  Solving partial differential equations (PDEs) on fine spatio-temporal scales
for high-fidelity solutions is critical for numerous scientific breakthroughs.
Yet, this process can be prohibitively expensive, owing to the inherent
complexities of the problems, including nonlinearity and multiscale phenomena.
To speed up large-scale computations, a process known as downscaling is
employed, which generates high-fidelity approximate solutions from their
low-fidelity counterparts. In this paper, we propose a novel Physics-Guided
Diffusion Model (PGDM) for downscaling. Our model, initially trained on a
dataset comprising low-and-high-fidelity paired solutions across coarse and
fine scales, generates new high-fidelity approximations from any new
low-fidelity inputs. These outputs are subsequently refined through
fine-tuning, aimed at minimizing the physical discrepancies as defined by the
discretized PDEs at the finer scale. We evaluate and benchmark our model's
performance against other downscaling baselines in three categories of
nonlinear PDEs. Our numerical experiments demonstrate that our model not only
outperforms the baselines but also achieves a computational acceleration
exceeding tenfold, while maintaining the same level of accuracy as the
conventional fine-scale solvers.
