标题： 渐近Lenia的滑翔机方程 显示英文标题

标题： The Glider Equation for Asymptotic Lenia

摘要： Lenia是康威生命游戏的连续扩展，表现出丰富的模式形成，包括称为滑翔机的自推进结构。 在本文中，我们关注渐近Lenia，这是一种用偏微分方程表述的变体。 通过利用这种数学公式，我们分析推导出滑翔机模式的条件，我们将其称为“滑翔机方程”。 我们证明，通过将此方程作为损失函数，梯度下降方法可以成功发现稳定的滑翔机配置。 这种方法使得能够优化更新规则以找到具有特定属性的新滑翔机，例如更快移动的变体。 我们还推导出一个与速度无关的方程，该方程可表征任何速度的滑翔机，扩展了寻找新图案的搜索空间。 虽然许多优化后的图案导致最终不稳定的各种瞬态滑翔机，但我们的方法能有效识别出通过传统方法难以发现的多样化模式。 最后，我们建立了渐近Lenia与神经场模型之间的联系，突出了连接这些系统的数学关系，并提出了分析连续动力系统中模式形成的新的研究方向。 显示英文摘要

摘要： 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.