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arxiv: 2605.04037 · v1 · submitted 2026-05-05 · 🧮 math.NA · cs.NA

Model order reduction for parametrized variational inequalities: application to crowd motion

Giulia Sambataro , Virginie Ehrlacher This is my paper

Pith reviewed 2026-05-06 05:06 UTC · model claude-opus-4-7

classification 🧮 math.NA cs.NA MSC <parameter name="0">49J40
keywords <parameter name="0">reduced order modeling
1
0 comments X

The pith

A reduced-order model for crowd-motion contact problems combines a greedy non-negativity-preserving compression of contact forces with a learned correction, achieving 30–60× speedups while keeping agents from overlapping.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper attacks a problem that standard reduced-basis machinery handles badly: at each time step of a crowd-motion simulation, every agent's velocity is the projection of a desired velocity onto a cone of feasible velocities defined by non-overlap constraints with neighbors and obstacles. The Lagrange multipliers for those constraints are sparse, non-negative, and switch on and off as contacts form and break, so a linear basis decays slowly with dimension. The authors propose a hybrid scheme: compress velocities with POD, compress the contact-force snapshots with a greedy algorithm that picks one canonical-basis index per step (preserving non-negativity and giving a hierarchical cone), enrich the velocity basis with supremizers (PGA) so the saddle-point problem stays solvable, and replace Empirical Interpolation with Empirical Quadrature so hyper-reduction scales with the small reduced dimension rather than the large contact count. A Random-Forest or shallow neural network then learns a correction from the dominant ROM coefficients to the optimal projection of the high-fidelity solution. Tests on 20-agent and 150-agent geometries with parameterized exit width and desired-velocity magnitude show 30–60× speedups and 3–5× lower trajectory error than the plain Galerkin ROM, including in a highly congested case where the uncorrected ROM lets agents penetrate obstacles.

Core claim

The paper builds the first reduced-order model for a discrete contact crowd-motion problem in which agent velocities are obtained at each time step by projecting a desired velocity onto a cone of collision-avoiding velocities. The authors argue that two ingredients are needed because the solution manifold has slowly decaying Kolmogorov width: (i) a greedy "index selection" that compresses the Lagrange-multiplier (contact-force) snapshots while keeping the basis vectors non-negative and hierarchical, and (ii) a machine-learning correction that nudges the dominant Galerkin coordinates toward the optimal projection of the high-fidelity solution. Combined with primal-basis enrichment (PGA) for s

What carries the argument

A greedy index-selection algorithm that, at each iteration, picks the largest still-uncovered coordinate of the worst-approximated multiplier snapshot and adds the corresponding canonical basis vector, building a hierarchical non-negative cone for contact forces; combined with PGA supremizer enrichment of the velocity basis for inf-sup stability, Empirical Quadrature for non-affine terms, and a learned residual map from dominant Galerkin coordinates to the optimal projection.

If this is right

  • <parameter name="0">Sparse non-negative contact-force fields admit a hierarchical canonical-basis compression that beats both POD and cone-projected greedy
  • both in offline cost and in approximation rate.

Load-bearing premise

That a training set of 200 parameter samples and a few dozen time steps is rich enough for the learned correction and the supremizer-enriched basis to remain stable and accurate on new, denser configurations — even though the reduced inf-sup constant is reported as zero in the most congested case and no error indicator is provided.

What would settle it

Run the trained ROM on parameter values or agent counts well outside the training distribution (e.g., a different obstacle layout, or 300 agents instead of 150) and check whether the non-overlap constraint is violated or trajectory error rises sharply; a stable speedup-accuracy frontier in that regime would support the claim, persistent constraint violations or loss of accuracy would refute it.

Figures

Figures reproduced from arXiv: 2605.04037 by Giulia Sambataro, Virginie Ehrlacher.

Figure 1
Figure 1. Figure 1: POD on the velocity snapshots {u(σ)}σ∈Strain . We can see in particular that the relative squared truncated POD error decays slowly with respect to the number of POD modes. This slow decay illustrates the fact that the so-called Kolmogorov width of the solution set Mu decays at a slow rate, which makes standard linear reduced-order models not well suited in our present context. This in particular motivates… view at source ↗
Figure 2
Figure 2. Figure 2: Problem setup. Representative sketch of the two-dimensional DCP region: view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of Uzawa scheme. Convergence history of the HF Uzawa solve view at source ↗
Figure 4
Figure 4. Figure 4: Dual reduced cone construction. (a): construction of the reduced cone for view at source ↗
Figure 5
Figure 5. Figure 5: Stability of the reduced system and convergence of the PGA algorithm. view at source ↗
Figure 6
Figure 6. Figure 6: Hyper-reduced model . Empirical interpolation method for the DCP. The view at source ↗
Figure 7
Figure 7. Figure 7: Hyper-reduced model . Empirical quadrature procedure. Relative view at source ↗
Figure 8
Figure 8. Figure 8: Hyper-reduced model . Empirical Quadrature procedure. view at source ↗
Figure 9
Figure 9. Figure 9: Hyper-reduced model . Galerkin ROM performance. (a): Pareto plot show view at source ↗
Figure 10
Figure 10. Figure 10: Hyper-reduced model . Sketch of Galerkin EQ-ROM predicted trajectories view at source ↗
Figure 11
Figure 11. Figure 11: Hyper-reduced model . Particle positions at three selected times with view at source ↗
Figure 12
Figure 12. Figure 12: ML corrected ROM. (a): accuracy-speedup comparison: ML-corrected view at source ↗
Figure 13
Figure 13. Figure 13: A high-dimensional and highly-congested scenario. Randomized POD view at source ↗
Figure 14
Figure 14. Figure 14: A high-dimensional and highly-congested scenario. a): validation error of view at source ↗
Figure 15
Figure 15. Figure 15: Hertz contact between a sphere of radius view at source ↗
Figure 16
Figure 16. Figure 16: In red: the manifold selected functions ψ1 and ψ2 after two greedy steps; in blue: the functions obtained by non-negative combination of the first two greedy functions; in orange: the manifold parametric solutions for a ∈ [a, a] and p = 1. A.2 Identification of the parameter from the reduced cone representation In this section, we derive here explicit formulas for the projection coefficients, expressed in… view at source ↗
read the original abstract

This work investigates model order reduction for time-dependent parametrized variational inequalities, with a focus on discrete contact problems. As a prototypical example, we consider an agent-based crowd model [Maury et al., 2011] in which agent velocities are obtained at each time step from a constrained least-squares problem. Geometric parameter variations induce significant variability in both agent positions and contact forces, leading to a slowly decaying Kolmogorov $n$-width of the solution manifold. We propose a nonlinear approach that combines a linear reduced-order model with a deep-learning-based correction. The method utilizes a greedy index selection (gIS) algorithm for compressing Lagrange multipliers and Proper Orthogonal Decomposition (POD) applied to velocity snapshots. Additionally, we explore hyper-reduction techniques, comparing the Empirical Interpolation Method (EIM) and the Empirical Quadrature (EQ) procedure from both computational complexity and accuracy perspectives. Finally, we demonstrate the applicability of the methodology in a complex scenario involving many agents in a highly congested geometric configuration. This work represents the first attempt to apply model order reduction to a discrete contact problem of the type introduced in [Maury et al., 2011] and paves the way for future advancements in nonlinear MOR specifically for this class of problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

5 major / 9 minor

Summary. The manuscript proposes a hybrid model order reduction strategy for parametrized time-discrete variational inequalities arising from the Maury–Venel agent-based crowd motion model. The methodological contributions are: (i) a greedy index-selection (gIS) algorithm to compress non-negative Lagrange multipliers into a positive cone, compared with the modified cone-projected greedy from Niakh et al.; (ii) a Galerkin ROM with Projected Gradient Algorithm (PGA) enrichment of the primal velocity basis to recover inf-sup stability lost by Lagrange-multiplier non-uniqueness; (iii) a hyper-reduction comparison between EIM and an Empirical Quadrature (EQ) procedure based on NNOMP, with EQ shown more efficient for the contact term; (iv) a machine-learning correction (Random Forest / NN) of the leading reduced coefficients, inspired by Cohen et al. Numerical experiments on a 20-agent case report 30–60× speedups and a 3–5× accuracy improvement from the ML correction, and a 150-agent congested test case is presented qualitatively. An appendix provides a theoretical analysis of the gIS for a sphere-plane Hertz contact problem, including identifiability proofs based on elliptic-integral computations.

Significance. If the central methodology holds up, this is a useful and timely contribution: it appears to be the first systematic MOR for the Maury–Venel discrete contact model, and it cleanly assembles several non-trivial ingredients (cone-preserving dual basis, PGA enrichment, sign-constrained EQ, ML correction) into a working pipeline. Specific strengths worth crediting: (a) the Hertz appendix contains an actual proof — strict monotonicity of the ratio λ₁/λ₂ via a density-crossing argument and standard elliptic-integral identities — establishing parameter identifiability from two reduced coordinates; this is a clean, independent theoretical result. (b) The EQ formulation with a sign constraint on the contact-matrix weights is well motivated physically and gives a measurable cost advantage over EIM for B_q. (c) The randomized POD comparison is documented quantitatively. (d) The 30–60× speedups on the 20-agent case are reproducible from the reported settings. The work paves the way for nonlinear MOR for non-smooth contact dynamics, a class where linear RB methods are known to be limited.

major comments (5)
  1. [§3.4 and §4.1 (Fig. 14)] The headline qualitative claim of the 150-agent experiment is that the ML-corrected ROM avoids the visible inter-penetration seen in the Galerkin ROM (Fig. 14c vs. 14d). However, the ML map Ψ is trained on residuals in coefficient space (MSE on Δα), and its inputs are (α^{N,R}, μ, ν) — it does not see B_q or d_q, and nothing in the construction enforces D_ℓ(q^{N,R}+hû) ≥ 0 after correction. Consequently, the loss being minimized is not the feasibility loss that Fig. 14d is being used to demonstrate. Please report, on the validation set, the worst-case constraint violation max_ℓ max_ν (-D_ℓ(q^{N,R}_corrected))_+ (and its time history) for both the Galerkin and ML-corrected ROMs. Without this, the claim that the ML correction restores non-overlap is not supported by the training objective.
  2. [§4.1, stability discussion around Eq. (15)] The reduced inf-sup constant is acknowledged to be 0 in the 150-agent regime, and PGA stability is asserted only on the training set. Yet validation parameters in P_valid are drawn from the same distribution and used to support the main claim. Please provide direct evidence that the PGA-enriched primal space is rich enough to satisfy the active constraints for out-of-sample parameters: e.g., the post-PGA supremizer residual sup_{v∈S_R(σ)} ||(I-P_{V_N+S_R})v||_V evaluated on P_valid (not just P_train), and the per-validation-parameter reduced inf-sup after enrichment (analogous to Fig. 5c) for the 150-agent case. As stated, the move from training to validation is an axiom rather than a tested property.
  3. [§4 (Fig. 12a) vs §4.1 (Fig. 14a)] The '3–5× accuracy improvement' headline is documented in the 20-agent regime where PGA gives strictly positive enriched stability and the system is comparatively sparse. The 150-agent figure (Fig. 14a) shows a much smaller relative gap between Galerkin and ML-corrected ROMs and operates in the regime where reduced inf-sup is 0. The two regimes have qualitatively different evidence, and the abstract/conclusion conflate them. Please either (i) restrict the quantitative '3–5×' claim to the regime where it is supported, or (ii) report the same accuracy metric and confidence bands for the 150-agent case so the comparison is on equal footing.
  4. [§3.4 and §5] The paper relies entirely on in-distribution validation (i.i.d. parameters from the same uniform box used for training, with 200 training × ~50 time steps). No a posteriori error indicator is provided, and the authors acknowledge this is postponed. For a method whose stated value is reliability of contact prediction in congested regimes, some indicator — even an inexpensive residual-based one in the form B_q α^{N,R} - d_q evaluated at a small set of EQ points — would substantially strengthen the recommendation. Please consider adding at least a simple online feasibility-residual indicator and reporting its correlation with the actual position error on P_valid.
  5. [Appendix A.2] The identifiability proof (Theorem 2) is clean for the two-mode reduced cone built on the endpoints (a̲, ā). It would be useful to clarify whether the strict monotonicity argument extends, even qualitatively, to the gIS algorithm actually used in the body of the paper (Algorithm 2), which selects canonical-basis vectors e_{i_r} rather than λ_{a̲,p̲}/λ_{ā,p̄}. As written, the appendix proves a fact about the mCPG-type construction (Remark 1) on a continuous Hertz family, not directly about Algorithm 2 applied to the discrete Maury–Venel multipliers. A sentence explicitly delineating the scope of the analytical result would prevent over-reading.
minor comments (9)
  1. [Eq. (3) vs. Eq. (6)] Equation (3) is referenced repeatedly in §2 as 'problem (6)' (e.g., 'numerical discretization of problem (6)' in §1, '§2 the high-fidelity discrete contact model... (6)'). The numbering is inconsistent; the ODE system is (3) and the KKT/Uzawa form is (6). Please harmonize.
  2. [Abstract / §1] The abstract says POD is applied to velocity snapshots while §3.1.1 describes the same POD; please mention explicitly that PGA enrichment (the part that actually delivers stability) is applied on top, since this is one of the load-bearing ingredients.
  3. [§3.3, Eq. (19)] The notation (b_i(σ) V̂_N) for the i-th row of B_q(σ) acting on V̂_N is slightly ambiguous; b_i is a row vector of length N (=2N_a) so b_i V̂_N ∈ R^N. Please add dimensions in the equation.
  4. [§4, Uzawa step] The choice ρ = 0.2/h² is stated without an estimate of ||B_q||₂² for the parameter range used, although §2.1 cites the convergence interval (0, 2/||B_q||₂²). A one-line check that 0.2/h² lies in this interval for the chosen h would help reproducibility.
  5. [§4.1] The 150-agent experiment uses N_T=50 time steps. Given that the FOM has 200 training parameters × 50 steps = 10⁴ snapshots, and that random forests/NNs are sensitive to the size of the training set, a brief sentence on the train/validation split for Ψ and the variance across retrainings would be useful.
  6. [§4, Fig. 9] The y-axis range (6×10⁻² to 6.4×10⁻²) is very narrow; the visual gap between curves overstates the accuracy spread. Please either widen the axis or annotate the absolute differences.
  7. [Algorithm 2, line 9] The next-snapshot selection criterion uses the current basis {e_{i_{r'}}}_{r'=1}^r but writes (λ(σ))_{i_{r'}} e_{i_{r'}} which is just zeroing-out already-selected coordinates; a sentence noting that this reduces to the L² norm of the residual on the unselected coordinates would aid the reader.
  8. [References] [Kep07] is cited as a 'Doctoral dissertation' for Uzawa-type methods; this is non-standard — please provide the canonical reference for Uzawa (Arrow–Hurwicz–Uzawa 1958) in addition. Several references appear truncated (e.g., [Pru+02], [Ver+03]).
  9. [Typos] p.1 'Ehrlarcher' (cover page) → Ehrlacher; p.2 'discretizaton' → discretization; p.5 caption 'foundu_ν' formatting; p.15 'straightfoward' → straightforward; p.20 'remind to Section 4' → refer to; p.24 'in list' → at least.

Simulated Author's Rebuttal

5 responses · 0 unresolved

We thank the referee for a careful and constructive report. The five major comments identify real gaps between what our training objectives and theoretical results actually establish and what the headline claims of the paper assert, particularly in the 150-agent regime. We accept all five points and will revise the manuscript accordingly. Specifically, we will (1) add an explicit feasibility metric max_ℓ(-D_ℓ)_+ on P_valid for both Galerkin and ML-corrected ROMs, replacing the purely qualitative non-penetration claim of Fig. 14; (2) report post-PGA supremizer residuals and per-validation-parameter reduced inf-sup diagnostics on P_valid for the 150-agent case, paralleling Fig. 5 of the 20-agent case, and reframe what PGA actually guarantees; (3) restrict the '3–5× accuracy' headline to the regime where it is supported and report matched accuracy metrics with confidence bands for the 150-agent case; (4) add a simple online residual-based feasibility indicator built from the EQ index set and report its correlation with the true position error on P_valid, while being explicit that this is an indicator and not a certified bound; (5) clarify the scope of Appendix A.2 — the identifiability theorem concerns the mCPG-type basis on the continuous Hertz family (Remark 1), not Algorithm 2 on discrete Maury–Venel multipliers — and add a qualitative discussion of the analogous statement for gIS. We believe these revisions address the referee's substantive concerns without changing the core

read point-by-point responses

  1. Referee: The ML correction Ψ is trained on MSE in coefficient space (Δα) and does not see B_q or d_q; nothing enforces D_ℓ(q^{N,R}+hû) ≥ 0. The Fig. 14d non-overlap claim is not supported by the training objective. Please report the worst-case constraint violation max_ℓ max_ν (-D_ℓ)_+ on P_valid for both Galerkin and ML-corrected ROMs.

    Authors: The referee is correct: the loss minimized by Ψ is a coefficient-space MSE and is not, by construction, a feasibility loss. The qualitative claim in Fig. 14 should therefore be backed by an explicit feasibility metric. In the revision we will add, for both the Galerkin and ML-corrected ROMs on P_valid (and on the 20-agent case for completeness), the time history and worst-case value of V(ν,μ) = max_ℓ ( -D_ℓ(q^{N,R}(ν,μ)) )_+, as well as its average over P_valid. We will report (i) the per-time-step maxima as curves with min/max bands and (ii) the global max over (ν,μ). We will also state plainly in the text that Ψ does not enforce feasibility a priori and that the observed reduction in penetration is an empirical effect: the ML-corrected primal coefficients lie closer to the FOM coefficients, which in turn satisfy the constraints, but no a posteriori guarantee follows from the training objective alone. A short discussion of constraint-aware loss functions (e.g. penalising negative D_ℓ at EQ points) will be added as a perspective for future work. revision: yes

  2. Referee: The reduced inf-sup is 0 in the 150-agent regime and PGA stability is asserted only on the training set, yet validation parameters are used to support the main claim. Please provide the post-PGA supremizer residual and per-validation-parameter reduced inf-sup on P_valid for the 150-agent case (analogous to Fig. 5c).

    Authors: We agree that the move from training to validation in §4.1 is currently presented as an assumption rather than a tested property. In the revision we will replicate, for the 150-agent test case, the diagnostics that were already shown for the 20-agent case in Fig. 5: namely (i) the worst-case post-PGA supremizer residual sup_{v∈S_R(σ)} ||(I-P_{V_N+S_R}) v||_V evaluated over σ ∈ {0,…,N_T}×P_valid, plotted against the PGA tolerance δ; and (ii) the time-averaged reduced inf-sup constant per validation parameter, with and without PGA enrichment, for the chosen pair (N,R)=(50,75). If, as we expect, the inf-sup remains 0 on parts of P_valid (consistent with multiplier non-uniqueness in the congested regime), we will state this explicitly and reframe the claim: PGA guarantees the primal space is rich enough to represent the active-constraint supremizers up to tolerance δ on P_valid, which is the property actually used, rather than positivity of the reduced inf-sup. We thank the referee for forcing this clarification. revision: yes

  3. Referee: The '3–5× accuracy improvement' is documented in the 20-agent regime where PGA gives positive enriched stability, while Fig. 14a (150-agent) shows a much smaller relative gap and operates with reduced inf-sup = 0. The abstract/conclusion conflate the two. Restrict the claim or report matched metrics for 150 agents.

    Authors: Accepted. The two regimes are quantitatively different and we should not conflate them. In the revised manuscript we will: (i) reword the abstract and conclusions so that the '3–5×' figure is attributed specifically to the 20-agent, PGA-stable regime (§4, Fig. 12a), and not to the 150-agent case; (ii) report, for the 150-agent test case, the same accuracy metric E_avg with min/max bands over P_valid for both Galerkin and ML-corrected ROMs at matched n, together with the projection error baseline (already in Fig. 14a) so the comparison is on equal footing; and (iii) add an explicit sentence noting that in the highly congested regime the relative improvement is more modest and that the qualitative non-penetration behavior (now quantified per Comment 1) is the principal observation. The headline numerical claim will be qualified accordingly. revision: yes

  4. Referee: The paper relies entirely on in-distribution validation and provides no a posteriori error indicator. Please consider adding an inexpensive residual-based online indicator (e.g. B_q α^{N,R}-d_q at EQ points) and report its correlation with the actual position error on P_valid.

    Authors: We agree this is a substantive weakness, and we had postponed it deliberately because a fully certified a posteriori bound for this saddle-point inequality is non-trivial. We accept the referee's milder request. In the revision we will add a simple, inexpensive online indicator based on the residuals already available from EQ: η(σ) = || ( B_q α^{N,R} - d_q )_+ || evaluated only at the EQ index set I_2, complemented by the dual feasibility residual β^{N,R} ⊙ (B_q α^{N,R}-d_q). We will report scatter plots of η(σ) versus the true position error E_μ on P_valid for the 20-agent case, together with Spearman/Pearson correlation coefficients, and we will discuss its behavior on the 150-agent case. We will be explicit that this is an indicator, not a certified bound, and that the construction of a rigorous a posteriori estimator remains future work. revision: yes

  5. Referee: Theorem 2 is proved for the two-mode reduced cone built on the endpoints (a̲, ā), i.e. the mCPG-type construction (Remark 1) on a continuous Hertz family. Please clarify whether the strict monotonicity argument extends, even qualitatively, to the gIS algorithm (Algorithm 2) actually used in the body, which selects canonical-basis vectors e_{i_r}.

    Authors: The referee is correct that, as written, Appendix A.2 establishes identifiability for the mCPG-type basis (ψ_1=λ_{a̲,p̲}, ψ_2=λ_{ā,p̄}) on the continuous Hertz family, not for Algorithm 2 applied to the discrete Maury–Venel multipliers. We did not intend to claim otherwise, but the scope is currently under-specified. In the revision we will: (i) add an explicit scope statement at the start of Appendix A clarifying that the analysis concerns the continuous Hertz family and the mCPG basis (Remark 1), and that the result is independent of, but motivates, the discrete gIS used in the main body; (ii) add a short paragraph discussing what the analogous statement would require for gIS — namely identifiability from two canonical coordinates (λ(σ))_{i_1}, (λ(σ))_{i_2} — and noting that for the Hertz family the canonical-basis selection by Algorithm 2 reduces to point evaluations of λ_{a,p}(r) at two radii r_{i_1}, r_{i_2}, whose ratio is again strictly monotone in a by an analogous (and simpler) elliptic-integral argument. We will indicate this as a qualitative extension and not claim a full theorem for the discrete Maury–Venel case. revision: yes

Circularity Check

0 steps flagged

No significant circularity: ML correction is standard supervised learning on HF residuals with held-out validation; PGA/greedy methods are cited from external prior work; Hertz appendix is an independent analytical derivation.

full rationale

This is a numerical methods paper whose central claims (greedy index selection for dual cones, EQ hyper-reduction, ML-corrected Galerkin ROM, 30–60× speedup, 3–5× accuracy gain) are evaluated against an external benchmark — a finite-element/Uzawa high-fidelity Maury–Venel solver implemented in SCoPI — on a validation parameter set P_valid disjoint from the training set P_train. That is the textbook setup for evaluating a ROM and is not circular. The ML correction Ψ is trained on (α^{N,R}, μ, ν) → Δα with Δα coming from HF projections, then evaluated on out-of-sample parameters. Training a regressor on HF residuals and reporting validation error is standard supervised generalization, not a self-definitional loop. The reader's caveat — that the loss (MSE on coefficients) is not the feasibility loss D_ℓ(q) ≥ 0 visualized in Fig. 14d — is a correctness/evaluation-metric concern, not a circularity concern (rule 5: "not standard consensus" or "wrong metric" goes under correctness, not circularity). PGA enrichment is taken from [Nia+23] (different first authors); it is an external method, not a self-citation chain. The mCPG comparison baseline is also from [Nia+23] and [BEE20]. The Hertz appendix derives identifiability of (a,p) from (λ_1, λ_2) via a clean monotonicity argument on elliptic integrals — a self-contained analytical result, not a renaming of inputs. The honest weakness — that the reduced inf-sup constant is 0 in the 150-agent regime and stability is asserted only on the training set — is acknowledged by the authors and postponed to future work for an a posteriori error indicator. This is a stability/reliability gap, not circularity: the headline "3–5× accuracy" number is computed against HF on validation parameters, not by definition. One minor self-referential element: stability of the ROM is enforced by PGA on training snapshots and then "preserved" on validation by assertion ("at least in the training parametric set, as guaranteed by PGA"), but this is openly stated, not concealed. It does not rise above score 1.

Axiom & Free-Parameter Ledger

5 free parameters · 3 axioms · 0 invented entities

The paper introduces no new physical entities; its load-bearing additions are algorithmic (gIS, ML correction wiring). The free-parameter list is mostly numerical-method hyperparameters, not data-fitted constants masquerading as predictions. Stability claims rest on the assumption that PGA enrichment from training transfers to validation; this is the most consequential ad-hoc element.

free parameters (5)
  • ROM dimensions (N, R) = (10,15), (15,20), (25,30), (40,65), (50,75) tested
    Chosen by hand to expose Pareto trade-off; not selected by an error indicator.
  • PGA tolerance delta = 1e-4 and 1e-8
    User-chosen primal-enrichment threshold.
  • EQ point counts (m_EQ_1, m_EQ_2, m_EQ_3) = 2N, R, 2R
    Heuristic choice; sensitivity not explored.
  • ML regressor hyperparameters = RF (default) or NN with hidden=[100,50], ReLU, Adam lr=0.01, 2000 epochs
    Chosen empirically; no ablation reported.
  • Uzawa step rho = 0.2/h^2
    Inside convergence interval but specific value chosen heuristically.
axioms (3)
  • domain assumption Maury-Venel discrete contact ODE (Eq. 3) faithfully models congested crowd motion.
    Inherited from [MV11]; not derived here.
  • standard math Uzawa iteration converges for the chosen step rho=0.2/h^2 within the analytic interval (0, 2/||B_q||^2).
    Cited from [CMT89]; standard saddle-point result.
  • ad hoc to paper PGA enrichment on training snapshots produces a primal basis that satisfies active constraints for validation parameters.
    Cited from [Nia+23] and assumed without an a posteriori bound; reduced inf-sup is 0 in the dense case.

pith-pipeline@v0.9.0 · 48965 in / 4964 out tokens · 86377 ms · 2026-05-06T05:06:34.419036+00:00 · methodology

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    u = P_{C_q^h}(υ_q) = argmin_{v ∈ C_q^h} |v − υ_q|² ... find (u,λ) ∈ ℝ^{2N_a} × ℝ^{N_cont}_+ solution to u = υ_q − B_q^T λ, λ ⊙ (B_q u − d_q) = 0, B_q u − d_q ≤ 0

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