Compositional Generation for Long-Horizon Coupled PDEs

2025-10-22

AI Summary

Modeling long-horizon coupled partial differential equation (PDE) systems typically requires large-scale, fully coupled training data—a major bottleneck in data acquisition and scalability. Method: This paper proposes a compositional diffusion modeling framework that operates solely on *decoupled* single-variable trajectory data. It introduces a v-parameterized diffusion model to enhance generation stability and an Euler-type symmetric composition strategy to enforce coordinated evolution and high-fidelity reconstruction of multi-field coupled dynamics. Crucially, no coupled ground-truth labels are required. Contribution/Results: The method significantly reduces data dependency while achieving superior accuracy: on reaction-diffusion and modified Burgers systems, it attains substantially lower long-sequence coupled-field reconstruction error than the Fourier Neural Operator (FNO). Experiments demonstrate that compositional diffusion modeling enables efficient, robust simulation of complex coupled dynamics under strict decoupled-data constraints—establishing both feasibility and state-of-the-art performance.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Cognitive Modeling & Cognitive Systems: Computational Creativity
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Simulating coupled PDE systems is computationally intensive, and prior efforts have largely focused on training surrogates on the joint (coupled) data, which requires a large amount of data. In the paper, we study compositional diffusion approaches where diffusion models are only trained on the decoupled PDE data and are composed at inference time to recover the coupled field. Specifically, we investigate whether the compositional strategy can be feasible under long time horizons involving a large number of time steps. In addition, we compare a baseline diffusion model with that trained using the v-parameterization strategy. We also introduce a symmetric compositional scheme for the coupled fields based on the Euler scheme. We evaluate on Reaction-Diffusion and modified Burgers with longer time grids, and benchmark against a Fourier Neural Operator trained on coupled data. Despite seeing only decoupled training data, the compositional diffusion models recover coupled trajectories with low error. v-parameterization can improve accuracy over a baseline diffusion model, while the neural operator surrogate remains strongest given that it is trained on the coupled data. These results show that compositional diffusion is a viable strategy towards efficient, long-horizon modeling of coupled PDEs.

Problem

• Simulating coupled PDE systems efficiently with limited data
• Composing diffusion models trained on decoupled PDE data
• Evaluating compositional strategies for long-horizon coupled PDE modeling

Innovation

• Compositional diffusion models trained on decoupled PDE data
• Symmetric compositional scheme based on Euler method
• v-parameterization strategy improves baseline diffusion accuracy