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

TL;DR

SVD-NO introduces a neural operator that explicitly models PDE solution operators using a low-rank SVD kernel, enhancing expressivity and efficiency, and achieves state-of-the-art results on diverse benchmarks.

Contribution

It proposes a novel SVD-based neural operator that explicitly parameterizes kernels, improving expressivity and computational efficiency for solving PDEs.

Findings

Achieves state-of-the-art performance on five benchmark PDEs.

Provides significant gains on PDEs with highly variable solutions.

Maintains reasonable computational complexity due to low-rank structure.

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.