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

Traditional numerical methods like the Finite Element Method (FEM) are computationally expensive when simulating path-dependent physical fields, such as plastic deformation in complex geometries. This paper addresses the need for a faster, generative approach capable of modeling these fields across entire loading-unloading sequences without the high computational overhead of traditional solvers or the slow, multi-step sampling typical of diffusion models.

The researchers propose a flow matching (FM) framework that treats stress field simulation as a video synthesis task. Key innovations include:

• Geometry-Informed Source Distribution: Instead of using standard Gaussian noise, the model uses an empirical distribution derived from the physical geometry (e.g., hole locations in a plate). This significantly reduces the intersection of conditional transport paths during training.
• Spatiotemporal DiT Backbone: A Diffusion Transformer (DiT) architecture is adapted for flow matching to capture long-range spatiotemporal dependencies in stress fields.
• Auxiliary Networks: A scientific Variational Autoencoder (VAE) is used for dimensionality reduction, and a ViT-Transformer handles token-level loading embeddings to inject physical context into the generation process.

By aligning the source distribution with the physical geometry, the model achieves nearly straight conditional transport paths. This allows for high-quality, one-step generation, bypassing the need for iterative refinement or distillation. The model demonstrates significant computational efficiency, providing a 6-7x speedup over FEM on CPUs and approximately two orders of magnitude speedup on consumer-grade GPUs, even when trained on limited datasets.