The Glider Equation for Asymptotic Lenia

TL;DR

This paper introduces the Glider Equation for Asymptotic Lenia, enabling analytical derivation and optimization of stable glider patterns through gradient descent, and explores connections to neural field models.

Contribution

It formulates the Glider Equation for Asymptotic Lenia, allowing analytical and optimization-based discovery of diverse glider patterns and establishing links to neural field models.

Findings

Successfully derived conditions for glider patterns

Optimized update rules to find novel gliders

Connected Asymptotic Lenia to neural field models

Abstract

Lenia is a continuous extension of Conway's Game of Life that exhibits rich pattern formations including self-propelling structures called gliders. In this paper, we focus on Asymptotic Lenia, a variant formulated as partial differential equations. By utilizing this mathematical formulation, we analytically derive the conditions for glider patterns, which we term the ``Glider Equation.'' We demonstrate that by using this equation as a loss function, gradient descent methods can successfully discover stable glider configurations. This approach enables the optimization of update rules to find novel gliders with specific properties, such as faster-moving variants. We also derive a velocity-free equation that characterizes gliders of any speed, expanding the search space for novel patterns. While many optimized patterns result in transient gliders that eventually destabilize, our approach effectively identifies diverse pattern formations that would be difficult to discover through traditional methods. Finally, we establish connections between Asymptotic Lenia and neural field models, highlighting mathematical relationships that bridge these systems and suggesting new directions for analyzing pattern formation in continuous dynamical systems.