用SVD分解核函数，提升偏微分方程求解器的表达能力与效率。
SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels
- 通过SVD显式参数化核函数，在低秩基上直接计算积分。
- 在5个基准方程上达到新最优性能，尤其擅长高空间变异性问题。
- 结构轻量，计算高效，适合需要高精度与实时性的工程场景。
神经算子已成为从数据中直接学习偏微分方程（PDE）解算子的有前景范式。现有方法如基于傅里叶或图结构的技术，对核积分算子的结构做出强假设，可能限制其表达能力。本文提出SVD-NO，显式以奇异值分解（SVD）参数化核函数，并在低秩基上直接执行积分。两个轻量网络分别学习左右奇异函数，对角参数矩阵学习奇异值，Gram矩阵正则项确保正交性。由于近似全核，SVD-NO具有高表达能力；同时因低秩结构，算子应用的计算复杂度仍合理，形成实用系统。在五个多样化基准方程的广泛评估中，SVD-NO实现新状态性能，尤其在解具有高度空间可变性的PDE时表现更优。代码已公开于https://github.com/2noamk/SVDNO.git。
原文摘要 · Abstract (English)
Neural operators have emerged as a promising paradigm for learning solution operators of partial differential equa- tions (PDEs) directly from data. Existing methods, such as those based on Fourier or graph techniques, make strong as- sumptions about the structure of the kernel integral opera- tor, assumptions which may limit expressivity. We present SVD-NO, a neural operator that explicitly parameterizes the kernel by its singular-value decomposition (SVD) and then carries out the integral directly in the low-rank basis. Two lightweight networks learn the left and right singular func- tions, a diagonal parameter matrix learns the singular values, and a Gram-matrix regularizer enforces orthonormality. As SVD-NO approximates the full kernel, it obtains a high de- gree of expressivity. Furthermore, due to its low-rank struc- ture the computational complexity of applying the operator remains reasonable, leading to a practical system. In exten- sive evaluations on five diverse benchmark equations, SVD- NO achieves a new state of the art. In particular, SVD-NO provides greater performance gains on PDEs whose solutions are highly spatially variable. The code of this work is publicly available at https://github.com/2noamk/SVDNO.git.
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