  Neural operators have emerged as a powerful tool for learning the mapping
between infinite-dimensional parameter and solution spaces of partial
differential equations (PDEs). In this work, we focus on multiscale PDEs that
have important applications such as reservoir modeling and turbulence
prediction. We demonstrate that for such PDEs, the spectral bias towards
low-frequency components presents a significant challenge for existing neural
operators. To address this challenge, we propose a hierarchical attention
neural operator (HANO) inspired by the hierarchical matrix approach. HANO
features a scale-adaptive interaction range and self-attentions over a
hierarchy of levels, enabling nested feature computation with controllable
linear cost and encoding/decoding of multiscale solution space. We also
incorporate an empirical $H^1$ loss function to enhance the learning of
high-frequency components. Our numerical experiments demonstrate that HANO
outperforms state-of-the-art (SOTA) methods for representative multiscale
problems.
