MgFNO: 面向参数化偏微分方程的多网格架构傅里叶神经算子
MgFNO: Multi-grid Architecture Fourier Neural Operator for Parametric Partial Differential Equations
AI总结：
本研究提出三级层次架构的多网格傅里叶神经算子MgFNO，通过解耦训练策略结合神经网络与多网格方法的互补特性，在三类PDE上大幅降低相对误差，还支持零样本超分辨率预测，为复杂PDE求解提供高效高精度新范式。
摘要：
神经算子是一类可在函数空间之间建立映射的新型模型，能让训练后的模型模拟偏微分方程（PDE）的解算子。本文提出一种多网格傅里叶神经算子（MgFNO），通过一种新颖的三级层次架构加速传统傅里叶神经算子的训练。MgFNO的核心创新在于其解耦训练策略，在不同分辨率层级采用三个独立网络：粗层级网络首先学习低分辨率近似解，中间网络对解进行细化，精细层级网络则实现高分辨率精度。通过将深度神经网络的频率原理与多网格方法相结合，MgFNO有效衔接了神经网络（低频到高频）与多网格方法（高频到低频误差缩减）的互补学习模式。实验结果表明，MgFNO在Burgers方程、Darcy流和Navier-Stokes方程上的相对误差分别为0.17%、0.28%和0.22%，与传统FNO相比分别降低了89%、71%和83%。此外，MgFNO支持零样本超分辨率预测，在粗网格上训练后可直接应用于高分辨率场景。本研究为求解以高频动力学为主的复杂偏微分方程建立了一种高效且高精度的新范式。所用代码和数据可在https://github.com/guozihao-hub/MgFNO/tree/master获取。
英文摘要：
Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator (MgFNO) that accelerates the training of traditional Fourier neural operators through a novel three-level hierarchical architecture. The key innovation of MgFNO lies in its decoupled training strategy employing three distinct networks at different resolution levels: a coarse-level network first learns low-resolution approximations, an intermediate network refines the solution, and a fine-level network achieves high-resolution accuracy. By combining the frequency principle of deep neural networks with multigrid methodology, MgFNO effectively bridges the complementary learning patterns of neural networks (low-to-high frequency) and multigrid methods (high-to-low frequency error reduction).Experimental results demonstrate that MgFNO achieves relative errors of 0.17%, 0.28%, and 0.22% on the Burgers' equation, Darcy flow, and Navier-Stokes equations, respectively, representing reductions of 89%, 71%, and 83% compared to the conventional FNO. Furthermore, MgFNO supports zero-shot super-resolution prediction, enabling direct application to high-resolution scenarios after training on coarse grids. This study establishes an efficient and high-accuracy new paradigm for solving complex PDEs dominated by high-frequency dynamics. Code and data used are available on https://github.com/guozihao-hub/MgFNO/tree/master.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
