Generative downscaling of PDE solvers with physics-guided diffusion models

TL;DR

The work reframes PDE downscaling as conditional sampling from the posterior $p(\boldsymbol{u}^f|\boldsymbol{u}^c)$ and introduces a two-stage physics-guided diffusion model (PGDM) that first uses a conditional diffusion generator to produce high-fidelity samples from a low-fidelity input and then refines them via a light physics-based optimization. It builds on DDPM and DDIM foundations, with a dedicated conditional diffusion model and a Gauss-Newton refinement to enforce the finer-scale PDE residuals, incorporating the source term when available. The approach is validated on nonlinear Poisson, Allen-Cahn, and Navier–Stokes equations, showing that PGDM achieves accuracy on par with fine-scale solvers while delivering order-of-magnitude speedups and data efficiency (as few as ~30 training samples). The results suggest that combining data-driven conditional generation with physics-based tightening yields practical, scalable PDE downscaling suitable for large-scale simulations and climate/modeling tasks.

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.

Figures (10)

• Figure 1: 2D nonlinear Poisson: Predictions and absolute errors generated by different solvers with $c=1.6$ and $N=100$ training samples.
• Figure 2: 2D nonlinear Poisson: Predictions and corresponding absolute errors generated by different solvers with $c=1.2$ and $N=100$ training samples.
• Figure 3: 3D nonlinear Poisson: Predictions and corresponding absolute errors generated by different solvers with $c=1.6$ and $N=100$ training samples.
• Figure 4: 3D nonlinear Poisson: Predictions and corresponding absolute errors generated by different solvers with $c=1.4$ and $N=100$ training samples.
• Figure 5: 2D Allen-Cahn: Predictions at $t=0.5$ and corresponding absolute errors generated by different solvers with $\gamma=5$, $c=1.6$ and $N=30$.