Wasserstein Evolution : Evolutionary Optimization as Phase Transition

TL;DR

This paper introduces Wasserstein Evolution, a novel theoretical framework for evolutionary algorithms that models their dynamics as a phase transition process using Wasserstein gradient flows, leading to improved diversity and stability.

Contribution

It reframes evolutionary optimization as a physical phase transition, establishing a Wasserstein gradient flow model and proposing a new algorithm with better diversity and convergence properties.

Findings

WE achieves higher solution diversity.

WE demonstrates improved stability over baselines.

Theoretical model links evolution to thermodynamic principles.

Abstract

Evolutionary algorithms (EAs) serve as powerful black-box optimizers inspired by biological evolution. However, most existing EAs predominantly focus on heuristic operators such as crossover and mutation, while usually overlooking underlying physical interpretability such as statistical mechanics and thermosdynamics. This theoretical void limits the principled understanding of algorithmic dynamics, hindering the systematic design of evolutionary search beyond ad-hoc heuristics. To bridge this gap, we first point out that evolutionary optimization can be conceptually reframed as a physical phase transition process. Building on this perspective, we establish the theoretical grounds by modeling the optimization dynamics as a Wasserstein gradient flow of free energy. Consequently, a robust and interpretable solver named Wasserstein Evolution (WE) is proposed. WE mathematically frames the trade-off between exploration and exploitation as a competition between potential gradient forces and entropic forces. This formulation guarantees convergence to the Boltzmann distribution, thereby minimizing free energy and maximizing entropy, which promotes highly diverse solutions. Extensive experiments on complex multimodal and physical potential functions demonstrate that WE achieves superior diversity and stability compared to established baselines.