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Distributionally Robust Shape and Topology Optimization

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper shows that distributionally robust shape and topology optimization—minimizing the worst expected cost over Wasserstein, moment, or CVaR ambiguity sets—can be rewritten as a single-level, augmented-variable problem, so…

desk verdict The framework is plausible, but the m=0 Wasserstein experiments solve an empty ambiguity set and the reported designs are artifacts; that needs fixing before the numerics can be trusted. read the letter →

arxiv 2507.21574 v1 pith:66G3NCWV submitted 2025-07-29 math.OC cs.NAmath.NA

Charles Dapogny , Julien Prando , Boris Thibert This is my paper
classification math.OCcs.NAmath.NA MSC 90C1549Q1090C47
keywords distributionallyrobustoptimizationtopologyshapeWassersteinambiguitysetentropyregularizationconvexdualityconditionalvalueatriskmoment
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to bring distributionally robust optimization, a machinery from convex optimization, into shape and topology optimization under uncertainty. The setting is that the physical model's parameters (loads, material coefficients) have an unknown law, known only through a nominal distribution reconstructed from a few samples. The paper formulates the design problem as minimizing the worst expected cost over an ambiguity set of nearby laws, and shows that for three ambiguity sets—entropy-regularized Wasserstein balls, moment-constrained sets, and conditional-value-at-risk criteria—the nested min-max program can be rewritten as a single minimization over an augmented set of variables. If correct, engineers can optimize structures that stay reliable under load and material scenarios close to, but absent from, the observed nominal law. Numerical examples in 2D and 3D topology and shape optimization illustrate the claim.

What carries the argument

The mechanism that carries the argument is entropy penalization of the ambiguity constraint. The entropy-regularized Wasserstein distance $W_\varepsilon(P,Q)$ is defined by minimizing coupling cost plus $\varepsilon$ times the relative entropy of the coupling against a reference coupling; this turns the worst-case expectation into a strictly concave problem whose dual can be solved in closed form, producing the log-sum-exp formula. The parameter $\varepsilon$ acts as a smoothing parameter, and the reference variance $\sigma^2$ controls how far the ambiguity set can spread the nominal law. For moment sets, a reference Gaussian $Q_0$ plays the same role; for CVaR, the infimum representation $\mathrm{CVaR}_\beta(C(h,\cdot))=\inf_{\alpha}[\alpha+(1-\beta)^{-1}\int [C(h,\xi)-\alpha]_+\,dP(\xi)]$ is the entry point. The Lagrange multiplier $\lambda$ and the matrix variable $S$ in the moment case reveal whether the worst-case law actually saturates the ambiguity bound.

What would settle it

Solve one of the paper's test problems, for instance the cantilever of Section 3.3.1, for a decreasing sequence of $\varepsilon$ at fixed $\sigma^2$ and $m$, and compare against the exact Wasserstein formulation obtained by replacing the log-sum-exp term with the formal supremum limit of Remark 2.6. If the optimal designs and worst-case costs do not converge as $\varepsilon\to 0$, or if they jump discontinuously, the regularized formulation is not a faithful proxy for the claimed Wasserstein ball.

Watch

Extended reading notes

Core claim

The central claim is that the bilevel structure of distributionally robust optimal design is removable for three practical classes of ambiguity. Using convex duality, the paper rewrites $\sup_{Q\in A}\int C(h,\xi)\,dQ(\xi)$ as an infimum over auxiliary variables: for the Wasserstein set this is $\lambda m + \lambda\varepsilon\int_\Xi \log\left(\int_\Xi e^{(C(h,\zeta)-\lambda c(\xi,\zeta))/(\lambda\varepsilon)}\,d\nu_\xi(\zeta)\right)dP(\xi)$, for the moment set it is an infimum over $(\lambda,\tau,S)$ with a Gaussian reference law, and for CVaR constraints it augments the same formula with the parameter $\alpha$. Because these formulas expose only the cost function and its derivative, the method is agnostic to the design parameterization: it is applied to SIMP density-based topology optimization and to geometric shape optimization by boundary variations, in two and three space dimensions. The out-of-sample tables show that the robust designs trade a few percent of nominal compliance for a large reduction of compliance under unobserved load scenarios.

Load-bearing premise

The load-bearing premise is that the entropy-regularized quantity $W_\varepsilon$ used to define the ambiguity set is a faithful stand-in for the true Wasserstein distance at the chosen $\varepsilon$; in reality $W_\varepsilon$ is not a distance, and the paper lists the $\varepsilon\to 0$ consistency proof as open future work.

Editorial extensions

If this is right

  • The same finite-element and adjoint machinery used for deterministic design can be reused: each robust objective evaluation only requires the cost and derivative for a handful of Monte Carlo parameter samples plus the scalar dual variables.
  • Increasing the Wasserstein radius $m$ changes designs in interpretable ways—diagonal reinforcements in cantilevers, thicker wind-facing regions in masts—so $m$ can be used by practitioners as a tuning knob for conservativeness.
  • Monitoring $\lambda$ during iterations diagnoses active constraints: when $\lambda$ goes to zero, enlarging the ambiguity set no longer affects the design, which can stop the optimization early.
  • Moment-based ambiguity sets output rounded re-entrant corners in L-shaped beams, a known stress-relief feature, and CVaR reliability constraints systematically increase structural volume as the threshold $\beta$ approaches 1.
  • The single-level reformulations carry to 3D shape optimization, so the computational overhead is not tied to dimension but to the cost of solving the underlying elasticity equations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: if the $\varepsilon\to 0$ consistency proof that the authors list as future work succeeds, the method would also compute exact Wasserstein robust designs; if it fails, $\varepsilon$ should be treated as a genuine modeling parameter chosen by calibration, not merely a numerical regularization.
  • Editorial extension: because the duality only needs the cost functional and its derivative, the same derivation should apply to thermal, fluid, and electromagnetic design problems wherever an adjoint state exists.
  • Editorial extension: the paper freezes design-dependent ambiguity sets; a nested scheme that updates the nominal law as the design changes could model uncertainty that grows with the structure, at the price of returning to a bilevel loop.
  • Editorial extension: with a single-sample nominal law $P=\delta_{\xi_0}$ and $m>0$, one could map the minimal radius $m$ needed to cover a target set of unobserved scenarios, giving a data-driven rule for choosing $m$ from the observed sample.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper proposes distributionally robust optimal design formulations under three types of ambiguity: entropy-regularized Wasserstein balls, moment-based ambiguity sets, and distributionally robust conditional value at risk. It derives single-level duality reformulations in an abstract optimal-design framework and applies them to density-based topology optimization and level-set geometric shape optimization, with numerical experiments in 2D and 3D. The central claim is that the resulting designs anticipate load or material scenarios that are close to, but absent from, the nominal empirical distribution.

Significance. The scope is broad and the paper is a timely application of distributionally robust optimization to structural design; the abstract framework and the combination of topology and shape optimization are genuine strengths. If the duality statements and the finite-epsilon method are correct, the m>0 formulations provide a useful practical tool. However, the paper does not prove the epsilon-to-zero consistency needed for the Wasserstein interpretation, and a substantial part of the numerical evidence uses m=0, where the ambiguity set is empty and the dual objective is unbounded below. These issues leave the central claims plausible but not fully established.

major comments (4)
  1. [§2.2, Eqs. (2.9)–(2.10); §3.3.1, §3.3.2, §3.5, §4.3] The m=0 Wasserstein experiments are not well-posed. As acknowledged in Remark 2.4, Wε(P,P)>0 for finite ε; in fact, for an atomic nominal P, any Q with finite Wε(P,Q) must be absolutely continuous with respect to the reference conditional νξ, so no Q satisfies Wε(P,Q)≤0. Hence the ambiguity set AW in (2.9) is empty for m=0, the supremum in (2.10) is −∞, and the dual in (2.11) is unbounded below as λ→∞. The λ histories in Figure 6 for m=0 show exactly this divergence, and the designs labeled h∗_{m=0} in Sections 3.3.1, 3.3.2, 3.5, and 4.3, as well as the corresponding rows of Table 1, are not solutions of a distributionally robust problem. The authors should state a positivity condition for m, preferably m>inf_Q Wε(P,Q), and remove or reinterpret the m=0 computations.
  2. [§5, and §2.2] The consistency of the entropy-regularized ambiguity sets with their exact Wasserstein counterparts as ε→0 is explicitly deferred to future work in Section 5. Until this limit is proved, the numerical designs are certified only for the regularized ambiguity set, not for the Wasserstein ball advertised in the title and abstract. This does not invalidate the finite-ε method, but the paper should state this condition clearly and use terminology such as 'entropy-regularized Wasserstein' rather than 'Wasserstein' when describing the robustness guarantee.
  3. [§2.4.3, Proposition 2.4] As typeset, Proposition 2.4 is internally inconsistent: the left-hand side has an integral of f with respect to Q inside the minimization over α, while the right-hand side has an exponent f(ζ)−λc(ξ,ζ) with no dependence on α. The subsequent reformulation for DC(h,λ,α) correctly uses [C(h,ζ)−α]_+, indicating that the proposition should read [f(ζ)−α]_+ in both the inner objective and the exponent. Please correct the statement and verify the displayed equality.
  4. [§2.3, Proposition 2.2; Appendix B] Proposition 2.2 is a central duality statement for the moment-based method, but its rigorous proof is deferred to [93], a PhD thesis by one of the authors, with only a formal sketch in Appendix B. For a journal submission this is not self-contained. The authors should either include a complete proof or provide a publicly available, refereed reference; otherwise Sections 3.4 and 4.4 rest on an unverified statement.
minor comments (4)
  1. [§3.3.1, Figure 3 caption] The caption repeats 'when m=0' twice and the phrase 'for different values of σ2 when m=0' is redundant; please clarify which row corresponds to which ε.
  2. [§3.3.1] The text says that for m=0 the problem 'may not coincide' with the nominal problem, but it should also state that the ambiguity set can be empty; please add a cross-reference to Remark 2.4 and a discussion of the required lower bound on m.
  3. [Appendix B] There is a typo in the last paragraph: 'sued' should be 'used'.
  4. [Abstract and Section 5] The abstract and conclusion use 'Wasserstein distance' for the quantity Wε; please use 'entropy-regularized Wasserstein' or add an explicit caveat that the consistency with the unregularized distance is not proved.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the dual reformulations are worked out through convex duality rather than assumed, and the numerical claims are forward evaluations; the main self-citation (Proposition 2.2 to [93]) is a supporting lemma with an in-paper proof sketch.

full rationale

Walking the derivation chain, the tractable reformulations are obtained by convex dual arguments that are not equivalent to their inputs: Proposition 2.1 is quoted from the external reference [14] and sketched in Appendix A, Proposition 2.4 is a variation of the same calculation, and Proposition 2.3 is the standard CVaR representation. The one in-group citation, Proposition 2.2 from the thesis [93] by one of the authors, is a supporting lemma for the moment-based reformulation and is accompanied by an independent formal sketch in Appendix B; it is not itself the conclusion being tested. No parameter is fitted to the designs or to the out-of-sample scenarios: the numerical claims are forward evaluations of compliance and stress for prescribed loads, after optimizing the reformulated objectives. The m=0 Wasserstein experiments raise a separate well-posedness concern, since Remark 2.4 already notes that W_epsilon(P,P) is often different from 0 and Section 5 lists consistency with the exact Wasserstein problem as future work; that is a correctness and interpretation issue rather than a definitional circularity. The self-citation to [93] and the reuse of the entropy-regularized Wasserstein framework from [14,33] modestly increase scrutiny, but no central claim reduces to its own inputs by construction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on user-chosen thresholds m, epsilon, sigma, m1, m2, beta and on formal convex-duality arguments. There is no fitted target parameter in the derivation, so the method is not a fit disguised as a prediction, but the consistency of the regularized formulation with exact Wasserstein robustness is unproved.

free parameters (5)
  • Wasserstein radius m = 0, 1, 5, 10 depending on experiment
    Hand-chosen threshold defining the size of the ambiguity set in Sections 3.3, 3.5, 4.3 and 3.6.3; it directly controls how much distributional uncertainty is allowed.
  • Entropy regularization epsilon = 1e-4 to 1e-2, also 0.01 and 0.001
    Chosen by hand to make reformulations tractable; results depend on it, and the paper leaves the epsilon-to-zero limit unproved.
  • Reference coupling variance sigma^2 = 2e-3, 2e-2, 1e-1, 1e-3, 1e-2.5, 1e-0.5
    Hand-chosen parameter in the definition of the reference coupling pi0; it controls the spreading of mass and therefore which laws are considered close to the nominal law.
  • Moment ambiguity bounds m1, m2 = e.g. (0,1), (1,1), (5,5), (10,10)
    Hand-chosen bounds on deviations of the first and second moments in the moment-based ambiguity set of Section 2.3 and numerical Sections 3.4 and 4.4.
  • CVaR threshold beta and safety level CT = beta in {0.01, 0.1, 0.5, 0.9, 0.99}, CT=40 in bridge example
    Hand-chosen probability level and safety threshold that define the failure-probability constraint in Section 3.6.
assumptions (5)
  • domain assumption Existence of a compact finite-dimensional parameter space Xi with continuous cost functions and integrability of all probabilistic integrals
    Invoked in Section 2.1 and in Propositions 2.1, 2.2, 2.4; the numerical settings satisfy it by construction, but the abstract framework relies on it.
  • standard math The infimum-supremum interchange in Appendices A and B is valid for the relevant functionals
    The paper says the arguments are formal; no Sion-von Neumann or measurable selection conditions are verified for the PDE-constrained cost functions.
  • ad hoc to paper The entropy-regularized Wasserstein ambiguity set with finite epsilon approximates the intended exact Wasserstein ambiguity set as epsilon tends to 0
    This is explicitly deferred to future work in Section 5; the numerical interpretations of distributional robustness rely on it.
  • domain assumption The physical model, SIMP interpolation, and linear elasticity boundary-value problems are accurate for the designs under consideration
    Used throughout Sections 3 and 4; optimality claims are relative to this model.
  • domain assumption A nominal law P reconstructed from one or a few samples is a reasonable reference
    Section 2.1 states P may be delta_xi0 from a single observation; the robustness guarantees should be with respect to a true law near P, which is the premise of DRO.

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Pith. "Pith review of Distributionally Robust Shape and Topology Optimization." pith.science (2026). https://pith.science/paper/66G3NCWV

@misc{pith2026250721574,
  author       = {Pith},
  title        = {Pith review of: Distributionally Robust Shape and Topology Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66G3NCWV}},
  note         = {Machine review of arXiv:2507.21574}
}
read the original abstract

This article aims to introduce the paradigm of distributional robustness from the field of convex optimization to tackle optimal design problems under uncertainty. We consider realistic situations where the physical model, and thereby the cost function of the design to be minimized depend on uncertain parameters. The probability distribution of the latter is itself known imperfectly, through a nominal law, reconstructed from a few observed samples. The distributionally robust optimal design problem is an intricate bilevel program which consists in minimizing the worst value of a statistical quantity of the cost function (typically, its expectation) when the law of the uncertain parameters belongs to a certain ``ambiguity set''. We address three classes of such problems: firstly, this ambiguity set is made of the probability laws whose Wasserstein distance to the nominal law is less than a given threshold; secondly, the ambiguity set is based on the first- and second-order moments of the actual and nominal probability laws. Eventually, a statistical quantity of the cost other than its expectation is made robust with respect to the law of the parameters, namely its conditional value at risk. Using techniques from convex duality, we derive tractable, single-level reformulations of these problems, framed over augmented sets of variables. Our methods are essentially agnostic of the optimal design framework; they are described in a unifying abstract framework, before being applied to multiple situations in density-based topology optimization and in geometric shape optimization. Several numerical examples are discussed in two and three space dimensions to appraise the features of the proposed techniques.

Figures

Figures reproduced from arXiv: 2507.21574 by the authors.

Figure 1
Figure 1. The β-value at risk VaRβ(C(h, ·)) of the cost C(h, ·) is the lowest level where the cumulative distribution function Ψ(h, ·) takes values greater than β; the β-conditional value at risk CVaRβ(C(h, ·)) is the integral of C(h, ·) over those events where it exceeds this value (red area). 2.4.2. Reliability-based optimal design using the conditional value at risk The notion of conditional value at risk paves the way to … view at source ↗
Figure 2
Figure 2. (a) Setting of the cantilever beam optimization problem considered in Sec￾tion 3.3.1; (b) Optimized structure for the ideal problem (3.7); (c) Convergence history of the objective C(h, ξ0 ); (d) Convergence history of the volume Vol(h). We now assume that the load ξ applied on ΓN is unknown: it belongs to a sufficiently large ball Ξ in R 2 , and its probability law is itself unknown: the latter is estimated solely f… view at source ↗
Figure 3
Figure 3. Optimized designs of the cantilever beam of Section 3.3.1 obtained by solving the distributionally robust problem (3.9) for m = 0 and different values of σ 2 when m = 0; (upper row) ε = 1e−2; (lower row) ε = 1e−4. Influence of the Wasserstein radius m. We now solve (3.9) for several values of the radius m of the Wasserstein ball; the results are depicted on [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (22 more)
Figure 4
Figure 4. Figure 4: Optimized designs of the cantilever beam of Section 3.3.1 obtained by solving the distributionally robust problem (3.9) for several values of the Wasserstein radius m; (upper row) σ 2 = 2e−2, ε = 1e−4; (middle and bottom rows) σ 2 = 1e−1, ε = 1e−4. 15 [PITH_FULL_IMAGE…
Figure 5
Figure 5. Figure 5: Convergence histories in the distributionally robust optimization of the cantilever considered in Section 3.3.1 for the value σ 2 = 1e−1 and several values of m; (a) Objective function; (b) Volume. As the Wasserstein radius m increases, the “bad” vertical perturbations…
Figure 6
Figure 6. Figure 6: Evolution of the Lagrange multiplier λ during the optimization of the cantilever conducted in Section 3.3.1 for various values of the Wasserstein radius m; (upper row) σ 2 = 2e−2, ε = 1e−4; (lower row) σ 2 = 1e−1, ε = 1e−4. ￾D ￾2 N ⇠1 1 1 2 2 0.1 0.5 ￾1 N ⇠2 ⇠3 ⇠4 ￾3 N…
Figure 7
Figure 7. Figure 7: (a) Setting of the mast problem considered in Section 3.3.2; (b) Optimized design for the problem (3.12), where the objective function is the mean value of the compliance under the nominal law. In this context, the displacement of the structure is the solution uh,ξ to …
Figure 8
Figure 8. Figure 8: Distributionally robust optimized designs h ∗ m=0, h ∗ m=5 and h ∗ m=10 of the electric mast considered in Section 3.3.2 associated to the values m = 0, 5 and 10. in addition to the weight of the cables hanging from the arms, horizontal stretching or compression effect…
Figure 9
Figure 9. Figure 9: Optimal design example of an L-shaped beam considered in Section 3.4; (a) Setting of the test-case; (b) Optimized design h ∗ det in the unperturbed situation; (c) Optimized design h ∗ mean for the mean value of the compliance C(h, ξ) under the nominal law P. We now ass…
Figure 10
Figure 10. Figure 10: Distributionally robust optimized designs obtained in the L-shaped beam exam￾ple of Section 3.4, associated to various values of the parameters m1, m2 of the moment-based ambiguity set AM. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Distributionally robust optimization under material uncertainties: (Upper row) First six eigenpairs (Ei , λi) of the covariance kernel (3.17); (Lower row) Realizations of the random field E(x, ξ) in (3.16), associated to five different instances of the parameter ξ. Re…
Figure 12
Figure 12. Figure 12: (a) Setting of the optimal design example of a gripping mechanism considered in Section 3.5; (b) Optimized design in the ideal situation; (c) Convergence history. where the adjoint state ph,ξ is the unique H1 (D) 2 solution to the following boundary-value problem: (3.…
Figure 13
Figure 13. Figure 13: Distributionally robust optimized designs for material uncertainties of the grip￾ping mechanism considered in Section 3.5 associated to different values of the Wasserstein radius m. 3.6. Reliability-based optimal design of a bridge using the conditional value at risk …
Figure 14
Figure 14. Figure 14: (a) Setting of the bridge problem considered in Section 3.6; (b) Optimized design h ∗ det in ideal conditions. where the threshold CT := 40 is close to the optimal value C(h ∗ det, ξ0 ) attained in the unperturbed minimiza￾tion problem (3.21). According to the discuss…
Figure 15
Figure 15. Figure 15: Optimized designs of the bridge under a safety constraint accounted for by the conditional value at risk, as considered in Section 3.6.2; (Upper row) σ 2 = 1e−2.5; (Lower row) σ 2 = 1e−2. of the Wasserstein type as in Section 2.2.2, the distributionally robust version…
Figure 16
Figure 16. Figure 16: Distributionally robust designs of the bridge considered in Section 3.6.3 for a Wasserstein radius m = 0.5, σ 2 = 1e−3 and various values of the parameter β. β = 0.01 β = 0.1 β = 0.5 β = 0.9 β = 0.99 [PITH_FULL_IMAGE:figures/full_fig_p028_16.png]
Figure 17
Figure 17. Figure 17: Distributionally robust designs of the bridge considered in Section 3.6.3for a Wasserstein radius m = 0.5, σ 2 = 1e−2.5 and various values of the parameter β. 4.1. Shape optimization of linear elastic structures We first recall a few basic facts about the optimization…
Figure 18
Figure 18. Figure 18: Variation Ωθ = (Id + θ)(Ω) of a shape Ω by the method of Hadamard. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: Two complementary representations of a shape Ω ⊂ D; (a) Graph of a level set function ϕ : D → R; (b) Corresponding meshed representation. which can be solved e.g. by the method of characteristics [91]; we rely to this end on the open-source code advection from [21]. S…
Figure 20
Figure 20. Figure 20: (a) Setting of the minimization of stress within a T-shaped beam considered in Section 4.3; (b) Mesh of the initial shape; (c) Optimized shape Ω ∗ det for the problem (4.5), assuming a perfect knowledge of the loads. We now assume that the applied load ξ is uncertain;…
Figure 21
Figure 21. Figure 21: Distributionally robust optimized designs obtained in the T-shaped beam ex￾ample of Section 4.3, associated to various values of the parameters m and σ 2 ; (1 st row) σ 2 = 1e−0.5; (2 nd row) σ 2 = 1e−1; (3 rd row) σ 2 = 1e−3. a b [PITH_FULL_IMAGE:figures/full_fig_p0…
Figure 22
Figure 22. Figure 22: Convergence histories of the objective function DW (Ω, λ) and Vol(Ω) of the distributionally robust optimal result of the T-shaped beam of [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: (a) Setting of the shape optimization example of a 3d cantilever beam considered in Section 4.4; (b) Front and (c) Back views of the optimized structure Ω ∗ det in the ideal situation where the applied load ξ is perfectly known. Now, we assume that the applied load ξ …
Figure 24
Figure 24. Figure 24: Optimized shapes of the 3d cantilever of Section 4.4 under distributional un￾certainties using a moment ambiguity set; front and back views for the parameters (a) m1 = 0, m2 = 1; (b) m1 = 1, m2 = 1; (c) m1 = 2, m2 = 1, and (d) m1 = 5, m2 = 1. 36 [PITH_FULL_IMAGE:figu…
Figure 25
Figure 25. Figure 25: Optimized shapes of the 3d cantilever of Section 4.4 under distributional un￾certainties using a moment ambiguity set; front and back views for the parameters (e) m1 = 0, m2 = 2; (f ) m1 = 2, m2 = 1, and (g) m1 = 5, m2 = 5. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_25.png]

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.