# Learning Distributions of Complex Fluid Simulations with Diffusion Graph Networks

arXiv: https://arxiv.org/abs/2504.02843 · HTML: https://mlsanity.com/paper/2504.02843/

## Key facts

- Cited by 25 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 19 Mar 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)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Improved Techniques for Training Score-Based Generative Models](https://mlsanity.com/paper/2006.09011/) (2020)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [DiGress: Discrete Denoising diffusion for graph generation](https://mlsanity.com/paper/2209.14734/) (2022)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [FreeU: Free Lunch in Diffusion U-Net](https://mlsanity.com/paper/2309.11497/) (2023)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)

## Built on by (most cited first, 25 of 25)

- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Diffusion-Based Hierarchical Graph Neural Networks for Simulating Nonlinear Solid Mechanics](https://mlsanity.com/paper/2506.06045/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Reframing Generative Models for Physical Systems using Stochastic Interpolants](https://mlsanity.com/paper/2509.26282/) (2025)
- [Generative Urban Flow Modeling: From Geometry to Airflow with Graph Diffusion](https://mlsanity.com/paper/2512.14725/) (2025)
- [Flow Matching and Diffusion Models via PointNet for Generating Fluid Fields on Irregular Geometries](https://mlsanity.com/paper/2601.03030/) (2026)
- [Geometric Autoencoder Priors for Bayesian Inversion: Learn First Observe Later](https://mlsanity.com/paper/2509.19929/) (2025)
- [Softly Constrained Denoisers for Diffusion Models Applied to Partial Differential Equations](https://mlsanity.com/paper/2512.14980/) (2025)
- [GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains](https://mlsanity.com/paper/2605.03497/) (2026)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Deterministic and probabilistic neural surrogates of global hybrid-Vlasov simulations](https://mlsanity.com/paper/2601.12614/) (2026)
- [Stable Long-Horizon Spatiotemporal Prediction on Meshes Using Latent Multiscale Recurrent Graph Neural Networks](https://mlsanity.com/paper/2602.18146/) (2026)
- [Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision](https://mlsanity.com/paper/2603.04431/) (2026)
- [Signal-Aware Conditional Diffusion Surrogates for Transonic Wing Pressure Prediction](https://mlsanity.com/paper/2604.11263/) (2026)
- [One Scale at a Time: Scale-Autoregressive Modeling for Fluid Flow Distributions](https://mlsanity.com/paper/2604.11403/) (2026)
- [Inpainting physics: self-supervised learning for context-driven fluid simulation](https://mlsanity.com/paper/2605.08832/) (2026)
- [Diffusion Graph Posterior Sampling for Nonlinear Inverse Problems with Application to Electrical Impedance Tomography](https://mlsanity.com/paper/2605.19621/) (2026)
- [PCGD: Physics-Guided Conditional Graph Diffusion for TCAD Device Simulation](https://mlsanity.com/paper/2606.29272/) (2026)
- [A Shortcut to Statistically Steady-State Turbulence with Flow Matching](https://mlsanity.com/paper/2607.13022/) (2026)
- [Scale-Aware Learning of Chaotic Dynamics on Unstructured Meshes via Binned Spectral Losses](https://mlsanity.com/paper/2607.19387/) (2026)
- [Fluid-DiT: Graph-Free Diffusion Transformers for Fluid Flow Simulations Learning](https://mlsanity.com/paper/2608.07161/) (2026)
- [Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers](https://mlsanity.com/paper/2610.03363/) (2026)
- [Cross-Domain Pretraining for Steady-State Neural CFD Surrogates](https://mlsanity.com/paper/2610.10398/) (2026)

## Abstract (arXiv)

Physical systems with complex unsteady dynamics, such as fluid flows, are often poorly represented by a single mean solution. For many practical applications, it is crucial to access the full distribution of possible states, from which relevant statistics (e.g., RMS and two-point correlations) can be derived. Here, we propose a graph-based latent diffusion (or alternatively, flow-matching) model that enables direct sampling of states from their equilibrium distribution, given a mesh discretization of the system and its physical parameters. This allows for the efficient computation of flow statistics without running long and expensive numerical simulations. The graph-based structure enables operations on unstructured meshes, which is critical for representing complex geometries with spatially localized high gradients, while latent-space diffusion modeling with a multi-scale GNN allows for efficient learning and inference of entire distributions of solutions. A key finding is that the proposed networks can accurately learn full distributions even when trained on incomplete data from relatively short simulations. We apply this method to a range of fluid dynamics tasks, such as predicting pressure distributions on 3D wing models in turbulent flow, demonstrating both accuracy and computational efficiency in challenging scenarios. The ability to directly sample accurate solutions, and capturing their diversity from short ground-truth simulations, is highly promising for complex scientific modeling tasks.