Deep learning has shown strong potential in modeling complex spatiotemporal dynamics. However, most existing methods depend on densely and uniformly sampled data, which is often unavailable in practice due to sensor and cost limitations. In many real-world settings, such as mobile sensing and physical experiments, data are burst-sampled with short high-frequency segments followed by long gaps, making it difficult to learn accurate dynamics from sparse observations. To address this issue, we propose Physics-Informed Multi-Scale Recurrent Learning (PIMRL), a novel framework specifically designed for burst-sampled spatiotemporal data. PIMRL combines macro-scale latent dynamics inference with micro-scale adaptive refinement guided by incomplete prior information from partial differential equations (PDEs). It further introduces a temporal message-passing mechanism to effectively propagate information across burst intervals. This multi-scale architecture enables PIMRL to model complex systems accurately even under severe data scarcity. We evaluate our approach on five benchmark datasets involving 1D to 3D multi-scale PDEs. The results show that PIMRL consistently outperforms state-of-the-art baselines, achieving substantial improvements and reducing errors by up to 80\% in the most challenging settings, which demonstrates the clear advantage of our model. Our work demonstrates the effectiveness of physics-informed recurrent learning for accurate and efficient modeling of sparse spatiotemporal systems.