We present Neural Spectral Methods, a technique to solve parametric Partial
Differential Equations (PDEs), grounded in classical spectral methods. Our
method uses orthogonal bases to learn PDE solutions as mappings between
spectral coefficients. In contrast to current machine learning approaches which
enforce PDE constraints by minimizing the numerical quadrature of the residuals
in the spatiotemporal domain, we leverage Parseval's identity and introduce a
new training strategy through a \textit{spectral loss}. Our spectral loss
enables more efficient differentiation through the neural network, and
substantially reduces training complexity. At inference time, the computational
cost of our method remains constant, regardless of the spatiotemporal
resolution of the domain. Our experimental results demonstrate that our method
significantly outperforms previous machine learning approaches in terms of
speed and accuracy by one to two orders of magnitude on multiple different
problems. When compared to numerical solvers of the same accuracy, our method
demonstrates a $10\times$ increase in performance speed.

我们提出了神经谱方法，一种在经典谱方法基础上解决参数化偏微分方程的技术。我们的方法使用正交基来学习 PDE 解作为谱系数之间的映射，通过谱损失实现更高效的神经网络微分和显著降低训练复杂度，在多个不同问题上，我们的方法在速度和准确性方面显著优于之前的机器学习方法，与相同准确性的数值解算器相比，我们的方法表现出了 10 倍的性能提升。

神经谱方法：频域自监督学习

Neural Spectral Methods: Self-supervised learning in the spectral domain

We propose a novel approach to Vietnamese word segmentation. Our approach is
based on the Single Classification Ripple Down Rules methodology (Compton and
Jansen, 1990), where rules are stored in an exception structure and new rules
are only added to correct segmentation errors given by existing rules.
Experimental results on the benchmark Vietnamese treebank show that our
approach outperforms previous state-of-the-art approaches JVnSegmenter,
vnTokenizer, DongDu and UETsegmenter in terms of both accuracy and performance
speed. Our code is open-source and available at:
this https URL

本文提出了一种基于单一分类涟漪向下规则方法的越南语词分割方法，该方法通过存储规则和只在现有规则给出的分割错误的基础上添加新规则来实现；在基准越南树库上的实验结果表明，与先前的 JVnSegmenter、vnTokenizer、DongDu 和 UETsegmenter 等最先进的方法相比，我们的方法在准确性和性能速度两方面均有所提高。