基于深度学习的多尺度问题LOD方法替代建模
Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems
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中文总结 AI 辅助
针对多尺度问题传统方法的局限，研究神经算子架构性能与局限，引入LOD-MSNO混合方法，结合LOD方法与数据驱动算子学习，克服计算瓶颈，给出理论误差估计，在多尺度输入精度上优于基线且保持计算效率。
中文摘要 AI 辅助
多尺度问题用传统数值方法难以解决，精确解析精细尺度特征需极精细离散化，在材料科学等应用中尤为突出。近期神经算子模型有潜力但在强异质或振荡系数时精度不足。本文聚焦含粗糙和高对比度输入的椭圆型偏微分方程，LOD方法虽成熟但计算成本高。研究了神经算子架构在多尺度问题中的性能及局限，引入LOD-MSNO混合方法，利用LOD作为多尺度先验并解决其计算瓶颈，给出理论误差估计，证明该方法在多尺度输入精度上优于现有神经算子基线且保持计算效率。
英文摘要
Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations. This challenge is particularly pronounced in applications such as materials science, fluid dynamics, climate systems, chemical processes, and complex networks. Recent neural operator models provide a promising data-driven alternative, but frequently struggle to achieve sufficient accuracy in the presence of strongly heterogeneous or oscillatory coefficients. In this work, we focus on the solution of elliptic PDEs with rough and high-contrast inputs. The Localized Orthogonal Decomposition (LOD) method is a well-established numerical approach for such problems, but it comes, however, at a substantial computational cost. We investigate the performance of popular neural operator architectures on these challenging multiscale problems and identify key limitations in their ability to resolve fine-scale structure. To overcome these challenges, we introduce LOD-MSNO (LOD-Multiscale Neural Operator), a hybrid approach that leverages the LOD method as a strong multiscale prior by building on its representation of the solution as a linear combination of problem-adapted basis functions, while addressing its main computational bottlenecks through data-driven operator learning. We further provide theoretical error estimates for the proposed coefficient-learning framework. Lastly, we demonstrate the potential of our proposed method to outperform current neural operator baselines in terms of accuracy for challenging multiscale inputs, while mainly retaining the computational efficiency of neural operator models.
