@inproceedings{
shysheya2024on,
title={On conditional diffusion models for {PDE} simulations},
author={Aliaksandra Shysheya and Cristiana Diaconu and Federico Bergamin and Paris Perdikaris and Jos{\'e} Miguel Hern{\'a}ndez-Lobato and Richard E. Turner and Emile Mathieu},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=nQl8EjyMzh}
}
Modelling partial differential equations (PDEs) is of crucial importance in science and engineering, and it includes tasks ranging from forecasting to inverse problems, such as data assimilation. However, most previous numerical and machine learning approaches that target forecasting cannot be applied out-of-the-box for data assimilation. Recently, diffusion models have emerged as a powerful tool for conditional generation, being able to flexibly incorporate observations without retraining. In this work, we perform a comparative study of score-based diffusion models for forecasting and assimilation of sparse observations.  In particular, we focus on diffusion models that are either trained in a conditional manner, or conditioned after unconditional training. We address the shortcomings of existing models by proposing 1) an autoregressive sampling approach, that significantly improves performance in forecasting, 2) a new training strategy for conditional score-based models that achieves stable performance over a range of history lengths, and 3) a hybrid model which employs flexible pre-training conditioning on initial conditions and flexible post-training conditioning to handle data assimilation. We empirically show that these modifications are crucial for successfully tackling the combination of forecasting and data assimilation, a task commonly encountered in real-world scenarios.
On conditional diffusion models for PDE simulations
Aliaksandra Shysheya Cristiana Diaconu Federico Bergamin Paris Perdikaris José Miguel Hernández-Lobato Richard E. Turner Emile Mathieu