Abstract: We propose a self-supervising learning framework for finding the dominant eigenfunction-eigenvalue pairs of linear and self-adjoint operators. We represent target eigenfunctions with coordinate-based neural networks and employ the Fourier positional encodings to enable the approximation of high-frequency modes. We formulate a self-supervised training objective for spectral learning and propose a novel regularization mechanism to ensure that the network finds the exact eigenfunctions instead of a space spanned by the eigenfunctions. Furthermore, we investigate the effect of weight normalization as a mechanism to alleviate the risk of recovering linear dependent modes, allowing us to accurately recover a large number of eigenpairs. The effectiveness of our methods is demonstrated across a collection of representative benchmarks including both local and non-local diffusion operators, as well as high-dimensional time-series data from a video sequence. Our results indicate that the present algorithm can outperform competing approaches in terms of both approximation accuracy and computational cost.

Key words: spectral learning, partial differential equation (PDE), neural network, slow features analysis