mlsanityData to 14 Oct 2026

Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations

arXiv 2512.14980 ↗First posted 17 Dec 2025cs.LGcited by 1

On the map: Diffusion models → Diffusion foundations and sampling

Abstract (arXiv)

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.

Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. Builds on 53 of its 73 references that are in the map. First posted on arXiv 17 Dec 2025.

Builds on

53 of 73 references.

Closest ancestors · 21

Same direction, newest first

  1. 2025Composition and Alignment of Diffusion Models using Constrained Learning26 Aug
  2. 2025Constrained Generative Modeling with Manually Bridged Diffusion Models27 Feb
  3. 2024Free Hunch: Denoiser Covariance Estimation for Diffusion Models Without Extra Costs15 Oct
  4. 2024Constrained Diffusion Models via Dual Training27 Aug
  5. 2024DEFT: Efficient Fine-Tuning of Diffusion Models by Learning the Generalised h-transform3 Jun
  6. 2024Constrained Synthesis with Projected Diffusion Models5 Feb
  7. 2023Analyzing and Improving the Training Dynamics of Diffusion Models5 Dec
  8. 2023Mirror Diffusion Models for Constrained and Watermarked Generation2 Oct
  9. 2023Metropolis Sampling for Constrained Diffusion Models11 Jul
  10. 2023Diffusion Models for Constrained Domains11 Apr
  11. 2023Reflected Diffusion Models10 Apr
  12. 2023I2SB: Image-to-Image Schr"odinger Bridge12 Feb
9 more
  1. 2023Star-Shaped Denoising Diffusion Probabilistic Models10 Feb
  2. 2023Information-Theoretic Diffusion7 Feb
  3. 2022Riemannian Diffusion Models16 Aug
  4. 2022Elucidating the Design Space of Diffusion-Based Generative Models1 Jun
  5. 2022Riemannian Score-Based Generative Modelling6 Feb
  6. 2021Variational Diffusion Models1 Jul
  7. 2021Maximum Likelihood Training of Score-Based Diffusion Models22 Jan
  8. 2020Score-Based Generative Modeling through Stochastic Differential Equations26 Nov
  9. 2020Denoising Diffusion Probabilistic Models19 Jun

Other references · 32

  1. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  2. 2023Adding Conditional Control to Text-to-Image Diffusion Modelscited by 5,503
  3. 2018Neural Ordinary Differential Equationscited by 3,779
  4. 2022DreamFusion: Text-to-3D using 2D Diffusioncited by 3,025
  5. 2022Video Diffusion Modelscited by 2,305
  6. 2022Diffusion Posterior Sampling for General Noisy Inverse Problemscited by 1,416
  7. 2023ProlificDreamer: High-Fidelity and Diverse Text-to-3D Generation with Variational Score Distillationcited by 1,219
  8. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  9. 2022Zero-Shot Image Restoration Using Denoising Diffusion Null-Space Modelcited by 583
  10. 2022Improving Diffusion Models for Inverse Problems using Manifold Constraintscited by 508
  11. 2023Denoising Diffusion Models for Plug-and-Play Image Restorationcited by 340
  12. 2023Flow Matching on General Geometriescited by 265
20 more
  1. 2023A Variational Perspective on Solving Inverse Problems with Diffusion Modelscited by 222
  2. 2023Protein Design with Guided Discrete Diffusioncited by 157
  3. 2024D-Flow: Differentiating through Flows for Controlled Generationcited by 122
  4. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 113
  5. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 108
  6. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 91
  7. 2024Physics-Informed Diffusion Modelscited by 90
  8. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 75
  9. 2023Direct Diffusion Bridge using Data Consistency for Inverse Problemscited by 66
  10. 2024On conditional diffusion models for PDE simulationscited by 60
  11. 2023CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problemscited by 56
  12. 2024Improving Diffusion Models for Inverse Problems Using Optimal Posterior Covariancecited by 44
  13. 2024Gradient-Free Generation for Hard-Constrained Systemscited by 38
  14. 2024Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modelingcited by 36
  15. 2025Guided Diffusion Sampling on Function Spaces with Applications to PDEscited by 36
  16. 2022A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element methodcited by 33
  17. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
  18. 2024Wavelet Diffusion Neural Operatorcited by 29
  19. 2025Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networkscited by 25
  20. 2022Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustrationcited by 21

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Modelscited by 0

Similar, not cited

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  1. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation
  2. 2026Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
  3. 2024Fast constrained sampling in pre-trained diffusion models
  4. 2024On the Asymptotic Mean Square Error Optimality of Diffusion Models
  5. 2022Diffusion models as plug-and-play priors
  6. 2024Statistical guarantees for denoising reflected diffusion models
  7. 2026The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problems
  8. 2024Denoising Diffusions with Optimal Transport: Localization, Curvature, and Multi-Scale Complexity