  Physics-informed deep learning has been developed as a novel paradigm for
learning physical dynamics recently. While general physics-informed deep
learning methods have shown early promise in learning fluid dynamics, they are
difficult to generalize in arbitrary time instants in real-world scenario,
where the fluid motion can be considered as a time-variant trajectory involved
large-scale particles. Inspired by the advantage of diffusion model in learning
the distribution of data, we first propose Pi-fusion, a physics-informed
diffusion model for predicting the temporal evolution of velocity and pressure
field in fluid dynamics. Physics-informed guidance sampling is proposed in the
inference procedure of Pi-fusion to improve the accuracy and interpretability
of learning fluid dynamics. Furthermore, we introduce a training strategy based
on reciprocal learning to learn the quasiperiodical pattern of fluid motion and
thus improve the generalizability of the model. The proposed approach are then
evaluated on both synthetic and real-world dataset, by comparing it with
state-of-the-art physics-informed deep learning methods. Experimental results
show that the proposed approach significantly outperforms existing methods for
predicting temporal evolution of velocity and pressure field, confirming its
strong generalization by drawing probabilistic inference of forward process and
physics-informed guidance sampling. The proposed Pi-fusion can also be
generalized in learning other physical dynamics governed by partial
differential equations.
