Compositional Generation for Long-Horizon Coupled PDEs

TL;DR

This paper explores a compositional diffusion approach that trains on decoupled PDE data and composes at inference to simulate coupled PDE systems over long time horizons, reducing data requirements.

Contribution

It introduces a novel compositional diffusion method for coupled PDEs trained on decoupled data, enabling efficient long-horizon simulations.

Findings

Compositional diffusion models recover coupled trajectories with low error.

V-parameterization improves diffusion model accuracy.

Neural operator remains strongest when trained on coupled data.

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.