Harmonic Analysis: Peter-Weyl Theorem and Machine Learning
The Peter-Weyl theorem is one of the most beautiful results in Harmonic Analysis, and it has found surprisingly wide applications in machine learning (drug discovery, molecular dynamics, robotics, particle physics, graphs, etc). In many real-world problems, data is scarce, complexity is high, or we simply want our models to generalize better and produce more robust solutions. In these cases, it makes sense to leverage any prior knowledge we have about the problem structure. For example, in meteorology the data lives on a sphere (the Earth), or we might know that the underlying dynamics roughly satisfy the Navier-Stokes equations. The desire to incorporate this kind of domain-specific knowledge into machine learning models gave rise to what we now call Physics-Informed ML and Geometry-Informed ML. The former led to Neural ODEs and Physics-Informed Neural Networks (PINNs), but in this post we focus on the latter. ...