mlsanityData to 14 Oct 2026

Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints

arXiv 2506.04171 ↗First posted 4 Jun 2025cs.LG cs.AI cs.CEcited by 52

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

Abstract (arXiv)

Deep generative models have recently been applied to physical systems governed by partial differential equations (PDEs), offering scalable simulation and uncertainty-aware inference. However, enforcing physical constraints, such as conservation laws (linear and nonlinear) and physical consistencies, remains challenging. Existing methods often rely on soft penalties or architectural biases that fail to guarantee hard constraints. In this work, we propose Physics-Constrained Flow Matching (PCFM), a zero-shot inference framework that enforces arbitrary nonlinear constraints in pretrained flow-based generative models. PCFM continuously guides the sampling process through physics-based corrections applied to intermediate solution states, while remaining aligned with the learned flow and satisfying physical constraints. Empirically, PCFM outperforms both unconstrained and constrained baselines on a range of PDEs, including those with shocks, discontinuities, and sharp features, while ensuring exact constraint satisfaction at the final solution. Our method provides a flexible framework for enforcing hard constraints in both scientific and general-purpose generative models, especially in applications where constraint satisfaction is essential.

Key facts. Cited by 52 papers in the map (data through 14 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 53 of its 73 references that are in the map. First posted on arXiv 4 Jun 2025.

Builds on

53 of 73 references.

Closest ancestors · 9

Same direction, newest first

  1. 2024Gradient-Free Generation for Hard-Constrained Systems2 Dec
  2. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observation25 Jun
  3. 2024D-Flow: Differentiating through Flows for Controlled Generation21 Feb
  4. 2023Functional Flow Matching26 May
  5. 2023Flow Matching on General Geometries7 Feb
  6. 2023A Denoising Diffusion Model for Fluid Field Prediction27 Jan
  7. 2022A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction26 Nov
  8. 2022Flow Matching for Generative Modeling6 Oct
  9. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 44

  1. 2017Attention Is All You Needcited by 60,361
  2. 2020Language Models are Few-Shot Learnerscited by 32,419
  3. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  4. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  5. 2020Score-Based Generative Modeling through Stochastic Differential Equationscited by 9,122
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  2. 2019Hamiltonian Neural Networkscited by 712
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  4. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 12; 2026: 40.

52 papers.

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  1. 2025Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraintscited by 28
  2. 2025Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problemscited by 10
  3. 2026Decoupled Diffusion Sampling for Inverse Problems on Function Spacescited by 10
  4. 2025End-to-End Probabilistic Framework for Learning with Hard Constraintscited by 9
  5. 2025Mirror Flow Matching with Heavy-Tailed Priors for Generative Modeling on Convex Domainscited by 8
  6. 2025Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generationcited by 7
  7. 2026Project and Generate: Divergence-Free Neural Operators for Incompressible Flowscited by 7
  8. 2026D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matchingcited by 6
  9. 2026FlowRefiner: Flow Matching-Based Iterative Refinement for 3D Turbulent Flow Simulationcited by 5
  10. 2026Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computingcited by 4
  11. 2025PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attentioncited by 3
  12. 2026Flow-based Extremal Mathematical Structure Discoverycited by 3
38 more
  1. 2026Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Samplingcited by 3
  2. 2025Reframing Generative Models for Physical Systems using Stochastic Interpolantscited by 2
  3. 2026ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidancecited by 2
  4. 2026Learning Flow Distributions via Projection-Constrained Diffusion on Manifoldscited by 2
  5. 2026Provably Safe Generative Sampling with Constricting Barrier Functionscited by 2
  6. 2026Conditional flow matching for physics-constrained inverse problems with finite training datacited by 2
  7. 2026The Right Measure for Physics-Constrained Generation: A Co-Area Correction for Posterior-Consistent PDE Inverse Problemscited by 2
  8. 2026Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewardscited by 2
  9. 2025Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracturecited by 1
  10. 2026Constrained Diffusion for Accelerated Structure Relaxation of Inorganic Solids with Point Defectscited by 1
  11. 2026AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Trainingcited by 1
  12. 2026Error-Conditioned Neural Solverscited by 1
  13. 2025GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometriescited by 0
  14. 2025FATHOMS-RAG: A Framework for the Assessment of Thinking and Observation in Multimodal Systems that use Retrieval Augmented Generationcited by 0
  15. 2025Flow-Opt: Scalable Centralized Multi-Robot Trajectory Optimization with Flow Matching and Differentiable Optimizationcited by 0
  16. 2025Prompt-to-Parts: Generative AI for Physical Assembly and Scalable Instructionscited by 0
  17. 2026Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulationscited by 0
  18. 2026Convolution Operator Network for Forward and Inverse Problems (FI-Conv): Application to Plasma Turbulence Simulationscited by 0
  19. 2026Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridgecited by 0
  20. 2026AMGenC: Generating Charge Balanced Amorphous Materialscited by 0
  21. 2026Flow Matching with Arbitrary Auxiliary Pathscited by 0
  22. 2026Inpainting physics: self-supervised learning for context-driven fluid simulationcited by 0
  23. 2026Enforcing Constraints in Generative Sampling via Adaptive Correction Schedulingcited by 0
  24. 2026Guided Flow Matching for Forward and Inverse PDE Problems with Sparse Observations: Algorithm and Theorycited by 0
  25. 2026SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modelingcited by 0
  26. 2026CCFM: Collision-Constrained Flow Matching for Safety-Critical Scenario Generationcited by 0
  27. 2026Constrained Flow Matching via Lagrangian Dual Flowscited by 0
  28. 2026Lantern: Conflict-Aware Gradient Blending for Physics-Guided Diffusion Models in Calorimeter Simulationcited by 0
  29. 2026Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fieldscited by 0
  30. 2026Self-Augmented Diffusion Guidance for Physics-Informed Generationcited by 0
  31. 2026Let It Go or Learn to Self-Correct: Continuous Diffusion for Constrained Discrete Taskscited by 0
  32. 2026Source Anchoring for Physical Consistency in Flow Matching Modelscited by 0
  33. 2026PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generationcited by 0
  34. 2026EP-Flow: Disordered Crystal Structure Prediction without Site-Level Annotationscited by 0
  35. 2026LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problemscited by 0
  36. 2026Xaurora: Generative Weather Forecasting with Denoising Stochastic Interpolants from a Foundation Model Priorcited by 0
  37. 2026Primal-Dual Flow Matching for Sample-Wise Constrained Generationcited by 0
  38. 2026When do explicit penalties improve flow matching?cited by 0

50 most cited of 52 All 52 (JSON)

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  4. 2026Physical Fidelity Reconstruction via Improved Consistency-Distilled Flow Matching for Dynamical Systems
  5. 2026PolyFlow: Safe and Efficient Polytope-Constrained Flow Matching with Constraint Embedding and Projection-free Update
  6. 2025Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation
  7. 2025Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling
  8. 2026From Euler to Dormand-Prince: ODE Solvers for Flow Matching Generative Models