SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915615041650688 |
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| author | Koren, Noam Mackenbach, Ralf J. J. van Sloun, Ruud J. G. Radinsky, Kira Freedman, Daniel |
| author_facet | Koren, Noam Mackenbach, Ralf J. J. van Sloun, Ruud J. G. Radinsky, Kira Freedman, Daniel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels Koren, Noam Mackenbach, Ralf J. J. van Sloun, Ruud J. G. Radinsky, Kira Freedman, Daniel Machine Learning 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. |
| title | SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2511.10025 |