SVD-NO: Learning PDE Solution Operators with SVD Integral Kernels
This work addresses limitations in neural operators for PDEs, offering a more expressive method that could benefit computational science and engineering applications, though it appears incremental as it builds on existing operator learning paradigms.
The authors tackled the problem of learning PDE solution operators by introducing SVD-NO, a neural operator that parameterizes kernel integrals via SVD for improved expressivity, achieving state-of-the-art results on five benchmark equations with notable gains on spatially variable PDEs.
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.