Eigenfunction Decomposition

Eigenfunction decomposition aims to represent complex systems or data using a set of orthogonal functions (eigenfunctions) and their corresponding weights (eigenvalues), revealing underlying structure and facilitating analysis. Current research focuses on developing robust algorithms, such as Rigged Dynamic Mode Decomposition and neural network-based approaches, to efficiently compute eigenfunctions, particularly for high-dimensional data and systems with continuous spectra, addressing challenges posed by biased data or complex geometries. These advancements are impacting diverse fields, including molecular dynamics, machine learning (e.g., kernel methods), and the solution of partial differential equations in physics and engineering, by enabling more accurate modeling and efficient computation.