  We develop a class of data-driven generative models that approximate the
solution operator for parameter-dependent partial differential equations (PDE).
We propose a novel probabilistic formulation of the operator learning problem
based on recently developed generative denoising diffusion probabilistic models
(DDPM) in order to learn the input-to-output mapping between problem parameters
and solutions of the PDE. To achieve this goal we modify DDPM to supervised
learning in which the solution operator for the PDE is represented by a class
of conditional distributions. The probabilistic formulation combined with DDPM
allows for an automatic quantification of confidence intervals for the learned
solutions. Furthermore, the framework is directly applicable for learning from
a noisy data set. We compare computational performance of the developed method
with the Fourier Network Operators (FNO). Our results show that our method
achieves comparable accuracy and recovers the noise magnitude when applied to
data sets with outputs corrupted by additive noise.
