Simulation of spatiotemporal systems governed by partial differential equations is widely applied in fields such as biology, chemistry, aerospace dynamics, and meteorology. Traditional numerical methods incur high computational costs due to the requirement of small time steps for accurate predictions. While machine learning has reduced these costs, long-term predictions remain challenged by error accumulation, particularly in scenarios with insufficient data or varying time scales, where stability and accuracy are compromised. Existing methods often neglect the effective utilization of multi-scale data, leading to suboptimal robustness in predictions. To address these issues, we propose a novel multi-scale learning framework, namely, the Physics-Informed Multi-Scale Recurrent Learning (PIMRL), to effectively leverage multi-scale data for spatiotemporal dynamics prediction. The PIMRL framework comprises two modules: the micro-scale module embeds physical knowledge into neural networks via pretraining, and the macro-scale module adopts a data-driven approach to learn the temporal evolution of physics in the latent space. Experimental results demonstrate that the PIMRL framework consistently achieves state-of-the-art performance across five benchmark datasets ranging from one to three dimensions, showing average improvements of over 9\% in both RMSE and MAE evaluation metrics, with maximum enhancements reaching up to 80%.

该研究针对传统数值方法在高计算成本和长期预测中存在的误差积累问题，提出了一种新颖的多尺度学习框架PIMRL。该框架通过微尺度模块嵌入物理知识和宏尺度模块进行数据驱动学习，有效利用多尺度数据进行时空动态预测。实验结果表明，PIMRL在多个基准数据集上显著提高了预测准确性，平均改进超过9%。

物理启发的多尺度递归学习用于时空预测