{"id":"35842899-fe88-46a0-b19c-f493d7d9cb47","arxiv_id":"2506.21104","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A space-time multicontinuum homogenization method is introduced for parabolic equations in shrinking perforated domains, validated by three numerical experiments with errors mostly below 10 percent.","lead":"This paper proposes a space-time multiscale method that upscales parabolic flow in porous domains whose channels shrink over time. It extends the authors' previous multicontinuum homogenization framework to evolving geometries, with numerical tests reporting errors near a few percent.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The derivation of (14) from the upscaled variational problem is asserted by ignoring cross terms 'by symmetric properties or the scaling in (10)' with no estimate; in the Section 4 experiments the RVE equals the coarse block (ε=H), so the O(ε/H) negligibility premise is violated.","rationale":"The reader's weakest assumption identifies the same core problem: the transition from the upscaled variational problem to (14) discards terms without a quantitative justification. My stress-test confirms this and sharpens it. The scaling argument in (10) gives, at best, O(ε/H) for the leading dropped terms, but the numerical experiments set the RVE equal to the coarse block, so ε=H and the smallness premise fails. This means the experiments validate a method in a regime where the formal derivation is silent. The paper has positive aspects: the multicontinuum framework is a natural extension of published work, the local problems with temporal derivatives and oversampling are a reasonable construction, and the reported errors are consistently small for the tested cases. No code or data are provided, so independent reproduction is not possible, which increases the need for the proposed diagnostic. The central claim is conditional on the neglected terms, so the CONDITIONAL verdict is appropriate; no change is needed.","tokens_in":9355,"tokens_out":13710,"duration_ms":159650,"concrete_test":"Using the same RVE geometries and local basis functions as in Example 1 (H=1/10 and H=1/20, with the stated oversampling), assemble the full upscaled variational problem without dropping any cross terms and solve it for the macroscopic variables U_i. Compare (a) the per-continuum averages of this full upscaled solution with those of the truncated model (14) and with the fine-scale reference, and (b) the magnitudes of the dropped bilinear forms relative to D_ji and B_ji. If the truncated and full solutions differ by more than the Table 1 errors, or if any dropped coefficient is not negligible compared with the retained coefficients, the neglect of these terms is not justified and (14) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the transition from the upscaled variational problem to the strong form (14). The paper states, without proof, that 'other terms will be ignored by the symmetric properties or the scaling in (10).' The ignored terms include cross terms such as ∫_{R_p} φ_i φ_j^n (∂U_i/∂t)(∂V_j/∂x_n), ∫_{R_p} φ_i^m φ_j V_j (∂U_i/∂t∂x_m), and the mixed stiffness terms. No symmetry in the local problems (8)-(9) implies these integrals vanish, and the scaling (10) only bounds them as O(ε/H) relative to the retained terms. Section 4 further states that 'Each coarse block is treated as an RVE,' so ε=H in the numerical experiments, making the O(ε/H) smallness assumption inapplicable. The reported 2-10% continuum-average errors therefore do not establish that (14) is the correct effective equation; the derivation gap is real and affects the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a space-time multicontinuum homogenization framework for parabolic equations posed on time-evolving perforated domains. The method introduces local cell problems on oversampled representative volume elements (RVEs) that include temporal derivatives and domain evolution, and derives a macroscopic system (14) for per-continuum averages. The framework is tested on three numerical examples with shrinking domains and heterogeneous permeability fields.","tokens_in":9620,"tokens_out":11075,"duration_ms":108519,"significance":"If the derivation is correct, the framework offers a computable coarse-scale model for a class of dynamic porous-media problems, and the numerical experiments provide non-circular evidence because they compare against an independent fine-scale reference solution. The main strengths are the conceptual novelty of combining multicontinuum ideas with time-evolving geometry and the breadth of the numerical tests. However, the central derivation of the effective equation contains an unproved negligibility step, the local cell problems are not fully specified, and the numerical setting does not satisfy the scale-separation assumption used in that derivation. These gaps prevent the paper from establishing its main claim as it stands.","major_comments":[{"comment":"The derivation of the strong form (14) from the displayed upscaled variational problem drops a number of cross terms with the statement that they are ignored 'by the symmetric properties or the scaling in the (10).' These include the mass coupling terms ∫ φ_i φ_j^n ∂U_i/∂t ∂V_j/∂x_n and ∫ φ_i^m φ_j V_j ∂^2 U_i/∂t∂x_m, as well as the mixed stiffness terms ∫ κ∇φ_i·∇φ_j^n and ∫ κ∇φ_i^m·∇φ_j. No symmetry property of the basis functions that would make these integrals vanish is established, and the scaling (10) only implies relative size O(ε/H) (or O(ε) in the unscaled integrals) compared with the retained terms. Since (14) is the paper's central output, this is a load-bearing gap. The authors should either provide a quantitative estimate for all omitted terms, justify their disappearance through explicit orthogonality properties, or numerically quantify their contribution.","section":"Section 3, transition to Eq. (14)"},{"comment":"The local cell problems (8)-(9) are not fully specified. No boundary condition on ∂R_p^+ is stated, and no initial condition for φ_i or φ_i^m is given beyond an elliptic equation at t=0. Without boundary conditions, the constrained variational problems are not well-posed: the bilinear form a(·,·) is not coercive on a space of functions without boundary constraints, and the constraints fix only averages. Since every effective coefficient in (14) is an integral of these basis functions, the well-posedness of (8)-(9) is a prerequisite for the method. The authors should specify the boundary condition on the oversampling boundary and provide a well-posedness argument, or at least state the discrete setting used in the numerical implementation.","section":"Section 3, Eqs. (8)-(9)"},{"comment":"The numerical experiments set 'each coarse block is treated as a RVE,' so the RVE scale equals the coarse scale H. This means the parameter ε in the scaling (10) coincides with the coarse-grid size, and the O(ε/H) negligibility premise in the derivation of (14) is not satisfied (indeed ε/H = 1). The reported relative L2 errors, which are 2-10%, therefore do not support the claim that (14) is the correct effective equation; they show only that some coarse model reproduces the continuum averages in the tested cases. To validate the derivation, the experiments should either use an RVE that is strictly smaller than the coarse block, or the derivation should be extended to the case ε = H with an explicit justification of the dropped terms.","section":"Section 4, first paragraph"},{"comment":"The approximation in (12) is written as ∫ ∂(φ_i U_i)/∂t v + ∫ ∂_m(φ_i^m U_i)/∂t v ≈ U_i ∫ ∂φ_i/∂t v + ∂_m U_i ∫ ∂φ_i^m/∂t v, but this omits the time-derivative terms ∂U_i/∂t ∫ φ_i v and ∂(∂_m U_i)/∂t ∫ φ_i^m v that arise from the product rule. The subsequent upscaled variational problem does contain a term V_j ∂U_i/∂t ∫ φ_i φ_j, so the display is inconsistent with the proceeding derivation. This notational inconsistency obscures which terms are being kept and which are being discarded, and it should be corrected.","section":"Section 3, Eqs. (12)-(13)"}],"minor_comments":[{"comment":"The displayed hierarchy 'h < ε < ε < H' uses the same symbol ε for two different scales (the heterogeneity scale and the RVE scale). Distinct symbols should be used to avoid confusion.","section":"Section 3, paragraph on scales"},{"comment":"The coefficient D^m_ji is defined, but the equation uses D^mn_ji. The index structure is inconsistent and should be aligned.","section":"Section 3, Eq. (15)"},{"comment":"The captions of Figures 7 and 8 state 'Example 1' but the corresponding text is for Example 3; the captions should be corrected.","section":"Section 4, Figures 7 and 8"},{"comment":"The text says the errors 'confirming the convergence' of the method, but only two coarse grid sizes are tested and no convergence rate is reported. A statement about the trend would be more cautious than a claim of confirmed convergence.","section":"Section 4, Tables 1-3"}],"recommendation":"major_revision","confidential_remarks":"The paper leans heavily on the authors' own prior works [14] and [29] for the scaling properties and the constraint structure, and the novelty lies mainly in adapting the multicontinuum framework to time-evolving perforated domains. The current version lacks a rigorous derivation of the effective equation and a complete specification of the local problems, which are prerequisites for a mathematical journal. If the authors can close the derivation gap and adjust the numerical claims accordingly, the paper would be a useful contribution; otherwise, the central claim is not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThis paper does something genuinely new: it moves multicontinuum homogenization into time-evolving perforated domains by building space-time cell problems with temporal derivatives and shrinking-domain effects. The numerical setup is clean, the experiments are benchmarked against a fine-scale reference rather than fitted, and the reported errors are small and shrink as H refines. That part is worth attention.\n\nThe soft spot is the derivation of the macroscopic equation. After writing the upscaled variational problem, the authors say 'other terms will be ignored by the symmetric properties or the scaling in (10)' and jump to the strong form (14). No estimate is given. The dropped cross terms are not zero by any symmetry in the local problems, and the scaling (10) only makes them O(ε/H) relative to the kept terms. Worse, in Section 4 each coarse block is treated as an RVE, so ε=H and that smallness is not available. The 2-10% errors are encouraging but they do not establish that (14) is the correct effective equation; they show the method performs reasonably as a numerical scheme.\n\nThere are other gaps in the same direction. The local cell problems (8)-(9) lack boundary conditions on the oversampling region and a proper initial condition for φ_i; the Lagrange multipliers are described only by words. There is no convergence theorem, and several displayed integrals are sloppy about whether κ multiplies both gradients. The scaling properties (10) are cited from prior work rather than proved.\n\nNone of this is fatal to the idea. The framework is coherent and the setting is physically motivated. What is missing is the analytical support for the central reduction. I would send this to peer review, but the referee's first request should be a real estimate for the dropped terms, or an explicit statement that the method is a coarse-grid numerical technique justified by experiments rather than a homogenization limit. With code and data released, the numerical side could be checked independently.\n\nI'd bring this to a reading group as an example of a promising extension with a derivable-but-missing rigorous core.","headline":"Useful space-time extension of multicontinuum homogenization, but the upscaled equation rests on an unproved term-dropping step and the numerics use ε=H, so the derivation gap is real.","tokens_in":10074,"tokens_out":7193,"would_cite":false,"duration_ms":70324,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B27","65M60","65N30","76S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A space-time multicontinuum upscaling procedure reduces parabolic flow in evolving perforated domains to a coupled macroscopic system with computable coefficients.","keywords":["multicontinuum homogenization","space-time upscaling","perforated domains","evolving domains","macroscopic equations","parabolic PDE","representative volume element","porous media flow"],"falsifier":"Take any one of the three test configurations and compute, at each time step and for each coarse block, the magnitudes of the dropped expressions $\\int_{R_p}\\phi_i\\phi_j^n \\partial_t U_i \\partial_n V_j$ and $\\int_{R_p}\\kappa\\nabla\\phi_i\\cdot\\nabla\\phi_j^n$ alongside the retained coefficients in (14). If the ratio of either class of dropped term to the corresponding retained term fails to stay small as $H\\to 0$ and the domain contracts, the effective equations will not reproduce the per-continuum averages measured by (16), and the paper's central claim collapses.","tokens_in":9161,"feed_emoji":"🕳️","tokens_out":9950,"duration_ms":94348,"temperature":0.7,"pith_summary":"The paper sets out to show that a diffusion-like parabolic flow in a pore structure that shrinks over time, through clogging, deposition, or dissolution, can be captured by a small system of macroscopic equations without resolving the moving fine-scale geometry. The construction separates the domain into continua by channel width and writes the fine solution as a truncated expansion $u \\approx \\phi_i U_i + \\phi_i^m \\frac{\\partial U_i}{\\partial x_m}$, with basis functions computed once on oversampled representative volume elements that include time-derivative terms. Inserting this expansion into the weak form and dropping cross terms yields the coupled macroscopic system $D_{ji}\\frac{\\partial U_i}{\\partial t}+B_{ji}U_i-\\partial_n(B^{mn}_{ji}\\partial_m U_i)=b_j$ with computable coefficients. A sympathetic reader would care because, if the upscaled model is right, large-scale simulations of reactive transport and pore clogging can run on a fixed coarse grid while still tracking the average behavior in each channel family.","feed_headline":"Coarse model reproduces flow in shrinking pores","feed_subtitle":"Space-time cell problems turn a moving perforated geometry into a computable macroscopic system.","key_machinery":"The carrying object is the two-term expansion $u \\approx \\phi_i U_i + \\phi_i^m \\partial_m U_i$ over a representative volume element $R_p$, with basis functions defined by the constrained local problems (8)-(9) on an oversampled region $R_p^+$. The constraints force $\\phi_i$ to have unit mean over continuum $i$ and zero mean over the others, and force $\\phi_i^m$ to match the mean of $x^m-c^m$ within continuum $i$; Lagrange multipliers $\\beta$ and $\\zeta$ enforce these averages, while the temporal terms in the cell problems transfer the domain evolution into the basis. The scaling estimates $\\|\\phi_i\\|=O(1)$, $\\|\\nabla\\phi_i\\|=O(1/\\varepsilon)$, $\\|\\phi_i^m\\|=O(\\varepsilon)$, $\\|\\nabla\\phi_i^m\\|=O(1)$ are the mechanism by which all unlisted cross terms are discarded, leaving the closed macroscopic system (14).","core_discovery":"The central claim is that the truncated multicontinuum expansion, together with the constrained space-time cell problems (8)-(9), reduces the original parabolic fine-scale model (1) to the macroscopic system (14) plus its initial-condition counterpart (15). The coarse variables $U_i$ are the per-continuum spatial averages of the fine solution, the coefficient tensors $D_{ji}$, $B_{ji}$, $B^{mn}_{ji}$, and $b_j$ are assembled from the local basis functions, and the time-dependent cell problems make the basis functions aware of the moving geometry. In the three numerical settings reported, with a two-continuum channel geometry, permeability contrast of $10^2$, and prescribed contraction of the two channels at different rates, the multiscale averages stay close to the fine-grid per-continuum averages, with relative errors in the few-percent range for $H=1/10$ and smaller for $H=1/20$.","pith_inferences":["Editorial extension: the paper does not estimate the dropped cross terms, but the natural check is to bound the ratios of $\\int_{R_p}\\phi_i\\phi_j^n \\partial_t U_i \\partial_n V_j$ and $\\int_{R_p}\\kappa\\nabla\\phi_i\\cdot\\nabla\\phi_j^n$ to the retained coefficients; if these ratios grow like $\\varepsilon/H$ as the domain contracts, the next-order basis functions $\\phi^{mn}$ would have to be retained r","Editorial extension: the geometry in the experiments is prescribed by a fixed contraction rate; a coupled extension would let the continuum indicator functions $\\psi_i$ evolve with the fine-scale solution, and the same local problems should still supply the coarse coefficients as long as the evolution is slow relative to the cell-problem time step.","Editorial extension: the oversampling depth is chosen globally and empirically; an a-posteriori indicator measuring the magnitude of the omitted cross terms on each coarse block could make the oversampling adaptive and cheaper on benign blocks."],"forward_implications":["Assembling $D_{ji}$, $B_{ji}$, $B^{mn}_{ji}$, and $b_j$ from the local cell problems gives a coarse model that runs on a fixed mesh even while the underlying pores shrink, so the cost of a time step no longer scales with the number of fine-grid cells that disappear.","The macroscopic unknowns $U_i$ are per-continuum averages, so quantities such as the mass remaining in thin low-permeability channels are available directly from the coarse solution without post-processing the fine geometry.","Although the paper implements two continua, the construction is stated for arbitrary characteristic functions $\\psi_i$, so channel networks with several width classes inherit the same local problems and the same coefficient formulas.","The initial-condition system (15) is derived by the same averaging, which lets the coarse model be started from the fine initial data without a separate fine-scale solve."],"supporting_citations":[{"why":"Supplies the multicontinuum homogenization construction and the scaling relations (10) used to discard cross terms.","marker":"[14]"},{"why":"Gives the multicontinuum expansion for perforated domains that this paper's ansatz (7) directly extends to time-evolving geometry.","marker":"[29]"},{"why":"Provides the general multicontinuum homogenization theory behind distinguishing continua by physical characteristics such as channel width.","marker":"[6]"},{"why":"Introduces the oversampled local-basis strategy that motivates solving the cell problems on $R_p^+$ instead of $R_p$.","marker":"[19]"},{"why":"Supplies the constrained-basis methodology whose averaging constraints the local problems (8)-(9) adapt.","marker":"[10]"},{"why":"Motivates the nonlocal multicontinuum averaging used to define the macroscopic variables $U_i$.","marker":"[11]"}],"fun_headline_variants":["Space-time upscaling tames evolving pore networks","Multicontinuum model for shrinking perforated media","Coarse equations track evolving pore geometries","Robust multiscale method for time-evolving porous media"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation moves from the upscaled variational problem to the strong form (14) by announcing that \"other terms will be ignored by the symmetric properties or the scaling in (10)\"; the whole coarse model rests on the unproved premise that every omitted cross term is genuinely negligible relative to the terms kept in (14).","fun_headline_variants_meta":{"raw":{"variants":["Space-time upscaling tames evolving pore networks","Multicontinuum model for shrinking perforated media","Coarse equations track evolving pore geometries","Robust multiscale method for time-evolving porous media"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000233,"raw_usage":{"total_tokens":1466,"prompt_tokens":890,"completion_tokens":576,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":506,"completion_tokens_details":{"reasoning_tokens":514}},"tokens_in":506,"tokens_out":576,"duration_ms":5863,"temperature":1.0,"reasoning_tokens":514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:34:31.056394+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any one of the three test configurations and compute, at each time step and for each coarse block, the magnitudes of the dropped expressions $\\int_{R_p}\\phi_i\\phi_j^n \\partial_t U_i \\partial_n V_j$ and $\\int_{R_p}\\kappa\\nabla\\phi_i\\cdot\\nabla\\phi_j^n$ alongside the retained coefficients in (14). If the ratio of either class of dropped term to the corresponding retained term fails to stay small as $H\\to 0$ and the domain contracts, the effective equations will not reproduce the per-continuum averages measured by (16), and the paper's central claim collapses.","supporting_citations":[{"cited_title":"Efendiev and W","cited_arxiv_id":null,"evidence_quote":"Supplies the multicontinuum homogenization construction and the scaling relations (10) used to discard cross terms."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the multicontinuum expansion for perforated domains that this paper's ansatz (7) directly extends to time-evolving geometry."},{"cited_title":"Chung, Y","cited_arxiv_id":null,"evidence_quote":"Provides the general multicontinuum homogenization theory behind distinguishing continua by physical characteristics such as channel width."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the oversampled local-basis strategy that motivates solving the cell problems on $R_p^+$ instead of $R_p$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the constrained-basis methodology whose averaging constraints the local problems (8)-(9) adapt."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates the nonlocal multicontinuum averaging used to define the macroscopic variables $U_i$."}],"review_version":1}