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REVIEW 5 major objections 5 minor 72 references

SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows

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

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Pith's one-line read SESaMo modifies normalizing flows with stochastic symmetry modulation plus a self-reparametrized KL objective, claiming near-perfect effective sample sizes on Gaussian-mixture, phi-4, and Hubbard benchmarks.

desk verdict SESaMo is a genuinely new plug-in for injecting symmetries into flows with strong empirical results, but the training objective is a joint ELBO whose fixed point is not shown to match the marginal target. read the letter →

arxiv 2505.19619 v3 pith:45AMSDU6 submitted 2025-05-26 cs.LG cond-mat.str-elhep-latphysics.comp-ph

Janik Kreit , Dominic Schuh , Kim A. Nicoli , Lena Funcke This is my paper · ORCID
classification cs.LGcond-mat.str-elhep-latphysics.comp-ph
keywords normalizingflowsstochasticmodulationsymmetrylearningbrokensymmetrieslatticefieldtheoryHubbardmodelvariationalinferenceself-reparametrizedKL
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

Normalizing flows turn a simple base density into a target density through invertible maps, but when the target has symmetries—multiple modes related by sign flips or rotations—a plain flow tends to drop modes and learn only one. SESaMo adds a stochastic modulation step: after the flow produces a sample, a random variable chooses one symmetry transformation from a fixed set to apply to it, and a learned parameter controls how often each transformation is used. Because that parameter is trainable, the model can represent exact symmetries, where all modes have equal mass, and broken symmetries, where the masses differ. On the benchmarks in the paper—an eight-Gaussian mixture with $Z_8$ symmetry, the complex $\phi^4$ theory on an $8\times 8$ lattice, and the Hubbard model on a $2\times 1$ lattice—SESaMo reaches effective sample sizes of $0.999(1)$, $0.951(3)/0.948(4)$, and $0.999(2)$, respectively, outperforming naive RealNVP and canonicalization. The authors conclude that stochastic modulation is a general recipe for injecting symmetry knowledge into flow-based samplers without requiring a fully equivariant architecture.

What carries the argument

The load-bearing object is the stochastic modulation map $S_{T,u}$ (Eq. 14): a finite set of symmetry transformations $T_0,\dots,T_M$ from which one is selected by a random variable $u$ and applied to the flow output, with probability $p_S(u)$. In the broken-symmetry case $p_S(u)$ is a Bernoulli probability depending on a learnable breaking parameter $b$, so the modulation can redistribute probability mass among modes of unequal weight. The second piece of machinery is the self-reparametrized KL divergence of Eqs. (17)–(18), which adds $\gamma\ln\hat{Z}$—an importance-weighted estimate of the log partition function, with $\gamma=0.5$ in all experiments—to the reverse KL. Appendix J shows this term is what supplies a nonzero gradient for $b$; the standard ELBO has identically zero gradient for $b$ in the large-sample limit. A penalty term $\Lambda(x)$ (Eq. 11) enforces numerical bijectivity by penalizing flow outputs that leave the canonical cell.

What would settle it

A concrete check: train SESaMo on the known two-Gaussian broken-$Z_2$ target of App. E and compare the learned marginal density $q_\theta(x)$ to the exact $p(x)$ pointwise, not only through the effective sample size; if ESS is high but the densities differ pointwise, the surrogate objective's fixed point is not the target.

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Extended reading notes

Core claim

The central claim is that stochastic modulation with the self-reparametrized KL objective teaches a single normalizing flow to cover all symmetry-related modes of a Boltzmann target density and to assign each mode the correct probability mass, for both exact and broken symmetries. The mechanism is to apply a bijective symmetry map $S_u$ to the flow output, where $u$ is sampled with probability $p_S(u)$; for broken symmetries $p_S$ is controlled by a learnable parameter $b$, so the model can produce unequal mode weights. To train $b$, the paper replaces the unknown partition function in the reverse KL divergence with a scaled importance-weighted estimate, giving a nonzero gradient where the standard ELBO has none. On the Hubbard model with broken $Z_4$ symmetry, the learned breaking ratio matches the analytic prediction and the effective sample size reaches $0.999(2)$, which the paper describes as establishing a new state of the art.

Load-bearing premise

Everything rests on the untested assumption that minimizing the modified loss in Eqs. (17)–(18), which includes a half-weighted estimate of the unknown normalization constant, drives the flow's overall density to the target; the paper does not prove that this surrogate objective has the same fixed point as the reverse KL.

Editorial extensions

If this is right

  • SESaMo should write all modes of an exactly symmetric multimodal target with equal mass; on the eight-Gaussian benchmark it reaches ESS $0.999(1)$, compared with $0.992(4)$ for canonicalization and $0.75(26)$ for naive RealNVP.
  • SESaMo should handle broken symmetries by learning $b$: in the broken-$Z_2$ real $\phi^4$ case and the broken-$Z_4$ Hubbard model, the estimated breaking ratio matches the analytically computed ratio within errors.
  • The same machinery extends to continuous symmetries: a trainable spline $h(u)$ parametrizes a rotation angle, letting SESaMo reach ESS around $0.95$ on exact and broken $U(1)$ complex $\phi^4$ theory, where canonicalization cannot be applied.
  • Training with SESaMo converges faster and more stably than the baselines in the reported benchmarks, as shown by ESS-versus-training-time curves.
  • Because the base density no longer needs to be invariant, SESaMo removes one constraint that equivariant-flow construction and canonicalization impose on the prior.

Reading between the lines

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

  • Because the modulation is a post-processing layer on flow outputs, it is agnostic to the internal flow architecture; a natural next step is to stack it on top of already-equivariant flows or gauge-equivariant samplers, where it could handle residual or softly broken components of the symmetry.
  • The reported performance likely depends on the choice $\gamma=0.5$: at $\gamma=0$ the symmetry-breaking parameter $b$ receives no gradient, so the method as stated has no canonical rule for setting $\gamma$, and the optimal value may shift with target complexity and lattice size.
  • As formulated, SESaMo uses declared symmetries rather than discovering them: the symmetry sectors must be known in advance, which suggests coupling it with symmetry detection or learning the set of transformations itself.
  • A stress test that follows from the paper's own limitation note is to shrink the spacing between modes in the Gaussian mixture until target mass sits on the canonical-cell boundary; quantifying the predicted drop in ESS would set the practical operating range of the method.
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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

5 major / 5 minor

Summary. The paper introduces SESaMo (Symmetry-Enforcing Stochastic Modulation), a method for incorporating discrete or continuous symmetry information into normalizing flows. Samples drawn from a base flow are transformed by a symmetry map S_u selected with probability p_S(u); the probability p_S is controlled by a learned breaking parameter b so that broken symmetries can be represented. Training uses a modified objective called the self-reparametrized KL divergence, which includes the unnormalized target log-density, an importance-weighted estimate of the partition function scaled by gamma, and a bijectivity penalty. Experiments on a Z8 Gaussian mixture, complex phi^4 theory with exact and broken U(1), and the Hubbard model with broken Z4 report effective sample sizes that are substantially higher than those of a plain RealNVP and, where applicable, canonicalization. The paper also provides analytical comparisons of the learned symmetry-breaking ratio for the Hubbard model and for real phi^4 theory.

Significance. If the proposed objective is a valid surrogate for reverse-KL training with a tractable marginal density, SESaMo would be a useful and general symmetry-enforcing mechanism for flow-based samplers, especially for broken symmetries where equivariant architectures and canonicalization are difficult to apply. The empirical work has real strengths: results are averaged over ten seeds, code is provided as a supplement, the Hubbard V=2x1 target is written exactly in closed form (Eqs. 51-52), and the learned mode fractions are checked against independent analytical integrals (Eq. 56). Those checks are genuine and are the most convincing part of the paper. However, the central training objective is used with the joint log-density over (x,u) rather than the marginal density q_theta(x), and the paper provides no fixed-point or consistency analysis for this surrogate. The reported high ESS may therefore be a property of the well-separated benchmarks tested rather than a guarantee that SESaMo learns p(x) in general.

major comments (5)
  1. [Sec. 3.2, Eqs. (15)-(18)] The quantity called "log q_theta(x)" in Eq. (15) is the log joint density of the augmented pair (x,u), not the marginal q_theta(x) = sum_u p_S(u) q_tilde(S_u^{-1} x). Substituting this joint log-density into Eq. (17) gives an objective equal to KL(q_marg(x) || p(x)) - H(u|x) up to constants. The extra conditional-entropy term rewards models in which u is uncertain given x; it vanishes only when the symmetry sectors are well separated. The paper never proves that the optimum of Eq. (17) coincides with the minimum of KL(q_theta(x)||p(x)), and no experiment probes overlapping or poorly separated modes. This is load-bearing because the abstract and Sec. 4.2 claim that SESaMo "allows to effectively learn a variety of exact and broken symmetries" as a general mechanism. Please either prove a fixed-point statement under explicit conditions, or characterize the bias of the joint-density objective and add benchmarks where the bias is visible.
  2. [Sec. 3.2 and App. J, Eqs. (17)-(18), (62)-(63)] The gamma ln Zhat term is introduced because the ordinary ELBO gradient with respect to b vanishes in expectation (Eq. 62), but the modified objective is not analyzed as an estimator of any divergence. The expected gradient of the self-normalized partition term is O(1/N) and its fixed point in b is never characterized. All experiments use gamma = 0.5, so the reported ESS values depend on this unanalyzed bias-variance tradeoff. In App. H.2-H.3, the comparison between SESaMo's sample mode fractions and the analytical target integrals is meaningful, but the additional statement that learned b matches R = 1 - 2 e^b is a consistency identity from Eq. (39), not independent evidence that the marginal q_theta equals the target. Please add either a bias analysis of Eq. (18), a comparison of the learned marginal density against the target on a diagnostic where ESS is not sufficient, or a study of the sensitivity of the final ESS and b to gamma.
  3. [Sec. 3.1, Eqs. (12)-(15)] The general stochastic-modulation density in Eqs. (12) and (15) omits both the marginalization over u and the Jacobian determinant of S_u. The correct marginal is q_theta(x) = sum_u p_S(u) q_tilde(S_u^{-1} x) |det J_{S_u^{-1}}(x)|; Eq. (15) is only the joint log-density and is exact only when det J_{S_u} = 1. All transformations tested in this paper (sign flips and rotations) are isometries, so the numerical results are not affected, but the paper advertises stochastic modulation as a general framework for arbitrary symmetries. The definitions should be corrected or the scope should be explicitly restricted to measure-preserving transformations.
  4. [Sec. 4.2 and Table 1] The claim in Sec. 4.2 of establishing "a new state-of-the-art" is stronger than the evidence supports. The comparison set is limited to a plain RealNVP and canonicalization; no comparison is made with Hamiltonian Monte Carlo, other flow architectures, or existing broken-symmetry samplers. I would recommend rewording this as state-of-the-art among the tested baselines.
  5. [Sec. 3.2 and App. I] The fixed choice gamma = 0.5 is stated without a sensitivity analysis. Since the self-reparametrized KL is not a standard divergence, the reader needs to know how much of the reported ESS improvement depends on the particular gamma value. A small ablation over gamma in [0,1] for at least the Gaussian mixture and the Hubbard model would substantially strengthen the empirical claim.
minor comments (5)
  1. [App. J, Eq. (62)] The notation is confusing because b denotes both the learned symmetry-breaking parameter in Eq. (36) and the partition-function estimator written as bZ in Eq. (17). Please rename one of them to avoid the collision.
  2. [Fig. 10 and App. H] There are several typos: 'breaking ration' in the caption of Fig. 10, 'the estimated ration' in App. H.3, 'continuos' in Apps. E/F/H.4, 'one a single A100' in App. I, and 'We will proof' in App. J. These should be corrected.
  3. [Eq. (12)] Equation (12) as written, q_theta(x) = S_u circ q_tilde(x) = q_tilde(x) p_S(u), is dimensionally inconsistent and should be replaced by an explicit pushforward formula for the modulated density, with the marginalization over u written out.
  4. [App. H, Eq. (39)] The analytical breaking-ratio comparison would be more persuasive if the paper reported the learned value of b itself alongside the sample-count estimate R and the analytical R, so that the reader can see which quantity is being compared with the target and which is an identity.
  5. [App. F, Eq. (41)] In the continuous-modulation section, p_S(u) in Eq. (41) is defined through a change of variables in u, but the formula already contains a determinant for the inverse rotation. This double-counting is not explained and should be clarified, since the chain rule in Eq. (43) appears to be used inconsistently with Eq. (41).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central results are empirical benchmarks against analytical ground truths; the only near-tautology (breaking ratio defined via b) is checked against independent integrals.

full rationale

The paper's derivation chain is not circular in the load-bearing sense. The training objective (Eqs. 17-18) is introduced as a modified ELBO with a self-normalized partition-function estimate; App. J derives the gradient with respect to the breaking parameter b and shows why gamma > 0 is needed. Whether this surrogate objective has the same fixed point as KL(q_theta || p) is an open correctness question, but an unproven surrogate is not a circular reduction of the kind this pass targets. The empirical ESS results are external benchmarks (Gaussian mixture, phi-4 theory, Hubbard model), not outputs of the fitted parameters. The breaking-ratio validation is the only place with a definitional flavor: Eq. (39) defines R = 1 - 2e^b, so the modulated model's mode imbalance is a deterministic function of the learned b. However, the paper does not stop at that identity; App. H.2 and H.3 compare the learned-breaking-ratio quantity to analytically integrated target probabilities (Eqs. 51 and 56), so the match is an independent check of b rather than a tautology. The 'analytical' curve in App. H.3 is calibrated with three parameters fitted to the alpha = 0 histogram, which weakens the prediction but does not make it circular. Self-citations (Refs. 14, 38, 58, 69) are used for benchmark definitions, code provenance, and baseline comparison, not as proof of the central claim; the equivariance theorem is cited from Köhler et al., an external source. Overall, the paper is self-contained against external targets, with only a minor presentational tautology around Eq. (39) that is anchored by independent analytical checks.

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

The central claim rests mainly on the learned symmetry-breaking parameter b and on the surrogate training objective. The free parameters are mostly hyperparameters or baseline choices; the most consequential is the unproven equivalence between the joint-density surrogate and the target reverse KL. No new physical entities are introduced. The analytical validation curves partly rely on fitted parameters, which should be read as consistency checks rather than fully independent predictions.

free parameters (5)
  • symmetry-breaking parameter b = learned during training, e.g., p_S = e^b for the flip probability
    Trained via the self-reparametrized KL; the paper validates it against analytical breaking ratios (Eqs. 39, 51-52, 56).
  • self-regularization weight gamma = 0.5 in all experiments
    Fixed by hand in App I; no sensitivity study is reported, and the gamma ln Z-hat term introduces a finite-N bias in the objective.
  • bijectivity penalty amplitude A and gradient scale B = not reported
    Enter Eq. (11) and control the numerical enforcement of bijectivity; their values are absent from the experimental details, limiting reproducibility.
  • parameters A, mu, sigma of the approximate phi-4 magnetization distribution = A = 0.499(2), mu = 2.126(3), sigma = 0.629(3)
    Fit to the alpha = 0 histogram in App H.3 and used to build the 'analytical' breaking ratio R(alpha); the label 'analytical' overstates the independence of this curve.
  • baseline prior variance for GMM RealNVP = (20, 10)
    Chosen to alleviate mode-dropping for the RealNVP baseline only (footnote to Tab. 3), introducing an asymmetric setup in the comparison.
assumptions (5)
  • ad hoc to paper The substitution of the joint log-density log q(x,u) = log q_tilde(x) + log p_S(u) for the marginal log-density log q(x) in the reverse KL objective, i.e., Eqs. (12)-(15) are used as the variational density.
    The paper never derives that minimizing E_q[ln q(x,u) - ln p(x)] is equivalent to minimizing KL(q(x) || p(x)); the objective is a surrogate whose fixed point may differ from the target marginal.
  • domain assumption The stochastic modulation map S_u is bijective so the change-of-variables formula (15) is valid; bijectivity is enforced only approximately through the penalty term Lambda(x) (Eq. 11).
    For discrete sign flips S_u is bijective on R^n, but the composition over the shared flow requires images of distinct sectors to be disjoint, which the penalty only penalizes heuristically; the paper itself notes bijectivity failures near cell borders (App A).
  • domain assumption The symmetry sectors, the group action {T_i} and the canonical cell Omega, are known a priori.
    Stated in App A as a limitation; holds for physical systems with well-defined symmetries but is not automatic for arbitrary targets.
  • domain assumption The importance-weighted estimator Z-hat_N (Nicoli et al. 2020) provides a valid self-regularizer with gamma in [0,1], and the modified objective (17) is used as a training loss.
    The estimator converges to the true Z, but the gamma-weighted term is a biased training objective for gamma > 0; the paper fixes gamma = 0.5 without studying the bias introduced.
  • standard math Exact change-of-variables formula for normalizing flows (Eq. 2).
    Standard background for likelihood computation in normalizing flows.

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Cite this review

Pith. "Pith review of SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows." pith.science (2026). https://pith.science/paper/45AMSDU6

@misc{pith2026250519619,
  author       = {Pith},
  title        = {Pith review of: SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/45AMSDU6}},
  note         = {Machine review of arXiv:2505.19619}
}
abstract

Deep generative models have recently garnered significant attention across various fields, from physics to chemistry, where sampling from unnormalized Boltzmann-like distributions represents a fundamental challenge. In particular, autoregressive models and normalizing flows have become prominent due to their appealing ability to yield closed-form probability densities. Moreover, it is well-established that incorporating prior knowledge - such as symmetries - into deep neural networks can substantially improve training performances. In this context, recent advances have focused on developing symmetry-equivariant generative models, achieving remarkable results. Building upon these foundations, this paper introduces Symmetry-Enforcing Stochastic Modulation (SESaMo). Similar to equivariant normalizing flows, SESaMo enables the incorporation of inductive biases (e.g., symmetries) into normalizing flows through a novel technique called stochastic modulation. This approach enhances the flexibility of the generative model, allowing to effectively learn a variety of exact and broken symmetries. Our numerical experiments benchmark SESaMo in different scenarios, including an 8-Gaussian mixture model and physically relevant field theories, such as the $\phi^4$ theory and the Hubbard model.

Figures

Figures reproduced from arXiv: 2505.19619 by the authors.

Figure 1
Figure 1. Visualization of the canonicalization approach making a flow-based model equivariant with respect to a Z2 symmetry. App. B and [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Visualization of the stochastic modulation approach for enforcing a Z2 symmetry in a flow-based model. 3 Proposed Method: SESaMo Crucially, certain symmetries may be difficult to incorporate through naive canonicalization strategies and are unlikely to be effectively captured by standard flow-based generative models. A representative case is a one-dimensional multimodal distribution with modes of unequal probability… view at source ↗
Figure 3
Figure 3. Gaussian mixture target density (exact Z8 symmetry). All flow-based models are trained until convergence. From left to right we show: the ground truth, RealNVP, canonicalization, and SESaMo (ours). We refer to App. I for more details on the experiments. The complex ϕ 4 scalar field theory in two dimensions The complex ϕ 4 theory offers a simple yet versatile framework for investigating interacting scalar fields. It … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Density for the Hubbard model (broken Z4 symmetry). All flow-based models are trained until convergence. From left to right we show: the ground truth, RealNVP, canonicalization, and SESaMo (ours). We refer to App. I for more details on the experiments. Note that despit…
Figure 5
Figure 5. Figure 5: Illustration of Symmetry-Enforcing Stochastic Modulation (SESaMo) (top row) and [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Example of a penalty term with λ(x) = |x| − π. The penalty term is zero for x ∈ [−π, π] and approaches A as x → ±∞. The parameter B controls the scaling of the penalty gradient. Additionally, one can compute T C−1 T ,z, T C−1 T ,z : x 7→  Tx, if z ∈ Ω T Tx, if Tz ∈ Ω …
Figure 7
Figure 7. Figure 7: Prior Gaussian distribution q0 with mean µ = 2 and standard deviation σ = 1. The transformation Su, implementing the Z2 symmetry, randomly flips the sign of a sample xi ∼ q0 with a probability determined by the breaking parameter b. When b = ln 0.5 (left), the resultin…
Figure 8
Figure 8. Figure 8: Illustration of a lattice described by the Hubbard model. Blue and red circles represent [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: ESS as a function of the GPU training time (minutes) for the Gaussian mixture (left) and [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Left: ESS for different values of the inverse temperature β. The blue and yellow markers correspond to canonicalization and SESaMo, respectively. Means and standard deviations are computed by averaging over three independently trained models (for each method) using th…
Figure 11
Figure 11. Figure 11: Histograms of the magnetization for real [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: Left: ESS for different values of the breaking factor α. The blue and yellow markers refer to canonicalization and SESaMo, respectively. Mean and standard deviations are computed by averaging three models (for both approaches) trained with three different seeds. Right…
Figure 13
Figure 13. Figure 13: Continuous Symmetries: Density plot for real and imaginary components of the complex￾valued fields of complex ϕ 4 scalar field theory, as introduced in Sec. 4, and sampled from trained generative models, i.e., RealNVP and SESaMo. The models have been trained to sample…
Figure 14
Figure 14. Figure 14: ESS as a function of the GPU training time (minutes) for the [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]

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Reference graph

Works this paper leans on

72 extracted references · 63 canonical work pages

  1. [1]

    Rosenbluth, Marshall N

    Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller, and Edward Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics , 21(6):1087–1092, 06 1953. 5This system can serve as a proxy to describe a quantum field theory with two flavors of differing masses [66]. 6Note that despite h...

    work page 1953

  2. [2]

    Statistical analysis of chemical transformation kinetics using markov-chain monte carlo methods

    Linus Görlitz, Zhenglei Gao, and Walter Schmitt. Statistical analysis of chemical transformation kinetics using markov-chain monte carlo methods. Environmental Science & Technology, 45(10):4429–4437, 05 2011

    work page 2011

  3. [3]

    Dragulescu and V

    A. Dragulescu and V . M. Yakovenko. Statistical mechanics of money.The European Physical Journal B - Condensed Matter and Complex Systems, 17(4):723–729, 2000

    work page 2000

  4. [4]

    Barahona

    F. Barahona. On the computational complexity of ising spin glass models. Journal of Physics A: Mathe- matical and General, 15(10):3241, oct 1982

    work page 1982

  5. [5]

    Christophe Andrieu, Nando de Freitas, Arnaud Doucet, and Michael I. Jordan. An introduction to mcmc for machine learning. Machine Learning, 50(1):5–43, 2003

    work page 2003

  6. [6]

    Fersht and Valerie Daggett

    Alan R. Fersht and Valerie Daggett. Protein folding and unfolding at atomic resolution. Cell, 108(4):573– 582, 2002

    work page 2002

  7. [7]

    Critical slowing down

    Ulli Wolff. Critical slowing down. Nuclear Physics B - Proceedings Supplements, 17:93–102, 1990

    work page 1990

  8. [8]

    Willcocks

    Sam Bond-Taylor, Adam Leach, Yang Long, and Chris G. Willcocks. Deep generative modelling: A comparative review of vaes, gans, normalizing flows, energy-based and autoregressive models. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(11):7327–7347, 2022

    work page 2022

Show all 72 references
  1. [9]

    Beyond EL- BOs: A large-scale evaluation of variational methods for sampling

    Denis Blessing, Xiaogang Jia, Johannes Esslinger, Francisco Vargas, and Gerhard Neumann. Beyond EL- BOs: A large-scale evaluation of variational methods for sampling. In Forty-first International Conference on Machine Learning, 2024

    2024

  2. [10]

    Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning

    Frank Noé, Simon Olsson, Jonas Köhler, and Hao Wu. Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science, 365(6457):eaaw1147, 2019

    2019

  3. [11]

    Solving statistical mechanics using variational autoregressive networks

    Dian Wu, Lei Wang, and Pan Zhang. Solving statistical mechanics using variational autoregressive networks. Phys. Rev. Lett., 122:080602, 2019

    2019

  4. [12]

    Nicoli, Shinichi Nakajima, Nils Strodthoff, Wojciech Samek, et al

    Kim A. Nicoli, Shinichi Nakajima, Nils Strodthoff, Wojciech Samek, et al. Asymptotically unbiased estimation of physical observables with neural samplers. Phys. Rev. E, 101:023304, 2020

    2020

  5. [13]

    M. S. Albergo, G. Kanwar, and P. E. Shanahan. Flow-based generative models for Markov chain Monte Carlo in lattice field theory. Phys. Rev. D, 100:034515, aug 2019

    2019

  6. [14]

    Nicoli, Christopher J

    Kim A. Nicoli, Christopher J. Anders, Lena Funcke, Tobias Hartung, et al. Estimation of thermodynamic observables in lattice field theories with deep generative models. Phys. Rev. Lett., 126:032001, 2021

    2021

  7. [15]

    Conditional image generation with pixelcnn decoders

    Aaron van den Oord, Nal Kalchbrenner, Lasse Espeholt, koray kavukcuoglu, et al. Conditional image generation with pixelcnn decoders. In Advances in Neural Information Processing Systems, volume 29, page 4797–4805, 2016

    2016

  8. [16]

    Pixel recurrent neural networks

    Aäron van den Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. Pixel recurrent neural networks. In Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pages 1747–1756, 2016

    2016

  9. [17]

    Variational inference with normalizing flows

    Danilo Rezende and Shakir Mohamed. Variational inference with normalizing flows. In Proceedings of the 32nd International Conference on Machine Learning, volume 37, pages 1530–1538, 2015

    2015

  10. [18]

    Normalizing flows: An introduction and review of current methods

    Ivan Kobyzev, Simon JD Prince, and Marcus A Brubaker. Normalizing flows: An introduction and review of current methods. IEEE transactions on pattern analysis and machine intelligence, 43(11):3964–3979, 2020

    2020

  11. [19]

    LeCun, B

    Y . LeCun, B. Boser, J. S. Denker, D. Henderson, R. E. Howard, et al. Backpropagation applied to handwritten zip code recognition. Neural Computation, 1(4):541–551, 1989

    1989

  12. [20]

    Group equivariant convolutional networks

    Taco Cohen and Max Welling. Group equivariant convolutional networks. In Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pages 2990–2999. PMLR, 20–22 Jun 2016

    2016

  13. [21]

    Cohen, Mario Geiger, Jonas Köhler, and Max Welling

    Taco S. Cohen, Mario Geiger, Jonas Köhler, and Max Welling. Spherical CNNs. In International Conference on Learning Representations, 2018

    2018

  14. [22]

    Gauge equivariant convolutional networks and the icosahedral CNN

    Taco Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolutional networks and the icosahedral CNN. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 1321–1330. PM...

    2019

  15. [23]

    Equivariant message passing for the prediction of tensorial properties and molecular spectra

    Kristof Schütt, Oliver Unke, and Michael Gastegger. Equivariant message passing for the prediction of tensorial properties and molecular spectra. In Proceedings of the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pa...

    2021

  16. [24]

    Lorentz group equivariant neural network for particle physics

    Alexander Bogatskiy, Brandon Anderson, Jan Offermann, Marwah Roussi, David Miller, et al. Lorentz group equivariant neural network for particle physics. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Resea...

    2020

  17. [25]

    E(n) equivariant graph neural networks

    Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E(n) equivariant graph neural networks. In ICML, pages 9323–9332, 2021

    2021

  18. [26]

    Clark, and James Stokes

    Di Luo, Giuseppe Carleo, Bryan K. Clark, and James Stokes. Gauge equivariant neural networks for quantum lattice gauge theories. Phys. Rev. Lett., 127:276402, dec 2021

    2021

  19. [27]

    A theoretical perspective on mode collapse in variational inference

    Roman Soletskyi, Marylou Gabrié, and Bruno Loureiro. A theoretical perspective on mode collapse in variational inference. arXiv preprint arXiv:2410.13300, 2024

    2024 arXiv

  20. [29]

    Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst

    Michael M. Bronstein, Joan Bruna, Yann LeCun, Arthur Szlam, and Pierre Vandergheynst. Geometric deep learning: Going beyond euclidean data. IEEE Signal Processing Magazine, 34(4):18–42, 2017

    2017

  21. [30]

    Gerken, Jimmy Aronsson, Oscar Carlsson, Hampus Linander, Fredrik Ohlsson, et al

    Jan E. Gerken, Jimmy Aronsson, Oscar Carlsson, Hampus Linander, Fredrik Ohlsson, et al. Geometric deep learning and equivariant neural networks. Artificial Intelligence Review, 56(12):14605–14662, 2023

    2023

  22. [31]

    Geometric deep learning: Grids, groups, graphs, geodesics, and gauges

    Michael M Bronstein, Joan Bruna, Taco Cohen, and Petar Veliˇckovi´c. Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv preprint arXiv:2104.13478, 2021

    2021 arXiv

  23. [32]

    Equivariant flows: Exact likelihood generative learning for symmetric densities

    Jonas Köhler, Leon Klein, and Frank Noe. Equivariant flows: Exact likelihood generative learning for symmetric densities. In Proceedings of the 37th International Conference on Machine Learning, volume 119 of Proceedings of Machine Learning Research, pages 5361–5370. PMLR, 13–...

    2020

  24. [33]

    E(n) equivariant normalizing flows

    Victor Garcia Satorras, Emiel Hoogeboom, Fabian Bernd Fuchs, Ingmar Posner, and Max Welling. E(n) equivariant normalizing flows. In Advances in Neural Information Processing Systems, 2021

    2021

  25. [34]

    Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018

    2018

  26. [35]

    Equivariant finite normalizing flows

    Avishek Joey Bose, Marcus Brubaker, and Ivan Kobyzev. Equivariant finite normalizing flows. arXiv preprint arXiv:2110.08649, 2021

    2021 arXiv

  27. [36]

    Albergo, Denis Boyda, Kyle Cranmer, Daniel C

    Gurtej Kanwar, Michael S. Albergo, Denis Boyda, Kyle Cranmer, Daniel C. Hackett, et al. Equivariant flow-based sampling for lattice gauge theory. Phys. Rev. Lett., 125:121601, sep 2020

    2020

  28. [37]

    Albergo, Kyle Cranmer, Daniel C

    Denis Boyda, Gurtej Kanwar, Sébastien Racanière, Danilo Jimenez Rezende, Michael S. Albergo, Kyle Cranmer, Daniel C. Hackett, and Phiala E. Shanahan. Sampling using SU (N ) gauge equivariant flows. Phys. Rev. D, 103(7):074504, 2021

    2021

  29. [38]

    Simulating the hubbard model with equivariant normalizing flows

    Dominic Schuh, Janik Kreit, Evan Berkowitz, Lena Funcke, Thomas Luu, Kim A Nicoli, and Marcel Rodekamp. Simulating the hubbard model with equivariant normalizing flows. arXiv:2501.07371, 2025

    2025 arXiv

  30. [39]

    Ballard, George Papamakarios, Stuart Abercrombie, Sébastien Racanière, et al

    Peter Wirnsberger, Andrew J. Ballard, George Papamakarios, Stuart Abercrombie, Sébastien Racanière, et al. Targeted free energy estimation via learned mappings. The Journal of Chemical Physics , 153(14):144112, oct 2020

    2020

  31. [40]

    Normalizing flows for atomic solids

    Peter Wirnsberger, George Papamakarios, Borja Ibarz, Sébastien Racanière, Andrew J Ballard, et al. Normalizing flows for atomic solids. Machine Learning: Science and Technology, 3(2):025009, may 2022

    2022

  32. [41]

    SE(3) equivariant augmented coupling flows

    Laurence Illing Midgley, Vincent Stimper, Javier Antoran, Emile Mathieu, Bernhard Schölkopf, et al. SE(3) equivariant augmented coupling flows. In Thirty-seventh Conference on Neural Information Processing Systems, 2023

    2023

  33. [42]

    Equivariant flow matching

    Leon Klein, Andreas Krämer, and Frank Noe. Equivariant flow matching. In Thirty-seventh Conference on Neural Information Processing Systems, 2023. 11

    2023

  34. [43]

    Diffusion models: A compre- hensive survey of methods and applications

    Ling Yang, Zhilong Zhang, Yang Song, Shenda Hong, Runsheng Xu, et al. Diffusion models: A compre- hensive survey of methods and applications. ACM Comput. Surv., 56(4), 2023

    2023

  35. [44]

    Equivariant diffusion for molecule generation in 3D

    Emiel Hoogeboom, Víctor Garcia Satorras, Clément Vignac, and Max Welling. Equivariant diffusion for molecule generation in 3D. In Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 8867–8887. PMLR...

    2022

  36. [45]

    Variational inference: A review for statisticians

    David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017

    2017

  37. [46]

    NICE: non-linear independent components estimation

    Laurent Dinh, David Krueger, and Yoshua Bengio. NICE: non-linear independent components estimation. In 3rd International Conference on Learning Representations, ICLR 2015, Workshop Track Proceedings, 2015

    2015

  38. [47]

    Density estimation using real NVP

    Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real NVP. In Interna- tional Conference on Learning Representations, 2017

    2017

  39. [48]

    Neural spline flows

    Conor Durkan, Artur Bekasov, Iain Murray, and George Papamakarios. Neural spline flows. In Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

    2019

  40. [49]

    Glow: Generative flow with invertible 1x1 convolutions

    Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018

    2018

  41. [50]

    Normalizing flows for probabilistic modeling and inference

    George Papamakarios, Eric Nalisnick, Danilo Jimenez Rezende, Shakir Mohamed, and Balaji Lakshmi- narayanan. Normalizing flows for probabilistic modeling and inference. J. Mach. Learn. Res., 22(1), jan 2021

    2021

  42. [51]

    Kullback and R

    S. Kullback and R. A. Leibler. On information and sufficiency. The Annals of Mathematical Statistics, 22(1):79 – 86, 1951

    1951

  43. [52]

    Invariant and equivariant graph networks

    Haggai Maron, Heli Ben-Hamu, Nadav Shamir, and Yaron Lipman. Invariant and equivariant graph networks. In International Conference on Learning Representations, 2019

    2019

  44. [53]

    On the universality of invariant networks

    Haggai Maron, Ethan Fetaya, Nimrod Segol, and Yaron Lipman. On the universality of invariant networks. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 4363–4371. PMLR, 09–15 Jun 2019

    2019

  45. [54]

    Expressive sign equivariant networks for spectral geometric learning

    Derek Lim, Joshua Robinson, Stefanie Jegelka, Yaron Lipman, and Haggai Maron. Expressive sign equivariant networks for spectral geometric learning. In ICLR 2023 Workshop on Physics for Machine Learning, 2023

    2023

  46. [55]

    Stochastic normalizing flows as non-equilibrium transformations

    Michele Caselle, Elia Cellini, Alessandro Nada, and Marco Panero. Stochastic normalizing flows as non-equilibrium transformations. JHEP, 07:015, 2022

    2022

  47. [56]

    Mathis Gerdes, Pim de Haan, Corrado Rainone, Roberto Bondesan, and Miranda C. N. Cheng. Learning lattice quantum field theories with equivariant continuous flows. SciPost Phys., 15:238, 2023

    2023

  48. [57]

    Albergo, et al

    Denis Boyda, Gurtej Kanwar, Sébastien Racanière, Danilo Jimenez Rezende, Michael S. Albergo, et al. Sampling using SU(n) gauge equivariant flows. Phys. Rev. D, 103:074504, apr 2021

    2021

  49. [58]

    Nicoli, Christopher J

    Kim A. Nicoli, Christopher J. Anders, Lena Funcke, Karl Jansen, Shinichi Nakajima, et al. NeuLat: a toolbox for neural samplers in lattice field theories. PoS, LATTICE2023:286, 2024

    2024

  50. [59]

    Progress in normalizing flows for 4d gauge theories.arXiv preprint arXiv:2502.00263, 2025

    Ryan Abbott, Denis Boyda, Daniel C Hackett, Gurtej Kanwar, Fernando Romero-López, Phiala E Shanahan, and Julian M Urban. Progress in normalizing flows for 4d gauge theories.arXiv preprint arXiv:2502.00263, 2025

    2025 arXiv

  51. [60]

    Nicoli, and Marco Panero

    Andrea Bulgarelli, Elia Cellini, Karl Jansen, Stefan Kühn, Alessandro Nada, Shinichi Nakajima, Kim A. Nicoli, and Marco Panero. Flow-based sampling for entanglement entropy and the machine learning of defects. Phys. Rev. Lett., 134:151601, Apr 2025

    2025

  52. [61]

    Lecture notes on normalizing flows for lattice quantum field theories

    Miranda CN Cheng and Niki Stratikopoulou. Lecture notes on normalizing flows for lattice quantum field theories. arXiv preprint arXiv:2504.18126, 2025

    2025

  53. [62]

    Critical Properties of Phi4-Theories

    Hagen Kleinert and Verena Schulte-Frohlinde. Critical Properties of Phi4-Theories. WORLD SCIEN- TIFIC, 2001

    2001

  54. [63]

    Path-gradient estimators for continuous normalizing flows

    Lorenz Vaitl, Kim Andrea Nicoli, Shinichi Nakajima, and Pan Kessel. Path-gradient estimators for continuous normalizing flows. In Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 21945–21959. PM...

    2022

  55. [64]

    Continual repeated annealed flow transport Monte Carlo

    Alex Matthews, Michael Arbel, Danilo Jimenez Rezende, and Arnaud Doucet. Continual repeated annealed flow transport Monte Carlo. In Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pages 15196–15219. ...

    2022

  56. [65]

    Fast and unified path gradient estimators for normalizing flows

    Lorenz Vaitl, Ludwig Winkler, Lorenz Richter, and Pan Kessel. Fast and unified path gradient estimators for normalizing flows. In The Twelfth International Conference on Learning Representations, 2024

    2024

  57. [66]

    Phases of n = 2 theories in two dimensions

    Edward Witten. Phases of n = 2 theories in two dimensions. Nuclear Physics B, 403(1):159–222, 1993

    1993

  58. [67]

    Arovas, Erez Berg, Steven A

    Daniel P. Arovas, Erez Berg, Steven A. Kivelson, and Srinivas Raghu. The hubbard model. Annual Review of Condensed Matter Physics, 13(1):239–274, March 2022

    2022

  59. [68]

    Thomas Luu and Timo A. Lähde. Quantum monte carlo calculations for carbon nanotubes. Phys. Rev. B, 93:155106, Apr 2016

    2016

  60. [69]

    Nicoli, Christopher J

    Kim A. Nicoli, Christopher J. Anders, Tobias Hartung, Karl Jansen, Pan Kessel, et al. Detecting and mitigating mode-collapse for flow-based sampling of lattice field theories. Phys. Rev. D, 108:114501, dec 2023

    2023

  61. [70]

    Neural spline flows

    Conor Durkan, Artur Bekasov, Iain Murray, and George Papamakarios. Neural spline flows. Advances in neural information processing systems, 32, 2019

    2019

  62. [71]

    Lähde, and Thomas Luu

    Jan-Lukas Wynen, Evan Berkowitz, Christopher Körber, Timo A. Lähde, and Thomas Luu. Avoiding ergodicity problems in lattice discretizations of the hubbard model. Phys. Rev. B, 100:075141, Aug 2019

    2019

  63. [72]

    Peskin and Daniel V

    Michael E. Peskin and Daniel V . Schroeder. An introduction to quantum field theory.Frontiers in Physics, 1995

    1995

  64. [73]

    An introduction to goldstone boson physics and to the coset construction

    Daniel Naegels. An introduction to goldstone boson physics and to the coset construction. arXiv preprint arXiv:2110.14504, 2021. 13 Target:p Prior:z∼q0 gθ Ω ST,u ex∼eqθ x∼qθ(x) Target:p Prior:z∼q0 CT,z gθ Ω zc ∼qzc C−1T,z ezc ∼qezc x∼qθ(x) Figure 5: Illustration of Symmetry-En...

    2021 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.