Explicit solutions for Thomson-like dynamical systems

Develop explicit analytical solutions for dynamical systems in mathematical physics that are analogous to the spectral dynamical systems discussed here (e.g., Thomson-type problems and related systems), for which explicit solutions are not yet known.

Background

The paper analyzes training dynamics by deriving a Gram-matrix gradient flow and associated spectral evolution equations. While it provides structural results (e.g., eigenvalue drift and projector dynamics), obtaining closed-form dynamic solutions is substantially harder.

The authors note that even closely related classical systems in mathematical physics—such as the Thomson problem and problems highlighted by Smale—do not admit explicit solutions in general, underscoring the difficulty of fully solving the analogous spectral dynamical systems they study.