mlsanityData to 7 Oct 2026

Physics-Informed Diffusion Models

arXiv 2403.14404 ↗First posted 21 Mar 2024cs.LG cs.CEcited by 88

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

Key facts. Cited by 88 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 44 of its 64 references that are in the map. First posted on arXiv 21 Mar 2024.

Builds on

44 of 64 references.

Closest ancestors · 7

Same direction, newest first

  1. 2023Aligning Optimization Trajectories with Diffusion Models for Constrained Design Generation29 May
  2. 2023Denoising diffusion algorithm for inverse design of microstructures with fine-tuned nonlinear material properties24 Feb
  3. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov
  4. 2022Diffusion Models Beat GANs on Topology Optimization20 Aug
  5. 2022Equivariant Diffusion for Molecule Generation in 3D31 Mar
  6. 2022GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation6 Mar
  7. 2020Permutation Invariant Graph Generation via Score-Based Generative Modeling2 Mar

Other references · 37

  1. 2017Attention Is All You Needcited by 59,932
  2. 2020Denoising Diffusion Probabilistic Modelscited by 20,851
  3. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  4. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  5. 2021High-Resolution Image Synthesis with Latent Diffusion Modelscited by 18,694
  6. 2020Denoising Diffusion Implicit Modelscited by 9,655
  7. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 8,964
  8. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  9. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,005
  10. 2022Classifier-Free Diffusion Guidancecited by 5,717
  11. 2016Layer Normalizationcited by 5,579
  12. 2021GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Modelscited by 3,595
25 more
  1. 2020Transformers are RNNs: Fast Autoregressive Transformers with Linear Attentioncited by 1,979
  2. 2023Consistency Modelscited by 1,778
  3. 2018Group Normalizationcited by 1,514
  4. 2020DiffWave: A Versatile Diffusion Model for Audio Synthesiscited by 1,339
  5. 2021Variational Diffusion Modelscited by 1,145
  6. 2022Diffusion Models: A Comprehensive Survey of Methods and Applicationscited by 1,107
  7. 2022Diffusion-LM Improves Controllable Text Generationcited by 1,006
  8. 2017Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learningcited by 935
  9. 2022Fast Sampling of Diffusion Models with Exponential Integratorcited by 526
  10. 2022PhysDiff: Physics-Guided Human Motion Diffusion Modelcited by 365
  11. 2021Crystal Diffusion Variational Autoencoder for Periodic Material Generationcited by 268
  12. 2023Efficient Diffusion Training via Min-SNR Weighting Strategycited by 242
  13. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  14. 2021Path Integral Sampler: a stochastic control approach for samplingcited by 175
  15. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  16. 2023Denoising Diffusion Samplerscited by 144
  17. 2017Sobolev Training for Neural Networkscited by 136
  18. 2023AR-Diffusion: Auto-Regressive Diffusion Model for Text Generationcited by 113
  19. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 100
  20. 2024Constrained Synthesis with Projected Diffusion Modelscited by 83
  21. 2020TopologyGAN: Topology Optimization Using Generative Adversarial Networks Based on Physical Fields Over the Initial Domaincited by 65
  22. 2023Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion modelscited by 48
  23. 2023From Zero to Turbulence: Generative Modeling for 3D Flow Simulationcited by 46
  24. 2023DiffuseBot: Breeding Soft Robots With Physics-Augmented Generative Diffusion Modelscited by 30
  25. 2020A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observablescited by 28

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 8; 2025: 31; 2026: 49.

88 papers.

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  1. 2024Understanding World or Predicting Future? A Comprehensive Survey of World Modelscited by 161
  2. 2024Diffusion Predictive Control with Constraintscited by 32
  3. 2025Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraintscited by 26
  4. 2025FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modelingcited by 19
  5. 2025I2-NeRF: Learning Neural Radiance Fields Under Physically-Grounded Media Interactionscited by 17
  6. 2025DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectoriescited by 14
  7. 2024Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operatorcited by 12
  8. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generationcited by 11
  9. 2025Source-Guided Flow Matchingcited by 11
  10. 2025Guiding diffusion models to reconstruct flow fields from sparse datacited by 11
  11. 2025PhyDA: Physics-Guided Diffusion Models for Data Assimilation in Atmospheric Systemscited by 9
  12. 2025Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problemscited by 8
38 more
  1. 2025Unifying and extending Diffusion Models through PDEs for solving Inverse Problemscited by 7
  2. 2024Conditional score-based diffusion models for solving inverse problems in mechanicscited by 6
  3. 2024Generating Physical Dynamics under Priorscited by 6
  4. 2024DiffBatt: A Diffusion Model for Battery Degradation Prediction and Synthesiscited by 6
  5. 2025Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspectivecited by 6
  6. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  7. 2025MoReGen: Multi-Agent Motion-Reasoning Engine for Code-based Text-to-Video Synthesiscited by 5
  8. 2024Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systemscited by 4
  9. 2025Dynamics-aware Diffusion Models for Planning and Controlcited by 4
  10. 2025FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulationcited by 4
  11. 2026PAD-Hand: Physics-Aware Diffusion for Hand Motion Recoverycited by 4
  12. 2025Pick-and-place Manipulation Across Grippers Without Retraining: A Learning-optimization Diffusion Policy Approachcited by 3
  13. 2025Surrogate Modeling of 3D Rayleigh-Benard Convection with Equivariant Autoencoderscited by 3
  14. 2026Physics-informed diffusion models in spectral spacecited by 3
  15. 2026Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systemscited by 3
  16. 2026Project and Generate: Divergence-Free Neural Operators for Incompressible Flowscited by 3
  17. 2026Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Samplingcited by 3
  18. 2025Physics-Informed Diffusion Models for Unsupervised Anomaly Detection in Multivariate Time Seriescited by 2
  19. 2025Inference-stage Adaptation-projection Strategy Adapts Diffusion Policy to Cross-manipulators Scenarioscited by 2
  20. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 2
  21. 2026PHDME: Physics-Informed Diffusion Models without Explicit Governing Equationscited by 2
  22. 2026PILD: Physics-Informed Learning via Diffusioncited by 2
  23. 2026Manifold-Aware Perturbations for Constrained Generative Modelingcited by 2
  24. 2026Learning Flow Distributions via Projection-Constrained Diffusion on Manifoldscited by 2
  25. 2026PH-Dreamer: A Physics-Driven World Model via Port-Hamiltonian Generative Dynamicscited by 2
  26. 2026Least-Action-Guided Diffusion for Physical Extrapolationcited by 2
  27. 2024InteractPro: A Unified Framework for Motion-Aware Image Compositioncited by 1
  28. 2025Latent Generative Modeling of Random Fields from Limited Training Datacited by 1
  29. 2025Generative Latent Diffusion Model for Inverse Modeling and Uncertainty Analysis in Geological Carbon Sequestrationcited by 1
  30. 2025PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Modelscited by 1
  31. 2026Ultra Fast PDE Solving via Physics Guided Few-step Diffusioncited by 1
  32. 2026Spectral Regularization for Diffusion Modelscited by 1
  33. 2026Physics-Informed Video Diffusion For Shallow Water Equationscited by 1
  34. 2025Censored Sampling for Topology Design: Guiding Diffusion with Human Preferencescited by 0
  35. 2025Towards Digital Twins for Optimal Radioembolizationcited by 0
  36. 2025GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometriescited by 0
  37. 2025Generative Models for Helmholtz Equation Solutions: A Dataset of Acoustic Materialscited by 0
  38. 2025Energy Loss Functions for Physical Systemscited by 0

50 most cited of 88 All 88 (JSON)

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  1. 2025The Principles of Diffusion Models
  2. 2024GUD: Generation with Unified Diffusion
  3. 2026A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
  4. 2023Diffusion Models for Constrained Domains
  5. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observation
  6. 2025Diffusion Models: A Mathematical Introduction
  7. 2025Training-Free Constrained Generation With Stable Diffusion Models
  8. 2026Efficient Diffusion Models under Nonconvex Equality and Inequality constraints via Landing