# Flow marching for a generative PDE foundation model

arXiv: https://arxiv.org/abs/2509.18611 · HTML: https://mlsanity.com/paper/2509.18611/

## Key facts

- Cited by 4 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 38 of its 53 references that are in the map.
- First posted on arXiv 23 Sep 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 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FlashAttention-2: Faster Attention with Better Parallelism and Work Partitioning](https://mlsanity.com/paper/2307.08691/) (2023)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion](https://mlsanity.com/paper/2407.01392/) (2024)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [Self Forcing: Bridging the Train-Test Gap in Autoregressive Video Diffusion](https://mlsanity.com/paper/2506.08009/) (2025)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [REPA-E: Unlocking VAE for End-to-End Tuning with Latent Diffusion Transformers](https://mlsanity.com/paper/2504.10483/) (2025)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [Why Exposure Bias Matters: An Imitation Learning Perspective of Error Accumulation in Language Generation](https://mlsanity.com/paper/2204.01171/) (2022)
- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Ca2-VDM: Efficient Autoregressive Video Diffusion Model with Causal Generation and Cache Sharing](https://mlsanity.com/paper/2411.16375/) (2024)
- [Progressive Autoregressive Video Diffusion Models](https://mlsanity.com/paper/2410.08151/) (2024)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)

## Built on by (most cited first, 4 of 4)

- [Advances in GRPO for Generation Models: A Survey](https://mlsanity.com/paper/2603.06623/) (2026)
- [Neural operator discovery from heterogeneous trajectories](https://mlsanity.com/paper/2607.23337/) (2026)
- [On Approaches to Building Surrogate ODE Models for Diffusion Bridges](https://mlsanity.com/paper/2512.12671/) (2025)
- [Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields](https://mlsanity.com/paper/2608.18004/) (2026)

## Abstract (arXiv)

Pretraining on large-scale collections of PDE-governed spatiotemporal trajectories has recently shown promise for building generalizable models of dynamical systems. Yet most existing PDE foundation models rely on deterministic Transformer architectures, which lack generative flexibility for many science and engineering applications. We propose Flow Marching, an algorithm that bridges neural operator learning with flow matching motivated by an analysis of error accumulation in physical dynamical systems, and we build a generative PDE foundation model on top of it. By jointly sampling the noise level and the physical time step between adjacent states, the model learns a unified velocity field that transports a noisy current state toward its clean successor, reducing long-term rollout drift while enabling uncertainty-aware ensemble generations. Alongside this core algorithm, we introduce a Physics-Pretrained Variational Autoencoder (P2VAE) to embed physical states into a compact latent space, and an efficient Flow Marching Transformer (FMT) that combines a diffusion-forcing scheme with latent temporal pyramids, achieving up to 15x greater computational efficiency than full-length video diffusion models and thereby enabling large-scale pretraining at substantially reduced cost. We curate a corpus of ~2.5M trajectories across 12 distinct PDE families and train suites of P2VAEs and FMTs at multiple scales. On downstream evaluation, we benchmark on unseen Kolmogorov turbulence with few-shot adaptation, demonstrate long-term rollout stability over deterministic counterparts, and present uncertainty-stratified ensemble results, highlighting the importance of generative PDE foundation models for real-world applications.