# One-Step Flow Matching for Generative Modeling of Path-Dependent Physical Fields

arXiv: https://arxiv.org/abs/2606.22752 · HTML: https://mlsanity.com/paper/2606.22752/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 24 of its 81 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## 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, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Stress field prediction in fiber-reinforced composite materials using a deep learning approach](https://mlsanity.com/paper/2111.05271/) (2021)
- [Eighty Years of the Finite Element Method: Birth, Evolution, and Future](https://mlsanity.com/paper/2107.04960/) (2021)
- [Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization](https://mlsanity.com/paper/2002.07600/) (2020)
- [Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence](https://mlsanity.com/paper/2108.02837/) (2021)
- [Physically recurrent neural networks for path-dependent heterogeneous materials: embedding constitutive models in a data-driven surrogate](https://mlsanity.com/paper/2209.07320/) (2022)
- [PH-Net: Parallelepiped Microstructure Homogenization via 3D Convolutional Neural Networks](https://mlsanity.com/paper/2201.09672/) (2022)
- [A single Long Short-Term Memory network for enhancing the prediction of path-dependent plasticity with material heterogeneity and anisotropy](https://mlsanity.com/paper/2204.01466/) (2022)
- [A micromechanics-based recurrent neural networks model for path-dependent cyclic deformation of short fiber composites](https://mlsanity.com/paper/2210.00842/) (2022)
- [Physically recurrent neural network for rate and path-dependent heterogeneous materials in a finite strain framework](https://mlsanity.com/paper/2404.17583/) (2024)
- [Heterogeneous Graph Representation of Stiffened Panels with Non-Uniform Boundary Conditions and Loads](https://mlsanity.com/paper/2510.02472/) (2025)
- [A hybrid global local computational framework for ship hull structural analysis using homogenized model and graph neural network](https://mlsanity.com/paper/2512.20020/) (2025)
- [Flow Matching and Diffusion Models via PointNet for Generating Fluid Fields on Irregular Geometries](https://mlsanity.com/paper/2601.03030/) (2026)
- [ViT-Transformer: Self-attention mechanism based constitutive modeling for nonlinear heterogeneous materials](https://mlsanity.com/paper/2510.16575/) (2025)
- [A Hybrid Conditional Diffusion-DeepONet Framework for High-Fidelity Stress Prediction in Hyperelastic Materials](https://mlsanity.com/paper/2603.18225/) (2026)

## Abstract (arXiv)

Physical simulations for intricate geometries with path-dependent constitutive models face difficulties due to the enormous computational cost they require. Recently, the emergence of generative AI models, which succeed in image and video synthesis tasks, has provided a promise to further improve simulations. Although U-Net-based denoising diffusion probabilistic models (DDPMs) have been adopted for elastic stress field generation, they typically require hundreds of sampling steps, and applications of generative models to path-dependent, e.g. plastic, stress fields remain very limited. In this work, we propose a novel flow matching (FM) model based on a transformer backbone for high-resolution path-dependent stress field generation with stochastic loading-unloading paths and geometry. The proposed model operates within the latent space of a variational autoencoder (VAE) and formulates the simulation of plastic fields as a video synthesis task, directly generating the stress fields across all time steps. Meanwhile, we design a non-Gaussian source distribution for flow matching, such that crossings among conditional transport paths are reduced during training. This enables our model to generate satisfactory samples in one step without relying on distillation. In addition, we introduce token-level loading embeddings and two auxiliary networks to further enhance the model performance in path-dependent simulation. The results demonstrate that, even with a limited training dataset, our model can accurately generate high-resolution path-dependent fields. It is much more computationally efficient than finite element analysis, providing a speedup of 6 to 7 times over FEM on CPUs and approximately two orders of magnitude speedup on consumer-grade GPUs.