Approximating Linear Operators for Machine Learning

Linear operators on function spaces appear in several branches of mathematics: In differential geometry, the Laplace-Beltrami operator captures the shape of manifolds. In dynamical systems theory, the Koopman operator globally linearizes systems that act non-linearly on their state. In scientific computing, matrices can be used to approximate solutions to partial differential equations. In machine learning, linear operators can be used to extract information from data sets. In particular, spectral decomposition into eigenvalues and eigenfunctions is an important tool that has yet to find its equal in the analysis of non-linear methods such as neural networks. In my talk, I will briefly introduce the operators that are most important for my research, and then outline recent projects with a focus on scientific machine learning in industrial applications: dimension reduction through Diffusion Maps and the construction of efficient surrogate models for dynamical systems.     «

Linear operators on function spaces appear in several branches of mathematics: In differential geometry, the Laplace-Beltrami operator captures the shape of manifolds. In dynamical systems theory, the Koopman operator globally linearizes systems that act non-linearly on their state. In scientific computing, matrices can be used to approximate solutions to partial differential equations. In machine learning, linear operators can be used to extract information from data sets. In particular, spect...     »