Climate downscaling is crucial for detailed small- scale analysis and for acquiring climate data in regions without weather stations. Operator learning has proven potential for this task. However, several challenges remain in operator learning, such as multimodal fusion, spatiotemporal fusion and input state and query adaptation. To address these challenges, we propose a Spatiotemporal Multimodal Fusion Operator with a State- Query Coupled Kernel (SMCK). This framework includes a latent space fusion encoder that encodes climate variables using position-wise multihead attention for multimodal fusion and integrates historical information to generate robust and precise representation. Additionally, we introduce a state-query coupled kernel that combines radial basis functions and discrete fourier encoding to enhance query location representation, while also adapting to the state to obtain the coupled kernel. Extensive experiments demonstrate that our method achieves state-of-the-art performance and provides strong support for climate downscaling and the planning of climate-related strategies.

The deep operator networks (DON), a class of neural operators that learn mappings between function spaces, have recently emerged as surrogate models for parametric partial differential equations (PDEs). However, their full potential for accurately approximating general black-box PDEs remains underexplored due to challenges in training stability and performance, primarily arising from difficulties in learning mappings between low-dimensional inputs and high-dimensional outputs. Furthermore, inadequate encoding of input functions and query positions limits the generalization ability of DONs. To address these challenges, we propose the Dynamical Coupled Operator (DCO), which incorporates temporal dynamics to learn coupled functions, reducing information loss and improving training robustness. Additionally, we introduce an adaptive spectral input function encoder based on empirical mode decomposition to enhance input function representation, as well as a hybrid location encoder to improve query location encoding. We provide theoretical guarantees on the universal expressiveness of DCO, ensuring its applicability to a wide range of PDE problems. Extensive experiments on real-world, high-dimensional PDE datasets demonstrate that DCO significantly outperforms DONs.