多阶段神经算子学习及其在卷积中的应用

中文摘要 AI 辅助

卷积积分广泛存在于各类应用中，为实现快速且准确的计算，本文提出两种通用的多阶段神经算子学习框架。第一种是深度配点神经算子（Deep Collocation Neural Operator, DCNO），属于监督学习方法，通过从输入-输出数据对中学习残差来迭代优化算子近似；第二种是深度伽辽金神经算子（Deep Galerkin Neural Operator, DGNO），属于无监督框架，适用于目标算子可由偏微分方程（PDE）表示的场景，训练时利用PDE残差的弱形式。两种方法均通过多个训练阶段逐步构建基算子以丰富近似空间，相比标准的单步算子学习，精度显著提升。本文对两种方法的近似能力进行了理论分析，并将其应用于卷积学习；大量数值实验表明，DCNO与DGNO在卷积问题上均达到高准确率，单精度浮点数下接近机器精度，且与传统求解器相比，在处理大量查询或参数变化时效率大幅提升。本文还将这些框架扩展至处理涉及卷积密度与核变化的多输入算子学习场景。

英文摘要

Convolution integrals widely exist in applications, and to enable fast and accurate computations, this paper introduces two general multi-stage neural operator learning frameworks. The first, Deep Collocation Neural Operator (DCNO), is a supervised approach that iteratively refines the operator approximation by learning residuals from input-output data pairs. The second, Deep Galerkin Neural Operator (DGNO), is an unsupervised framework applicable when the target operator can be represented by a PDE, leveraging the weak form of the PDE residual for training. Both methods progressively construct basis operators through multiple training stages to enrich the approximation space, leading to significantly improved accuracy over standard one-shot operator learning. We provide theoretical analysis for their approximation capabilities and implement them for learning convolutions. Extensive numerical experiments demonstrate that both DCNO and DGNO achieve high accuracy, approaching machine precision under single float for convolution problems, and offer substantial efficiency gains for numerous queries or parametric variations compared to traditional solvers. We also extend these frameworks to handle multi-input operator learning scenarios involving variations in both the density and kernel of a convolution.

发表机构

• Eastern Institute of Technology（东方理工学院）
• Wuhan University（武汉大学）
• Tianjin University（天津大学）

机构由 AI 辅助整理，请以论文原文为准。