Mitigating spectral bias for the multiscale operator learning
This work solves the spectral bias issue in neural operators for multiscale PDEs, benefiting applications like reservoir modeling and turbulence prediction, though it is an incremental improvement over existing methods.
The authors tackled the spectral bias problem in neural operators for multiscale PDEs, which causes poor learning of high-frequency components, and proposed HANO with hierarchical attention and an H¹ loss to address this, achieving state-of-the-art performance in numerical experiments.
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.