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.

翻译：神经算子已成为学习偏微分方程（PDE）无限维参数空间与解空间映射的强大工具。本文聚焦于具有重要应用背景的多尺度PDE，如油藏模拟和湍流预测。我们证明，对于此类PDE，神经网络对低频分量的谱偏差给现有神经算子带来了显著挑战。为解决该问题，受层次矩阵方法启发，我们提出层次注意力神经算子（HANO）。HANO具有尺度自适应交互范围和层级上的自注意力机制，能够以可控线性成本实现嵌套特征计算，并对多尺度解空间进行编码/解码。我们还引入经验$H^1$损失函数以增强对高频分量的学习。数值实验表明，在代表性多尺度问题中，HANO的性能超越了现有最优（SOTA）方法。