SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels

• URL: http://arxiv.org/abs/2511.10025v1
• Date: Fri, 14 Nov 2025 01:26:58 GMT
• Title: SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels
• Authors: Noam Koren, Ralf J. J. Mackenbach, Ruud J. G. van Sloun, Kira Radinsky, Daniel Freedman,
• Abstract summary: We present SVD-NO, a neural operator that parameterizes the kernel by its singular-value decomposition (SVD) and then carries out the integral directly in the low-rank basis.<n>As SVD-NO approximates the full kernel, it obtains a high de- gree of expressivity.
• Score: 35.16133249685271
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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