Multi-stage neural operator learning with application for convolutions

2026-08-19

AI Summary

本文提出两种多阶段神经算子学习框架DCNO和DGNO，以解决卷积积分的快速准确计算问题，通过迭代优化算子近似提高精度。

Abstract

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.

Problem

• convolution integrals
• fast and accurate computations
• multi-stage neural operator learning

Innovation

• multi-stage neural operator learning
• convolution integrals
• Deep Collocation Neural Operator
• Deep Galerkin Neural Operator
• unsupervised framework