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Compositional Generation for Long-Horizon Coupled PDEs

Somayajulu L. N. Dhulipala, Deep Ray, Nicholas Forman

TL;DR

This work tackles the high computational cost of simulating coupled PDEs by training diffusion surrogates on decoupled components and composing them at inference to recover coupled fields. The authors develop a PoE-based compositional diffusion framework with both $\varepsilon$-prediction and $v$-parameterization, and employ an Euler-inspired symmetric update to assemble the joint solution. Evaluations on long-horizon Reaction–Diffusion and modified Burgers systems show that the approach yields low coupling error despite decoupled training, with $v$-parameterization consistently improving accuracy over the baseline. While a Fourier Neural Operator trained on coupled data remains the strongest baseline, the results demonstrate that compositional diffusion is a viable, data-efficient strategy for scalable, long-horizon modeling of 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.

Compositional Generation for Long-Horizon Coupled PDEs

TL;DR

This work tackles the high computational cost of simulating coupled PDEs by training diffusion surrogates on decoupled components and composing them at inference to recover coupled fields. The authors develop a PoE-based compositional diffusion framework with both -prediction and -parameterization, and employ an Euler-inspired symmetric update to assemble the joint solution. Evaluations on long-horizon Reaction–Diffusion and modified Burgers systems show that the approach yields low coupling error despite decoupled training, with -parameterization consistently improving accuracy over the baseline. While a Fourier Neural Operator trained on coupled data remains the strongest baseline, the results demonstrate that compositional diffusion is a viable, data-efficient strategy for scalable, long-horizon modeling of 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.
Paper Structure (6 sections, 5 equations, 2 figures, 1 table)

This paper contains 6 sections, 5 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Methodology
  4. Results
  5. Conclusions and Future Work
  6. Acknowledgments

Figures (2)

  • Figure 1: Comparison of the compositional DDPM strategies (epsilon baseline and v-parameterization trainings) with the ground truth for predicting the coupled fields in the reaction-diffusion equation: (a) u-field and (b) v-field. Note that the two DDPM models for each test case were only trained with the de-coupled PDE data and were composed together using a PoE strategy to recovery the joint or coupled fields.
  • Figure 2: Comparison of the compositional DDPM strategies (epsilon baseline and v-parameterization trainings) with the ground truth for predicting the coupled fields in the modified Burgers equation: (a) c-field and (b) v-field. Note that the two DDPM models for each test case were only trained with the de-coupled PDE data and were composed together using a PoE strategy to recovery the joint or coupled fields.