  This paper explores the efficacy of diffusion-based generative models as
neural operators for partial differential equations (PDEs). Neural operators
are neural networks that learn a mapping from the parameter space to the
solution space of PDEs from data, and they can also solve the inverse problem
of estimating the parameter from the solution. Diffusion models excel in many
domains, but their potential as neural operators has not been thoroughly
explored. In this work, we show that diffusion-based generative models exhibit
many properties favourable for neural operators, and they can effectively
generate the solution of a PDE conditionally on the parameter or recover the
unobserved parts of the system. We propose to train a single model adaptable to
multiple tasks, by alternating between the tasks during training. In our
experiments with multiple realistic dynamical systems, diffusion models
outperform other neural operators. Furthermore, we demonstrate how the
probabilistic diffusion model can elegantly deal with systems which are only
partially identifiable, by producing samples corresponding to the different
possible solutions.
