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.

翻译：在精细时空尺度上求解偏微分方程以获取高保真解，对众多科学突破至关重要。然而，由于问题固有复杂性（包括非线性和多尺度现象），这一过程可能代价高昂。为加速大规模计算，采用降尺度技术，即从低保真近似解生成高保真近似解。本文提出一种新颖的物理引导扩散模型（PGDM）用于降尺度。该模型首先在跨粗-细尺度的低-高保真配对解数据集上训练，随后可从任意新的低保真输入生成高保真近似解。这些输出通过微调进一步优化，旨在最小化由细尺度离散偏微分方程定义的物理偏差。我们在三类非线性偏微分方程上评估并对比模型与其他降尺度基准方法的性能。数值实验表明，该模型不仅优于基准方法，还能在保持与传统细尺度求解器相同精度水平的同时，实现超过十倍的加速。