# Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints

arXiv: https://arxiv.org/abs/2506.08604 · HTML: https://mlsanity.com/paper/2506.08604/

## Key facts

- Cited by 28 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 44 of its 68 references that are in the map.
- First posted on arXiv 10 Jun 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 44)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Video Diffusion Models](https://mlsanity.com/paper/2204.03458/) (2022)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Structure-based Drug Design with Equivariant Diffusion Models](https://mlsanity.com/paper/2210.13695/) (2022)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [Generator Matching: Generative modeling with arbitrary Markov processes](https://mlsanity.com/paper/2410.20587/) (2024)
- [Deep Dynamical Modeling and Control of Unsteady Fluid Flows](https://mlsanity.com/paper/1805.07472/) (2018)

## Built on by (most cited first, 28 of 28)

- [CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators](https://mlsanity.com/paper/2512.05297/) (2025)
- [Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation](https://mlsanity.com/paper/2509.25157/) (2025)
- [Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria](https://mlsanity.com/paper/2508.07102/) (2025)
- [Enforcing governing equation constraints in neural PDE solvers via training-free projections](https://mlsanity.com/paper/2511.17258/) (2025)
- [Physics-Constrained Adaptive Flow Matching for Climate Downscaling](https://mlsanity.com/paper/2604.03459/) (2026)
- [Reframing Generative Models for Physical Systems using Stochastic Interpolants](https://mlsanity.com/paper/2509.26282/) (2025)
- [ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance](https://mlsanity.com/paper/2601.20227/) (2026)
- [Ultra Fast PDE Solving via Physics Guided Few-step Diffusion](https://mlsanity.com/paper/2602.03627/) (2026)
- [Stabilizing Physics-Informed Consistency Models via Structure-Preserving Training](https://mlsanity.com/paper/2602.09303/) (2026)
- [Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow](https://mlsanity.com/paper/2605.08915/) (2026)
- [Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions](https://mlsanity.com/paper/2605.16118/) (2026)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [Gradient-Update Mismatch: Rethinking Conflict-Free Training of Physics-Informed Neural Networks](https://mlsanity.com/paper/2609.01558/) (2026)
- [On the flow matching interpretability](https://mlsanity.com/paper/2510.21210/) (2025)
- [ScatterPrism: convergence for generative simulation and inverse problems in particle and nuclear physics](https://mlsanity.com/paper/2604.01313/) (2026)
- [Physics-Based Flow Matching for Full-Field Prediction of Silicon Photonic Devices](https://mlsanity.com/paper/2605.06929/) (2026)
- [(HB-ARFM) History-Bootstrapped Flow Matching for Inverse Boiling Reconstruction](https://mlsanity.com/paper/2606.00349/) (2026)
- [Drifting Models for Surrogate Flow Modeling](https://mlsanity.com/paper/2606.07481/) (2026)
- [PCGD: Physics-Guided Conditional Graph Diffusion for TCAD Device Simulation](https://mlsanity.com/paper/2606.29272/) (2026)
- [Constrained Flow Matching via Lagrangian Dual Flows](https://mlsanity.com/paper/2607.04513/) (2026)
- [Multilevel Graph Wavelet Compressed Sensing with Scale-Aware Neural Recovery](https://mlsanity.com/paper/2607.20857/) (2026)
- [Fourier-Latent Diffusion for Constrained Generation of Triply Periodic Minimal Surfaces](https://mlsanity.com/paper/2608.02151/) (2026)
- [Generative Diffusion Surrogates with Analytical Variance Schedule](https://mlsanity.com/paper/2609.01705/) (2026)
- [Source Anchoring for Physical Consistency in Flow Matching Models](https://mlsanity.com/paper/2609.33510/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)
- [PMosFM: Preconditioned Manifold Matching for One-Step Physics-Constrained Generation](https://mlsanity.com/paper/2609.40287/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)
- [Characterizing and Mitigating Distributional Bias in Physics-Informed Generative Models](https://mlsanity.com/paper/2610.16582/) (2026)

## Abstract (arXiv)

Physics-constrained generative modeling aims to produce high-dimensional samples that are both physically consistent and distributionally accurate, a task that remains challenging due to often conflicting optimization objectives. Recent advances in flow matching and diffusion models have enabled efficient generative modeling, but integrating physical constraints often degrades generative fidelity or requires costly inference-time corrections. Our work is the first to recognize the trade-off between distributional and physical accuracy. Based on the insight of inherently conflicting objectives, we introduce Physics-Based Flow Matching (PBFM) a method that enforces physical constraints at training time using conflict-free gradient updates and unrolling to mitigate Jensen's gap. Our approach avoids manual loss balancing and enables simultaneous optimization of generative and physical objectives. As a consequence, physics constraints do not impede inference performance. We benchmark our method across three representative PDE benchmarks. PBFM achieves a Pareto-optimal trade-off, competitive inference speed, and generalizes to a wide range of physics-constrained generative tasks, providing a practical tool for scientific machine learning. Code and datasets available at https://github.com/tum-pbs/PBFM.