发表机构
Oklahoma State University（俄克拉荷马州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出AFDONet，首个由自适应傅里叶分解理论完全引导设计的神经算子PDE求解器，兼具数学可解释性、高精度与高效性，在任意流形和尖锐梯度数据集上表现优越。
AI 中文摘要
偏微分方程（PDE）的精确数值解在许多科学和工程应用中至关重要。在这项工作中，我们引入了一种名为AFDONet的新型神经PDE求解器，它首次将神经算子学习与自适应傅里叶分解（AFD）理论结合到一个专门设计的变分自编码器（VAE）结构中，用于求解光滑流形上的一般非线性PDE。AFDONet是第一个其架构和组件设计完全由既定数学框架（此处为AFD理论）引导的神经PDE求解器，将神经算子设计从艺术转变为科学。因此，AFDONet还表现出卓越的数学可解释性和扎实性，并具有若干理想性质。此外，AFDONet在多个基准问题中取得了出色的求解精度和有竞争力的计算效率。特别是，由于与AFD理论的深刻联系，AFDONet在求解i）任意（黎曼）流形上的PDE和ii）具有尖锐梯度的数据集上的PDE时表现出优越性能。总体而言，这项工作为设计可解释的神经算子框架提供了一种新范式。
英文摘要
Accurate numerical solutions of partial differential equations (PDEs) are crucial in numerous science and engineering applications. In this work, we introduce a novel neural PDE solver named AFDONet, which incorporates neural operator learning and adaptive Fourier decomposition (AFD) theory for the first time into a specifically designed variational autoencoder (VAE) structure, to solve a general class of nonlinear PDEs on smooth manifolds. AFDONet is the first neural PDE solver whose architectural and component design is fully guided by an established mathematical framework (in this case, AFD theory), turning neural operator design from an art to a science. Thus, AFDONet also exhibits exceptional mathematical explainability and groundness, and enjoys several desired properties. Furthermore, AFDONet achieves outstanding solution accuracy and competitive computational efficiency in several benchmark problems. In particular, thanks to its deep connections with AFD theory, AFDONet shows superior performance in solving PDEs on i) arbitrary (Riemannian) manifolds, and ii) datasets with sharp gradients. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.
Comments37 pages
