Geometric flow regularization in latent spaces for smooth dynamics with the efficient variations of curvature

Andrew Gracyk

REVIEW 5 major objections 5 minor 57 references

Geometric flow regularization in latent spaces for smooth dynamics with the efficient variations of curvature

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that adding curvature-flow regularization to the latent space of encoder-decoder PDE surrogates improves zero-shot and adversarial robustness, as measured by lower relative L1 errors under out-of-distribution noise and…

desk verdict Genuinely new heuristic curvature-flow losses, honestly labeled as such, but the robustness claim rests on one noisy table and several derivation gaps; worth a serious referee, not a citation yet. read the letter →

arxiv 2506.09679 v4 pith:EM72KIQV submitted 2025-06-11 math.NA cs.NA

Andrew Gracyk This is my paper · ORCID

classification math.NAcs.NA MSC 53C4465M9968T07

keywords geometricflowslatentspaceregularizationcurvatureencoder-decoderPDEsurrogatesrobustnessvariationalautoencoderphysics-informedlearning

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

The paper is trying to establish that imposing curvature-driven geometric flows on the latent space of encoder-decoder PDE surrogates creates smoother, better-structured representations that withstand out-of-distribution and adversarial perturbations. Its four regularization losses—Gaussian curvature via closed-path integration, a Taylor-expanded Gauss equation flow, a Perelman-functional flow, and a harmonic-map energy flow—all lower mean relative L1 error against an extended-VAE baseline on Burger's equation across most of the nine noise and scaling scenarios. The paper is explicit that the methods trade exact geometric meaning for computational efficiency while preserving nondegeneracy, curvature awareness, and volume distortion in the latent geometry. If the claim holds, geometric regularization of the latent metric is a cheap route to robustness for learned dynamics.

What carries the argument

The machinery is a family of latent geometric flows of the form $\partial_t g + F_K[g] = 0$, where $g$ is a neural-network-approximated Riemannian metric on the encoded manifold and $F_K$ is a curvature-dependent differential operator. Four computable losses instantiate it: a Gaussian-curvature loss that uses Stokes' theorem to replace curvature integrals over sampled balls with closed line integrals of first-order Christoffel symbols; a flow built from a Taylor-linearized Gauss equation in which the Weyl tensor is replaced by a second-fundamental-form proxy; a non-parametric Perelman-functional flow for conformally changed metrics with scalar curvature in closed form; and a harmonic-map energy derivative loss. What carries the argument is that each loss is engineered to keep the metric nondegenerate, distortive, and curvature-aware, rather than to solve the geometric PDE exactly.

What would settle it

Monitor the quantity $(g_{ij}\partial_t g_{ij} - 2\tilde{K} g^2_{ij})$ during a training run of the closed-path Gaussian-curvature method; if it changes sign or reaches zero, the loss is not implementing the claimed curvature flow. A sharper test: run the method on the sphere, where Ricci flow has the known radius formula $r(t)=\sqrt{R^2 - 2(d-1)t}$, and check whether the learned metric evolution tracks it.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that augmenting the latent space of an encoder-decoder with curvature-flow losses improves zero-shot and adversarial fidelity for smooth dynamics. For Burger's equation with 300 out-of-distribution samples per scenario, the regularized methods' means are lower than the extended-VAE baseline in most of the nine scenarios; for instance, the Perelman-functional loss gives 0.482 versus 0.765 in Scenario 1, and the path-integration loss gives 0.610 versus 1.310 in Scenario 8. The paper attributes this to latent geometries that are canonical, smooth, curvature-aware, geodesic-aware, and non-topologically void or sparse, and it maintains that these properties persist through the flow despite the losses being 'diminished analytically and mostly heuristic.'

Load-bearing premise

The load-bearing premise is that a certain integrand in the Gaussian-curvature loss is nonnegative, which makes division by it acceptable; the paper itself concedes that nonnegativity is not guaranteed once a curvature proxy is used, and in real training the loss is never zero, so the denominator can cross zero and the loss landscape can become ill-defined.

Editorial extensions

If this is right

Encoder-decoder PDE surrogates trained with any of the four curvature-flow losses should degrade less when evaluation inputs are noised, scaled, or locally perturbed relative to training.
The closed-path Gaussian-curvature loss offers a cheap way to impose surface Ricci flow without differentiating Christoffel symbols, which could make curvature regularization practical in settings where explicit curvature computation is prohibitive.
Because the flows are designed to prevent near-trivial or degenerate metrics, regularized latent spaces should maintain volume and representational capacity over training, which the paper connects to better inference quality.
The same four regularizers apply to both variational and deterministic encoder-decoder architectures, so the gains are not tied to the VAE objective alone.

Reading between the lines

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

If the gains come from the curvature losses' implicit smoothing rather than from accurate geometric evolution, then simpler spectral or Laplacian regularizers might match the robustness at lower cost; the paper does not test this.
A natural next experiment is to run the same four losses on other PDE families, such as wave or reaction-diffusion equations, and on higher intrinsic latent dimensions, to see whether the Burger's-error pattern generalizes.
The paper's own caveat about denominator zero suggests checking whether the closed-path loss actually implements the claimed flow during training; if it does not, the observed robustness may still be real but would need a different explanation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's my read. The paper introduces four latent-space geometric-flow losses: Stokes-loop Gaussian curvature, a linearized-Gauss-equation flow with a Weyl-replacement tensor, a Perelman-functional time-derivative, and a harmonic-energy time-derivative. Those exact terms are new to the literature, and the author explicitly labels them heuristic ("invented," "mostly heuristic," "slightly informal"). That honesty is a real plus: the design goals (nondegeneracy, volume distortion, curvature lower bounds) are coherent, and the reported training overhead is modest. The citation pattern is fine, and the one self-citation is a genuinely related baseline.

The problem is that the central claim, that these flows improve zero-shot and adversarial fidelity, is not supported strongly enough. The only quantitative evidence is Table 1: nine Burger's OOD scenarios with 300 samples each. The standard deviations are large and heavily overlapping; no significance tests or effect sizes are reported. None of the scenarios is an actual adversarial attack, so "adversarial fidelity" is not directly measured. Loss-scale coefficients are tuned per method with "a little bit of freedom," which can inflate apparent gains. No code or data is shipped.

On the derivation side, the paper concedes most of the damage itself. Appendix F divides by a quantity that can pass through zero; the text admits nonnegativity "is not guaranteed." The Navier-Stokes experiment uses an expanding sphere radius with the wrong sign relative to the Ricci-flow formula it cites (Eq. 176 vs Eq. 37). Theorem 5's proof ends by saying the Ricci term might cancel out, so the theorem's only claimed content is unproved. The nondegeneracy results (Theorems 2-4) are largely artifacts of how the losses and tensors are constructed, not discoveries about curvature flows.

Where does that leave us? The idea is plausible and the losses are worth examining, but the empirical table does not establish the headline claim, and the math has enough loose ends that I would not rely on any single result. This is a classic conditional-accept situation: the paper deserves a serious referee because it proposes concrete, testable new losses in an active subfield and is candid about what is speculative. The referee should demand code/data, at least one more dataset with significance tests, and fixes to the division-by-zero step and the sphere-flow inconsistency. I would send it out, not desk-reject; I would not cite it for the empirical claim in its current form.

Referee Report

5 major / 5 minor

Summary. The manuscript proposes four geometric-flow regularizers for latent spaces of encoder-decoder models used as PDE surrogates: a Gaussian-curvature closed-path-integral loss, a second-fundamental-form surrogate flow derived from a linearized Gauss equation, a Perelman-functional conformal flow, and a harmonic-map energy flow. The central empirical claim is that these curvature-flow regularizations improve the zero-shot and adversarial fidelity of latent-space dynamics. Quantitative support is a single Burger's equation experiment (Table 1, nine out-of-distribution scenarios, 300 samples each), supplemented by qualitative Navier-Stokes figures and several appendices containing formal derivations and informal heuristics.

Significance. If the empirical claim held, the paper would make a practical contribution by offering cheap curvature proxies that avoid Christoffel-symbol differentiation and by connecting geometric flows to robust latent-space learning. The manuscript is unusually transparent about its heuristic character: it explicitly labels steps as non-rigorous (Appendix F, Eqs. 72-74), admits that a key nonnegativity assumption is not guaranteed, and acknowledges that one theorem is only 'slightly informal.' These admissions are a genuine strength. However, the advertised conclusion is empirical, and the current evidence—one quantitative dataset, no significance testing, no code or data release, and no actual adversarial attack—is too weak to establish the robustness improvement claimed in the abstract.

major comments (5)

[Table 1 and §J.2] The only quantitative support for the abstract's robustness claim is Table 1, which reports relative L1 errors over nine Burger's out-of-distribution scenarios with 300 samples each. The reported standard deviations are very large relative to the mean differences (e.g., Scenario 1: 0.765±2.704 vs. 0.482±0.962; Scenario 6: 0.237±0.881 vs. 0.152±0.302), no paired significance tests, confidence intervals, or effect sizes are reported, and none of the scenarios is an actual adversarial attack. The per-method loss scaling coefficients are explicitly given 'a little bit of freedom' in §J.2, which permits tuning that could inflate apparent gains. The headline improvement could be within noise, so the central empirical claim is not established.
[Appendix F, Eqs. (58)-(61)] The derivation of the Gaussian-curvature closed-path loss divides the integrand by (g_ij ∂_t g_ij - 2 K̃ g^2_ij) and justifies the division by claiming that E[integrand]=0 implies the integrand is zero almost everywhere by nonnegativity. The paper itself states that 'the nonnegativity is not guaranteed but holds with a sufficient error tolerance' because the Gaussian-curvature proxy K̃ introduces error. In actual training the loss is never zero, the denominator can vanish, and the loss landscape is ill-defined. As written, the path-integration loss does not implement the claimed curvature flow ∂_t g = K g, so this load-bearing derivation collapses.
[Eq. (37) vs. Eq. (176), §3.1 and §J.1] The spherical Ricci-flow radius is r(t) = sqrt(R^2 - 2(d-1)t) in Eq. (37), a contracting sphere, whereas the Navier-Stokes manifold in Eq. (176) is defined with sqrt(50^2 + 50(d-1)t), an expanding sphere. This sign mismatch is never reconciled, and the sign choice is quantitatively relevant to the Navier-Stokes experiment described in §J.1.
[Theorems 2-4, Appendix G/H] The nondegeneracy results are largely consequences of the loss construction rather than properties of the geometric flows. Theorem 4 assumes the loss of Eq. (136) is exactly satisfied and then concludes that ||g||_F cannot tend to 0; because the loss contains an arbitrary positive constant target and the metric-dependent terms vanish in the limit, the conclusion is nearly a restatement of the assumption. Theorems 2 and 3 similarly build in the Γ term to make the contracted flow nonzero. These theorems are presented as guarantees, but they do not establish that the proposed regularizers produce nondegenerate latent geometry in practice.
[Theorem 5 and Appendix I, Eq. (166)-(173)] Theorem 5 claims the harmonic-map loss has nontrivial dependence on Ricci curvature, but the proof ends by conceding that 'we do not show the Ricci curvature term is not canceled out.' This is a load-bearing gap because the harmonic flow is advertised as one of the four main contributions. Additionally, the Monte-Carlo equivalence in Appendix F, Eqs. (72)-(74), is explicitly called non-rigorous and contains an unexplained sign inconsistency (the fourth expression equals the negative of the integral in the fifth expression unless the integral vanishes), so the computational justification for replacing surface integrals with sampled circle integrals is incomplete.

minor comments (5)

[§2] The word 'tradtional' should be 'traditional'.
[§J.2] The text refers to 'Table 2' for the Burger's results, but the quantitative table is labeled Table 1; the cross-reference should be corrected.
[§J.1] In the Navier-Stokes experiment, d denotes the extrinsic dimension of the sphere (d=100), while in the rest of the paper d denotes the intrinsic dimension; this notation conflict should be disambiguated.
[§J.2] The word 'datatset' should be 'dataset'.
[§1] The phrase 'We refer to 3 for an illustration' should read 'We refer to Figure 3 for an illustration.'

Circularity Check

0 steps flagged · score 2.0 of 10

No material circularity: the central robustness claim is an empirical comparison, and the theoretical appendices are explicitly heuristic or conditional; the only self-citation is minor and not load-bearing.

full rationale

The paper's central claim (zero-shot and adversarial fidelity gains from curvature-flow regularization) is supported by Table 1, a measured comparison over OOD Burger's scenarios. No parameter is fitted to the OOD targets and then reported as a prediction; the per-method loss scaling coefficients in Sec. J.2 are tuning choices, not a forced statistical fit. The theoretical appendices are explicitly labeled heuristic or invented (e.g., Sec. 3.3: 'not as much derived as it is invented'; App. F: 'valid under a (non-rigorous) Monte-Carlo-type argument'). Theorems 2-4 are conditional consistency statements about the constructed losses and tensors, not first-principles predictions; for instance, Theorem 2's nondegeneracy condition is the expression produced after the Gamma-term is added, and Theorem 4 assumes exact loss satisfaction, so these are properties of the design rather than independent results. This is self-consistent construction, not circular reasoning. The one self-citation, [18], is used only to justify the extended-encoder baseline choice ('this architecture generally has better out-of-distribution robustness than a baseline encoder D∘E (see [18])'); it is not load-bearing for the comparison among the paper's four methods or against the baseline, and removing it would not change Table 1. Genuine weaknesses are present but are not circularity: App. F's division by a possibly zero denominator with the admission that 'nonnegativity is not guaranteed,' the App. F 'non-rigorous' Monte-Carlo equivalence, and Theorem 5's admitted failure to show that the Ricci term is not canceled. These are correctness and rigor risks, and they weaken the causal interpretation of the empirical gains, but they do not make the empirical claim reduce to its own inputs.

Assumptions & free parameters 5 free parameters · 7 assumptions · 4 invented entities

The central empirical claim rests on a small set of assumptions: a latent manifold with a learnable metric, the validity of four ad-hoc curvature surrogates, and the well-posedness of the losses. Most theoretical claims are conditional or informally proved.

free parameters (5)

Loss scaling coefficients per geometric flow = unspecified; 'given a little bit of freedom'
Section J.2: each method's loss has different terms and the scaling coefficients are tuned, which affects all reported comparisons.
Positive constant targets in Perelman and harmonic losses = e.g., 0.5 in Eq. 189
The losses are |C - d/dt functional|; C is hand-chosen. Section 3.4 and Eq. 189 use 'positive constant' and 0.5 respectively.
Sphere radius and expansion coefficient in Navier-Stokes = 50; noise scale 0.375
Eq. 176 uses r(t)=sqrt(50^2 + 50(d-1)t), with hand-picked radius 50 and added Gaussian noise scale, even though Ricci flow contraction (Eq. 37) has a different form.
Sampling circle radius r for Gaussian curvature line integral = not enumerated
Appendix F Eqs. 63-66 parameterize circles with radius r; the choice of r and number of increments affects the curvature proxy but is not specified.
KL regularization coefficient beta = 0.001
Section J.2 fixes beta=0.001; it is a hyperparameter of the variational objective, though not of the geometric flow itself.

assumptions (7)

ad hoc to paper The integrand of the Gaussian curvature loss is nonnegative a.e., so E[...]=0 implies the integrand is 0 a.e., permitting division by (g_ij ∂_t g_ij - 2 K̃ g^2_ij)
Appendix F Eqs. 58-61; the paper admits nonnegativity is not guaranteed with the curvature proxy and says 'holds with a sufficient error tolerance'.
domain assumption Latent data lies on a smooth immersed manifold and the network metric g stays in an epsilon-neighborhood of the true Riemannian metric
Section 2, F_epsilon definition: 'our simulated Riemannian is near the true Riemannian metric in the function space in an epsilon-neighborhood'.
standard math Whitney immersion theorem applies to embed the d-manifold in 2d-1 Euclidean space
Appendix C: 'We will employ the Whitney immersion theorem in all of our experiments'.
domain assumption Time differentiation and integration commute (Clairaut) and boundary terms vanish for neural fields ψ and metric g
Appendices H and I use exchange of d/dt and integral; Appendix H justifies it with 'Clairaut's theorem' and compact support.
ad hoc to paper The Taylor-linearized Gauss equation around a uniform-curvature metric (Π0 = κ g) is a valid surrogate for Ricci curvature
Section 3.3 and Appendix G, Eqs. 5-7, 83-89; the paper states the flow is 'invented' and 'slightly ad hoc'.
ad hoc to paper Monte-Carlo sampling over balls reproduces the manifold integral of the curvature proxy
Appendix F Eqs. 72-74 are described as a 'non-rigorous Monte Carlo-type argument'.
domain assumption Eigenvalue decay rates are regular enough that eigenvalue bounds in Theorem 4 hold
Appendix H: 'it is crucial to assume regularity on the decay rates of the eigenvalues'; the theorem's conclusion depends on this unverified condition.

invented entities (4)

Weyl-replacement tensor Γ_ijkl = (1/d) Ĥ_ij δ_ij g_kl
purpose: Added to the curvature-tensor surrogate so the contracted flow has a nonvanishing diagonal, preventing metric collapse
Appendix G Eqs. 93-95; introduced ad hoc; the paper says it is the 'Weyl tensor replacement' with no external justification.
Second fundamental form surrogate Π_ij = ||∂_ij E - Σ g^kl ⟨∂_ij E, ∂_k E⟩ ∂_l E||^2
purpose: Cheap stand-in for the true second fundamental form in the Gauss-equation flow
Section 3.3 Eq. 90; replaces a normal-bundle quantity with a Hessian norm; the paper says 'we will use a second fundamental form surrogate'.
Gaussian curvature proxy K̃ via closed-path line integrals
purpose: Estimates Gaussian curvature without differentiating Christoffel symbols
Appendix F Eqs. 62-70; the proxy eK is a computational surrogate and the paper states it introduces error.
Non-parametric conformal factor ψ network for the Perelman-type flow
purpose: Drives metric evolution through a learned conformal transformation without a closed-form flow
Section 3.4: 'we do not have a closed form for the evolution, thus we describe the corresponding g as non-parametric'.

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Pith. "Pith review of Geometric flow regularization in latent spaces for smooth dynamics with the efficient variations of curvature." pith.science (2026). https://pith.science/paper/EM72KIQV

@misc{pith2026250609679,
  author       = {Pith},
  title        = {Pith review of: Geometric flow regularization in latent spaces for smooth dynamics with the efficient variations of curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EM72KIQV}},
  note         = {Machine review of arXiv:2506.09679}
}

read the original abstract

We design strategies in nonlinear geometric analysis to temper the effects of adversarial learning for sufficiently smooth data of numerical method-type dynamics in encoder-decoder methods, variational and deterministic, through the use of geometric flow regularization. We augment latent spaces with geometric flows to control structure, relying on adaptations of curvature and Ricci flow. All of our flows are solved using physics-informed learning. Traditional geometric meaning is traded for computing ability, but we maintain key geometric invariants, the primary of which are maintained, intrinsically-low structure, nontriviality due to sufficient lower bounds on curvature, distortion of volume element, that develop quality in the inference stage. We instill representations that are canonical, smooth, curvature-aware, geodesic-aware, and non-topologically void or sparse. The primary bottleneck of a Ricci curvature flow is that Ricci curvature is high order, thus expensive to compute, so we will attempt to overcome this with properly justified proxies. Our primary contributions are fourfold. We develop a loss based on Gaussian curvature using closed path circulation integration for surfaces, bypassing automatic differentiation of the Christoffel symbols through use of Stokes' theorem. We invent a new parametric flow valid under a Taylor expansion derived from the Gauss equation. We develop two strategies based on time differentiation of functionals, one with a special case of scalar curvature for conformally-changed metrics, and another with harmonic maps, their energy, and induced metrics. Our losses are diminished analytically and mostly heuristic but maintain overall integral latent structure. We showcase that curvature flows and the formulation of geometric structure in intermediary encoded settings enhance learning and overall zero-shot and adversarial fidelity.

Figures

Figures reproduced from arXiv: 2506.09679 by the authors.

**Figure 1.** We illustrate scenarios of noised data in our NavierStokes experiment. We will be studying the auxiliary qualities of PDE learning via mitigating the adversarial effects of irregular data, such as that which is outof-distribution and noised, by availing the development of geometric structure in the latent stages in learning tasks. Our methods will emphasize computational efficiency in the training stage for scal… view at source ↗

**Figure 2.** We plot the means and standard deviations of errors on a specific outof-distribution setting with a (a) vanilla VAE and (b) the Perelman’s functional experiment. With our hyperparameters, results are consistent, especially means. We propose a physics-informed neural network (PINN) framework [48] [49] [41] for finding the metric solution g to the geometric flow. In a tradtional objective, the L 2 integral loss can … view at source ↗

**Figure 3.** We illustrate the manifold latent space in the extended case where all u, E, D are active. (a) corresponds to a curvature-regularized method, and (b) corresponds to a latent space that forms naturally. As we can see, ours are more canonical with respect to a curvature-uniform metric. Red corresponds to out-of-distribution data, and green is noised data. These results are consistent with our Gaussian curvature experi… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: We compare curvature-type flows to alternative flows that are easy to compute, which are typically trivial in derivative order. Algorithm 1: Geometric flow-regularized encoder-decoder training Data: A1 = {ϕ˜ (i) 0 }i , A2 = {ϕ˜ (i) tj }ij evaluated at (x1, . . . , xn, …

**Figure 5.** Figure 5: We illustrate our results on a single example of Burger’s equation in the variational setting. The OOD scenario here is noise with σ = 1.5 is injected into 101 locations of the initial condition. We highlight this is only on a single example, so please refer to [PITH_…

**Figure 6.** Figure 6: We present uncurated (except the first) results on our heavily-regularized autoencoder on Navier-Stokes data using Ricci flow on the sphere. This figure illustrates in-distribution data with heavy dropout and latent Gaussian noise in the test setting, not training. We …

**Figure 7.** Figure 7: We compare a geometric flow of the form ∂tg = −Λ to the steady geometric flow of [18], which produces both a more canonical geometry and one that is of large measure. We plot 3 trajectories with severely noised initial condition and sample discrete relative L 2 error a…

**Figure 8.** Figure 8: We provide empirical evidence that curvature-based flows help support nondegeneracy. This figure represents the means (over the batch, so with respect to both PDE data and time) of the 2 × 2 Riemannian metric learned in our Gaussian curvature experiment with respect to…

**Figure 9.** Figure 9: We provide a similar figure to that of 8 but on our custom geometric flow based on the Riemannian decomposition with the inexpensive second fundamental form. It is noteworthy that we have initialized our metric as g(u, 0) = 3gunit sphere here, which helped slightly. Bl…

**Figure 10.** Figure 10: This figure goes hand-in-hand with 9 and metric belonging to the custom Riemannian decomposition experiment, but we instead initialize our metric at t = 0 with cos(u 1 )du1 ⊗ du3 in the (3, 1),(1, 3) off-diagonal. Note that this helps support a nondegenerate off-diago…

**Figure 11.** Figure 11: We provide a similar figure to that of 8 but of our Perelman functional experiment with a conformally changed metric. As we can see, nondegeneracy is also supported in this experiment. Here, we have clipped ψ (not its derivatives) to be sufficiently large, which we ha…

**Figure 12.** Figure 12: We plot the means and standard deviations of the parametric map in the harmonic map experiment with respect to training iteration mod 20. The standard deviation especially helps show nontriviality, since the manifold can be immersed anywhere in space. 31 [PITH_FULL_I…

**Figure 13.** Figure 13: We plot the 5 × 5 mean over the batch (which includes over the time interval) diagonal elements of a learned Riemannian metric with a geometric flow of the form ∂tg(u, t) = −Λ(u, t) (both are positive semi-definite) in the variational setting on Burger’s equation data…

**Figure 14.** Figure 14: We plot the 5 × 5 mean over the batch (which includes over the time interval) diagonal elements of a learned Riemannian metric with a geometric flow of the form ∂tg(u, t) = −Λ(u, t) in the encoder-decoder setting on Burger’s equation data with respect to training iter…

**Figure 15.** Figure 15: We demonstrate more uncurated (curated for complexity but not for inference quality) results on our Navier-Stokes experiment. We have found our method behaves significantly better with the geometric flow than without under reasonable noise. Eventually, if the OOD effe…

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