why":"Provides the minimum-norm least-squares solution used in the closed-form SVD solution and in the truncated system.","marker":"[34]"},{"why":"Hints that spatial and temporal discretization trade off in sparse FOM inference, motivating the sampling CFL analysis.","marker":"[2]"},{"why":"Shows stability-promoting constraints for sFOM inference via Gershgorin, informing the sufficient stability condition used here.","marker":"[27]"},{"why":"Documents similar sampling-induced instabilities in another data-driven dynamical-system method, framing the paper's contribution.","marker":"[35]"}],"fun_headline_variants":["Sampling CFL condition stabilizes inferred discrete-time PDE models","Stable data-driven advection: sampling CFL derived analytically","Sparse full-order model inference needs a CFL-like sampling bound","Sampling ratio bound for stable discrete-time model inference","Data-driven PDE stability: a sampling CFL from linear advection"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bound is derived from a first-order Taylor approximation of the training data; if the snapshots are not smooth or $\\Delta x$ and $\\Delta t$ are not small enough for that approximation, the actual least-squares solution can have a different stability threshold.","fun_headline_variants_meta":{"raw":{"variants":["Sampling CFL condition stabilizes inferred discrete-time PDE models","Stable data-driven advection: sampling CFL derived analytically","Sparse full-order model inference needs a CFL-like sampling bound","Sampling ratio bound for stable discrete-time model inference","Data-driven PDE stability: a sampling CFL from linear advection"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000259,"raw_usage":{"total_tokens":1575,"prompt_tokens":924,"completion_tokens":651,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":565}},"tokens_in":540,"tokens_out":651,"duration_ms":6786,"temperature":1.0,"reasoning_tokens":565,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:23:03.830593+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Solve the exact least-squares problem for 1D linear advection over a sweep of $\\Delta x$ and $\\Delta t$, first with the smooth cosine initial condition and then with a sharp or non-smooth profile, and compare the empirical instability boundary of the inferred sFOM with $\\Delta t/\\Delta x \\le (m+1)/(3c)$. A systematic discrepancy for non-smooth data, or an exact-solution threshold that differs from the Taylor-based bound even for smooth data, would show that the sampling CFL condition is only an approximation, not the true stability threshold of the inference problem.","supporting_citations":[{"cited_title":"¨Uber die Abgren- zung der Eigenwerte einer Matrix","cited_arxiv_id":null,"evidence_quote":"Supplies the Gershgorin circle theorem that converts eigenvalue stability into the sufficient row-sum condition $\\|\\beta_i\\|_1 \\le 1$."},{"cited_title":"Non-intrusive reduced-order modeling for dynamical systems with spatially localized features","cited_arxiv_id":"2501.04400","evidence_quote":"Prior work establishing the $\\ell_2$-regularization/stability connection for sFOM and data augmentation; the discrete-time argument in Section 2.2 follows it."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the minimum-norm least-squares solution used in the closed-form SVD solution and in the truncated system."}],"review_version":1}