Climate change presents challenges to crop productivity, such as increasing the likelihood of heat stress and drought. Solar-Induced Chlorophyll Fluorescence (SIF) is a powerful way to monitor how crop productivity and photosynthesis are affected by changing climatic conditions. However, satellite SIF observations are only available at a coarse spatial resolution (e.g. 3-5km) in most places, making it difficult to determine how individual crop types or farms are doing. This poses a challenging coarsely-supervised regression task; at training time, we only have access to SIF labels at a coarse resolution (3 km), yet we want to predict SIF at a very fine spatial resolution (30 meters), a 100x increase. We do have some fine-resolution input features (such as Landsat reflectance) that are correlated with SIF, but the nature of the correlation is unknown. To address this, we propose Coarsely-Supervised Regression U-Net (CSR-U-Net), a novel approach to train a U-Net for this coarse supervision setting. CSR-U-Net takes in a fine-resolution input image, and outputs a SIF prediction for each pixel; the average of the pixel predictions is trained to equal the true coarse-resolution SIF for the entire image. Even though this is a very weak form of supervision, CSR-U-Net can still learn to predict accurately, due to its inherent localization abilities, plus additional enhancements that facilitate the incorporation of scientific prior knowledge. CSR-U-Net can resolve fine-grained variations in SIF more accurately than existing averaging-based approaches, which ignore fine-resolution spatial variation during training. CSR-U-Net could also be useful for a wide range of "downscaling'" problems in climate science, such as increasing the resolution of global climate models.