Compositional Generation for Long-Horizon Coupled PDEs

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.