论文标题
减轻多尺度操作员学习的光谱偏差
Mitigating spectral bias for the multiscale operator learning
论文作者
论文摘要
神经操作员已成为一种有力的工具，用于学习偏微分方程（PDES）的无限二维参数和解决方案空间之间的映射。在这项工作中，我们专注于具有重要应用的多尺度PDE，例如储层建模和湍流预测。我们证明，对于此类PDE，对低频组件的频谱偏差给现有的神经操作员带来了重大挑战。为了应对这一挑战，我们提出了一个受层次矩阵方法启发的层次关注神经操作员（Hano）。 Hano在级别的层次结构上具有量表自适应的互动范围和自我附件，从而实现了可控的线性成本和多尺度解决方案空间的编码/解码的嵌套特征计算。我们还结合了经验$ H^1 $损失功能，以增强高频组件的学习。我们的数值实验表明，对于代表性的多尺度问题，河内的表现优于最先进的方法（SOTA）方法。
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.
