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REVIEW 3 major objections 5 minor 19 references

Diffusion models can respect PDE constraints through a soft denoiser adjustment instead of loss regularizers — and stay unbiased when the physics is imperfect.

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2026-08-03 15:53 UTC pith:ZPK57SMF

load-bearing objection A practical architecture-level trick for softly constraining diffusion denoisers, with a real but unresolved distributional-bias question in the Darcy results. the 3 major comments →

arxiv 2512.14980 v4 pith:ZPK57SMF submitted 2025-12-17 cs.LG

Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations

Victor M. Yeom-Song , Severi Rissanen , Arno Solin , Samuel Kaski , Mingfei Sun This is my paper
classification cs.LG
keywords diffusion modelsconstrained generationphysics-informedPDE constraintsdenoiser architectureguidancesoft inductive biasmisspecification
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The authors argue that the right place to inject physics knowledge into a diffusion model is the denoiser architecture, not the training loss or the sampling loop. Their softly constrained denoiser adds a learnably scaled gradient of the constraint residual to the denoiser output, creating an inductive bias toward constraint-compliant samples that the model can override if the constraint is misspecified. Because the standard denoising loss and sampler are untouched, the population-level guarantee that the model converges to the true data distribution is preserved. The paper proves that regularizer-based approaches necessarily shift the optimal denoiser away from the conditional expectation and degrade the evidence lower bound, and it shows empirically that the new method improves constraint residuals on circle, Darcy flow, and Helmholtz benchmarks while keeping negative log-likelihood close to that of a vanilla model. If correct, SCD offers a drop-in, low-overhead upgrade for physics-constrained generation that is robust to model misspecification.

Core claim

The central discovery is that guidance-style constraint corrections can be folded directly into the denoiser parameterization, with a learnable scale controlling how much the constraint is followed. The proposed softly constrained denoiser is Dθ(xt, t) = Dorig_θ(xt, t) + γθ(xt, t) σ(t)² ∇_Dorig log l_c(Dorig), where l_c = exp(−‖R(·)‖) is a relaxed constraint function derived from a PDE residual. The training loss and sampling procedure remain identical to standard diffusion models, so in the infinite-data limit the optimal denoiser still equals E[x0|xt] and the score estimate stays valid. The authors prove that adding a residual regularizer to the loss, as in prior physics-informed diffusion

What carries the argument

The central object is Eq. (9), the softly constrained denoiser: Dθ(xt, t) = Dorig_θ(xt, t) + γθ(xt, t) σ(t)² ∇ log l_c(Dorig_θ(xt, t)), where l_c is a relaxed constraint function (e.g., exp(−|R(·)|)) and γθ is a small learnable MLP that outputs a nonnegative scaling. This parameterization is derived from a guidance-style approximation to the tilted score for the distribution l_c(x0)p(x0), replacing the expensive vector-Jacobian product of full guidance with a direct gradient of the constraint evaluated at the denoiser output. Because the standard denoising score-matching loss is used unchanged, the architecture carries the inductive bias without forfeiting the Tweedie‑based connection betwee

Load-bearing premise

The load-bearing premise is that the learned scaling γθ can zero out or correctly weight the constraint gradient so that the SCD parameterization remains capable of representing the true conditional expectation E[x0|xt] when the constraint is misspecified — a property that depends on finite network capacity and optimization succeeding.

What would settle it

Train an SCD model on a dataset while deliberately using a constraint residual that is pure noise (e.g., a random scalar field unrelated to the data), then measure the trained γθ outputs and compare the sampled distribution to a vanilla model; if γθ stays large and the distribution visibly shifts, the flexibility guarantee fails in practice.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Constraint knowledge can be incorporated into a diffusion model without changing the loss or sampler, preserving the standard distribution-modeling guarantee in the population limit.
  • Regularizer-based physics-informed diffusion models are provably biased: their optimal denoiser is shifted from E[x0|xt] and their ELBO is strictly worse than vanilla, explaining their sensitivity to misspecified constraints.
  • SCD delivers lower PDE residuals than vanilla diffusion and guidance baselines while maintaining NLL close to vanilla on Helmholtz and within a few bits per dim on Darcy flow.
  • The method is agnostic to the base denoiser architecture and adds modest training/sampling overhead, making it a practical drop-in for diffusion models applied to PDEs and other differentiable constraints.
  • Because any differentiable constraint can define l_c, the same soft inductive bias could be applied beyond PDEs, e.g., to hard physical bounds or learned quality scores.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the learned γθ behaves as intended, an SCD model trained with a completely uninformative random constraint should learn to set γθ near zero and recover the vanilla distribution; that would be a direct, testable confirmation of the flexibility claim.
  • The unbiasedness guarantee is asymptotic; the Darcy Flow results (NLL −5.8 vs. vanilla −10.5 bits/dim) suggest that finite capacity and optimization can leave a residual bias, so practitioners should monitor NLL alongside residuals.
  • The derivation relies on a diagonal covariance approximation for p(x0|xt); using better covariance estimates or Monte Carlo integration, as the limitations section hints, could yield improved SCD variants with stronger constraint compliance.
  • SCD's soft bias could be combined with a final hard-constraint projection for applications requiring exact satisfaction, while keeping the training-phase distribution unbiased.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes Softly Constrained Denoisers (SCD), an architecture modification for diffusion models that appends a learned, noise-level-dependent multiple of the constraint gradient to the denoiser output (Eq. 9). The training loss and sampling procedure are left unchanged, which the authors argue preserves the standard diffusion-model guarantee of converging to the data distribution while providing a soft inductive bias toward constraint satisfaction. The paper proves that regularizer-based training biases the optimal denoiser away from the conditional mean (Props. 3.1–3.2), and reports experiments on toy circle data, Darcy flow, and the Helmholtz equation, measuring residual compliance and NLL. The central claims are that SCD improves constraint satisfaction over vanilla diffusion and guidance baselines while maintaining low distributional bias and robustness to misspecified constraints.

Significance. If the claims are substantiated, SCD would be a practically attractive alternative to physics-informed regularization and inference-time guidance for PDE-constrained generation, with low overhead and an architecture-agnostic construction. The formal result that residual regularizers bias the optimal denoiser is clean and a useful contribution to the literature. The toy and Helmholtz experiments show clear benefits for SCD in terms of residual reduction without NLL degradation. However, the Darcy Flow results, which are the main PDE benchmark, show a substantial NLL regression for SCD, directly undercutting the core 'low distributional bias' claim. The paper also ships reproducible details and a transparent limitation statement, which is commendable.

major comments (3)
  1. [Table 2, §5.2] The Darcy Flow NLL results directly contradict the paper's central claim of low distributional bias. At the correctly specified constraint level (fmax=7.5), SCD achieves a validation NLL of −5.8 bits/dim versus −10.5 for vanilla diffusion, a degradation of 4.7 bits/dim. Under misspecification (fmax=12.5/15), SCD remains at −5.9/−5.8, whereas vanilla stays at −10.5. This is not a small gap, and it is the paper's own metric for distributional fidelity. The Limitations section (Sec. 6) concedes that 'the parameterization may push the model towards particular biases in practice.' This is evidence that the learned γθ does not reliably down-weight the constraint gradient when it conflicts with data, so the 'flexibility to deviate' that distinguishes SCD from regularizers/guidance is not demonstrated on this benchmark.
  2. [Eq. (9), §3] The claim that condition (ii) — convergence to the data distribution — is 'achieved by keeping the standard diffusion loss function and sampling loop' is only an argument about the global optimum in the infinite-data, infinite-capacity limit. In practice, the parameterization Dθ = Dorig + γθ σ² ∇ log l_c(Dorig) may have local minima or optimization paths that do not reach the vanilla solution even though γθ=0 is representable. The paper provides no analysis of the learned γθ values, no ablation setting γθ to zero, and no diagnostic of the optimization trajectory. Given the Darcy NLL gap, the assertion that SCD 'maintains enough flexibility to deviate' is unsupported without such evidence. Please report statistics of γθ or a counterfactual where the correction is disabled.
  3. [§3, Eqs. (6)–(9)] The derivation of the SCD parameterization rests on a sequence of approximations: Dirac-delta approximation to p(x0|xt), diagonal covariance Λ_t, and finally replacing Λ_t by a learned scalar γθ. The paper frames this as a 'principled way to embed the constraint residual,' but the final form is heuristic and its relationship to the exact guidance update (Eq. 4) is loose. This is acceptable if the empirical results are strong, but given the Darcy NLL regression, the heuristic nature should be acknowledged more prominently and the failure modes analyzed. For instance, the constraint gradient ∇ log l_c(Dorig) may have large magnitude at certain noise levels, and the scalar γθ may be unable to modulate it appropriately across the whole trajectory.
minor comments (5)
  1. [Abstract] Typo: 'thesesoftly constrained denoisers' should be 'these softly constrained denoisers'.
  2. [§5.1, Fig. 4] The figure reports DPS results, but Table 1 does not include DPS. Please clarify whether DPS is a guidance baseline and why it is omitted from the table.
  3. [Eq. (5)] The residual function R is used in the regularizer definition before being formally introduced in Sec. 3. Consider defining it earlier or adding a forward reference.
  4. [Table 2/3 captions] The captions say 'Left: Illustration of the Darcy Flow field' but the table layout combines an illustration with the results; please reformat or clarify the caption.
  5. [Sec. F.5] The runtime table reports only vanilla and SCD; it would be useful to include PIDM and guidance runtimes for context, since the paper claims minimal overhead over the baseline.

Circularity Check

0 steps flagged

No significant circularity: the SCD construction is an architecture inspired by guidance approximations, with the scaling γθ learned from the standard diffusion objective; key claims are empirical and self-citations are not load-bearing.

full rationale

The paper's central construction, Eq. (9), is presented as an architecture that embeds a constraint gradient into the denoiser output, motivated by a chain of acknowledged approximations to the guidance formula Eq. (4). This is a design choice, not a derivation whose conclusion is identical to its input: γθ is a learnable scaling trained with the standard denoising objective Eq. (3), and the paper does not claim to derive the empirical constraint-compliance improvements from the parameterization alone. The flexibility claim rests on the possibility γθ can down-weight the correction, which is an empirical/architectural question, not a circular definition. The proofs in Propositions 3.1 and 3.2 analyze the regularizer baseline mathematically and are not circular. The self-citations to Rissanen et al. (2025) appear in contextual lists of denoiser-covariance and Tweedie-formula approximations; they are not used as a uniqueness theorem or as the sole justification for a central claim, and the paper explicitly treats the covariance approximation as one of 'many alternative formulations.' The Limitations section's admission that 'the parameterization may push the model towards particular biases in practice' is an honest empirical caveat about possible distributional bias, not evidence that any prediction reduces by construction to a fitted quantity. The experimental evaluation uses residuals for constraint compliance and NLL for distributional fidelity; the residual metric coincides with the constraint used in training, but that is standard evaluation of constraint satisfaction, not a renamed prediction. Overall, the derivation chain is self-contained and the central claims are empirically testable, so no circular step is established.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central claim rests on a learned scalar γθ, a hand-designed constraint relaxation l_c, a tuned noise schedule, and several approximation choices in the derivation (Dirac posterior, diagonal covariance). No new physical entities are introduced. The unbiasedness guarantee is an assumption, not a theorem.

free parameters (4)
  • γθ scaling network parameters = learned (not reported)
    The magnitude of the constraint correction is fit by minimizing the standard denoising loss; this is the main learned free component of the SCD architecture (Eq. 9, Alg. 1).
  • Constraint relaxation design l_c = exp(-||R||) and norm choice = varies per experiment
    The choice of norm and the exp(-residual) relaxation, plus the constant c in R_circle (Eq. 12), determine what gradient information the denoiser receives; it is a design choice, not derived.
  • Truncated log-normal noise schedule parameters = μ=-2, σ=1.7, lower limit a=-4
    Chosen after observing poor low-noise refinement; this modified loss improves residuals for all methods and is not derived from first principles (Sec. F.1, Table 4).
  • Guidance scale for guided vanilla baseline = 0.03
    Tuned by grid search on residuals in the Darcy Flow experiment (Sec. 5.2); a baseline hyperparameter, but still a fit.
axioms (4)
  • domain assumption R(x) is continuously differentiable and l_c = exp(-||R||) supplies useful gradient information at the denoiser output
    Required for ∇ log l_c to be well-defined; if R is flat or discontinuous, the correction is uninformative or undefined (Sec. 3, Eq. 9).
  • ad hoc to paper Approximate p(x0|xt) as a Dirac at Dθ and covariance as diagonal (Λ_t), then replace Λ_t by learned scalar γθ
    The 'derivation' of Eqs. (8)-(9) relies on these two approximation choices from the guidance literature; they are not shown to be valid for PDE constraints.
  • domain assumption Standard denoising loss with the SCD parameterization preserves convergence to E[x0|xt] as data and optimization go to infinity
    The unbiasedness guarantee is asserted rather than proven; it requires that γθ can learn to zero out the correction and that global optimization is reached (Sec. 3 opening paragraph).
  • domain assumption Finite-difference stencils approximate PDE residuals accurately for both training and evaluation
    Residuals and l_c for Darcy/Helmholtz use finite differences on n×n grids; the discretization is taken from Bastek et al. (2025), and discretization error is not quantified (Sec. 5.2).

pith-pipeline@v1.3.0-alltime-deepseek · 20893 in / 11208 out tokens · 112917 ms · 2026-08-03T15:53:50.329902+00:00 · methodology

0 comments
read the original abstract

Diffusion models have become a powerful generative prior for solutions of partial differential equations (PDEs). Existing approaches enforce physical constraints either by adding the PDE residuals as loss regularizers or through inference-time adjustments. These methods bias the model away from the true data distribution, which is especially problematic when the governing PDE is misspecified. To circumvent these issues while making the most out of the PDE constraint, we introduce soft inductive biases into the denoiser architecture derived from the PDEs. We show that these softly constrained denoisers exploit constraint knowledge to improve compliance over standard denoisers, while maintaining enough flexibility to deviate from it in case of misspecification with respect to observed data.

Figures

Figures reproduced from arXiv: 2512.14980 by Arno Solin, Mingfei Sun, Samuel Kaski, Severi Rissanen, Victor M. Yeom-Song.

Figure 1
Figure 1. Figure 1: Mathematical constraints, even if partially inaccurate, are useful to learn an underlying data distribution (a). Purely data￾driven learning may deviate from the true distribution (b), while regularizer-based methods struggle to balance between observed data and the constraint (c). Our method (d) strikes a balance between constraint and observed data, using the former when useful while favoring the latter … view at source ↗
Figure 2
Figure 2. Figure 2: Previous works have focused on the training (Bastek et al., 2025) and sampling (Huang et al., 2024) axes of the design space to enforce constraints on Diffusion Models. We present an orthogonal approach to enforce soft constraints through the denoiser architecture. time adjustments, we incorporate constraint knowledge into the forward pass of the network. The distinction is concep￾tually illustrated in [P… view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of SCD. By “connecting” the constraint feedback to the denoiser output, we introduce an inductive bias in the denoiser towards constraint satisfaction. The scaling netwrk γθ allows it to modulate how much to follow the constraint. by a separate network. The correction term in Eq. (9) only evaluates the gradient of the constraint lc until Dθ, avoid￾ing a costly vector-Jacobian product. While we… view at source ↗
Figure 4
Figure 4. Figure 4: Top left: Illustration of the “chop” misspecification. Av￾erage Wasserstein-1 distances on the circles examples with varying degrees of misspecification on “Chop”. Vanilla and SCD keep steady values of W1 distance, indicating their flexibility to learn the true data distribution, whereas PIDM and DPS consistently increases with higher levels of misspecification. Means drawn with two standard deviations. ti… view at source ↗
Figure 5
Figure 5. Figure 5: Residuals and samples produced by vanilla diffusion, DPS guidance, SCD and PIDM. The residuals obtained from PIDM are on average lower, but regions with low residual magnitude are mode localized. SCD has an overall more uniform lower magnitude across the residual. 5.3. Helmholtz Equation The Helmholtz Differential Equation is used to model the propagation of waves through (possibly heterogeneous) me￾dia. I… view at source ↗
Figure 6
Figure 6. Figure 6: shows a histogram of the learned distribution of values for pressure and permeability using each of the compared methods. Particularly, as noted by (Bastek et al., 2025), PIDM presents excessive bias compared to the other methods. 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Pressure p 10 5 10 4 10 3 10 2 10 1 10 0 10 1 D e n s i t y o f p 0 10 20 30 40 Permeability K 10 6 10 5 10 4 10 3 10 2 10 1 10 0 D e n s i t y o f K … view at source ↗
Figure 7
Figure 7. Figure 7: Additional samples from the “chop” example. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗

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Reference graph

Works this paper leans on

19 extracted references · 1 linked inside Pith

  1. [2]

    Karnakov, P., Litvinov, S., and Koumoutsakos, P

    doi: 10.1137/24M1636071. Karnakov, P., Litvinov, S., and Koumoutsakos, P. Solving inverse problems in physics by optimizing a discrete loss: Fast and accurate learning without neural networks.PNAS Nexus, 3(1):pgae005, January

    doi:10.1137/24m1636071

  2. [4]

    Following these results, we use the truncated log-normal distribution for all our experiments

    We use a mean of -1.5 and standard deviation of 1.2 for the log-normal loss and a mean of -2, standard deviation of 1.7 and truncation lower limit of -4 for the truncated log-normal loss. Following these results, we use the truncated log-normal distribution for all our experiments. 17 Softly Constrained Denoisers for Diffusion Models Table 4.Residuals obt...

    2022

  3. [5]

    Helmholtz Equation For the Helmholtz Equation we used the UNet implementation by Karras et al

    Table 5.Architecture hyperparameters for the Darcy Flow experiments Hyperparameter Value Model channels 24 Number of residual blocks 8 Per-resolution multipliers [1, 2, 3, 4] Attention resolutions [16, 8] F.4. Helmholtz Equation For the Helmholtz Equation we used the UNet implementation by Karras et al. (2023). We use the Heun sampler implementation by Ka...

    2023

  4. [7]

    Table 7.Runtimes of vanilla and our method on training and sampling on an NVIDIA H200 GPU

    We note that our method sees the most impact at sampling time, since the overhead duplicates per each sampling iteration because the Heun sampler makes two neural function evaluations per iteration. Table 7.Runtimes of vanilla and our method on training and sampling on an NVIDIA H200 GPU. Sampling is done for eight samples at a time using the Heun sampler...

    2022

  5. [9]

    Bastek et al

    and an inference-time adjustment where the final samples are optimized to have a small PDE residual. Bastek et al. (2025) present Physics-Informed Diffusion Models (PIDM), a framework to train DDPM-based diffusion models with a PDE residual as a regularizer term to minimize along the loss function. Several works utilize DPS-like guidance (Chung et al., 20...

    2025

  6. [10]

    Injecting Measurement Structure for Training Inverse Problem SolversMathematically, the closest work is the likelihood-informed Doob’s h-transform by Denker et al

    or GNNs (Valencia et al., 2025), as it remains compatible with any base architecture. Injecting Measurement Structure for Training Inverse Problem SolversMathematically, the closest work is the likelihood-informed Doob’s h-transform by Denker et al. (2024), who finetune adapters using observation gradients ∇x0 p(y|x 0), similar to our constraint-informed ...

    2025

  7. [11]

    dual path

    for Bayesian inference from noisy observations, whereas wedefine lc(x0) to restrict generation to a constrained subset. Their goal is an alternative to inference-time adjustment, while we seek to 13 Softly Constrained Denoisers for Diffusion Models alleviate distributional biases and constraint misspecification. Furthermore, we train from scratch, whereas...

    2022

  8. [12]

    We take p(xT ) to be approximately N(0, σ2 maxI), i.e

    = logp(x T ) + Z T 0 ∇xt ·g(x t, t)dt,(20) where g(t) =∇ xt logp(x t) is the drift of the probability flow ODE, as well as the score function, which is approximated as ∇xt logp(x t)≈ Dθ (xt,t)−xt σ2 t for score matching diffusion models (Song et al., 2021b; Karras et al., 2022), and T=σ 2 max the maximum noise level of the diffusion process. We take p(xT ...

    2022

  9. [13]

    fine-grained

    and f is the net flow of fluid through a given point.Netflow means that if there is the same amount fluid entering and exiting at a given point, then the net flow is zero. As a more concrete example, we can use Darcy flow to describe how water will flow through a body of sand. We can expect more water to flow at the areas where we apply more pressure to s...

    2017

  10. [14]

    We then take the approximation: p(x0|xt)≈ N(x0|µ,Σ),(34) with mean parameter µ and covariance parameter Σ, as this allows us to use Tweedie’s formula, connecting the score function with the exact moments ofp(x 0|xt) (Efron, 2011; Rissanen et al., 2025): µ=E[x 0|xt] =x t +σ 2 t ∇xt logp(x t)(35) Σ =Cov[x 0|xt] =σ 2 t  σ2 t ∇2 xt logp(x t)| {z } Hessian ...

    2011

  11. [100]

    intermediate

    The input for either task is the denoiser output Dθ(xt, t)and the diffusion time t. Between each layer there is an ELU activation function to ensure that the scaling factor remains positive. F.1. Modified Loss Function The loss function by Karras et al. (2022) has the form seen in Eq. (3) with the choice of distributions t∼ LogNormal(µtrain, σ2 train) and...

    2022

  12. [128]

    For the “Dent” variant of misspecification, we use the following parametric curve: C(θ) =(r(θ) cos(θ), r(θ) sin(θ))(39) r(θ) =1−0.25·β wrap(θ− π 2 ) 1.2 ,5 · 1 + 0.6 1−2 wrap(θ− π 2 ) 1.2 2!! (40) wrap(θ) = ((θ+π) mod 2π)−π(41) β(u,5) = ( (1−u 2)5 :|u|<1 0otherwise, (42) where C defines the coordinates of every point in the curve in polar coordinatese. F....

    2023

  13. [2015]

    Song, J., Vahdat, A., Mardani, M., and Kautz, J

    PMLR. Song, J., Vahdat, A., Mardani, M., and Kautz, J. Pseudoinverse-Guided Diffusion Models for Inverse Prob- lems. InInternational Conference on Learning Represen- tations (ICLR), 2023a. Song, J., Zhang, Q., Yin, H., Mardani, M., Liu, M.-Y ., Kautz, J., Chen, Y ., and Vahdat, A. Loss-guided diffu- sion models for plug-and-play controllable generation. I...

    2024

  14. [2020]

    C., Azizzadenesheli, K., and Anandkumar, A

    Yao, J., Mammadov, A., Berner, J., Kerrigan, G., Ye, J. C., Azizzadenesheli, K., and Anandkumar, A. Guided dif- fusion sampling on function spaces with applications to pdes.arXiv preprint arXiv:2505.17004,

    arXiv

  15. [2021]

    Kong, X., Brekelmans, R., and Steeg, G. V . Information- theoretic diffusion. InThe Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5,

    2023

  16. [2022]

    Analyzing and Improving the Training Dy- namics of Diffusion Models.2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp

    Karras, T., Aittala, M., Lehtinen, J., Hellsten, J., Aila, T., and Laine, S. Analyzing and Improving the Training Dy- namics of Diffusion Models.2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pp. 24174–24184,

    2024

  17. [2023]

    Oommen, V ., Bora, A., Zhang, Z., and Karniadakis, G. E. Integrating neural operators with diffusion models im- proves spectral representation in turbulence modeling. arXiv preprint arXiv:2409.08477,

    Pith/arXiv arXiv

  18. [2024]

    Proofs Proposition 3.1.Let D∗ reg(xt, t)be the denoiser that minimizes the regularized objective Lreg

    12 Softly Constrained Denoisers for Diffusion Models Appendices A. Proofs Proposition 3.1.Let D∗ reg(xt, t)be the denoiser that minimizes the regularized objective Lreg. The optimal denoiser output is shifted as follows D∗ reg(xt, t) =E[x0|xt]−λ[∇ yR(y)]⊤ R(y) y=D∗reg ,(11) meaning that the optimal denoiser output D∗ reg(xt, t)is shifted from the conditio...

    2021

  19. [2025]

    Bastek, J.-H., Sun, W., and Kochmann, D

    preprint: arXiv:2506.08604. Bastek, J.-H., Sun, W., and Kochmann, D. Physics-informed diffusion models. InInternational Conference on Learn- ing Representations (ICLR),

    arXiv