  In this work, we present an arbitrary-scale super-resolution (SR) method to
enhance the resolution of scientific data, which often involves complex
challenges such as continuity, multi-scale physics, and the intricacies of
high-frequency signals. Grounded in operator learning, the proposed method is
resolution-invariant. The core of our model is a hierarchical neural operator
that leverages a Galerkin-type self-attention mechanism, enabling efficient
learning of mappings between function spaces. Sinc filters are used to
facilitate the information transfer across different levels in the hierarchy,
thereby ensuring representation equivalence in the proposed neural operator.
Additionally, we introduce a learnable prior structure that is derived from the
spectral resizing of the input data. This loss prior is model-agnostic and is
designed to dynamically adjust the weighting of pixel contributions, thereby
balancing gradients effectively across the model. We conduct extensive
experiments on diverse datasets from different domains and demonstrate
consistent improvements compared to strong baselines, which consist of various
state-of-the-art SR methods.
