An Energy-Based Steepest Descent Evolution Principle: Applications in Continuum and Graph Theory

This work introduces the Steepest Descent Evolution Principle (SDEP), a general variational framework that explains how systems evolve by following the path of steepest energy decrease. The principle is formulated in a broad mathematical setting, encompassing normed vector spaces, Banach spaces, and Hilbert spaces, where functional derivatives and gradients provide the foundation for its dynamics. Using Dirichlet energies as test cases, we show that the SDEP naturally recovers classical diffusion laws: the heat equation in the continuum and diffusion equations on graphs governed by the graph Laplacian. These results highlight the unifying power of the principle, offering a simple recipe for deriving dynamical equations across different contexts. Beyond classical physics, the framework opens avenues for applications in data science, network dynamics, and optimization, where energy-based models and steepest descent play a central role.