Generative downscaling of PDE solvers with physics-guided diffusion models

TL;DR

This paper introduces a physics-guided diffusion model for efficient downscaling of PDE solutions, significantly accelerating high-fidelity computations while preserving accuracy across various nonlinear PDEs.

Contribution

The paper presents a novel physics-guided diffusion model that generates high-fidelity PDE solutions from low-fidelity inputs and refines them by minimizing physical discrepancies.

Findings

Outperforms baseline downscaling methods in accuracy.

Achieves over tenfold computational speedup.

Maintains high fidelity comparable to traditional solvers.

Abstract

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.