论文标题
莱尼亚生活的存在
Existence of Life in Lenia
论文作者
论文摘要
莱尼亚（Lenia）是对康威（Conway）生活游戏的持续概括。 Bert Wang-Chak Chan自2019年以来在Lenia模拟中发现并发表了许多看似有机的动态。这些模拟从功能空间初始条件开始遵循Euler曲线算法。 Banach空间上Lipschitz向量场的积分曲线存在的Picard-Lindelöf定理无法保证解决方案，因为与定义Lenia的无数差异方程相关的向量场是不连续的。但是，我们证明了动态木用于生成仿真实际上是弧场场，而不是源自整体差异方程的向量场的传统欧拉方法。使用ARC场理论，我们证明Euler曲线会收敛到求解原始插差方程的独特流。探索了扩展，并讨论了熵的建模。 关键字：弧字段；不连续的矢量场；整数差异方程；熵模型
Lenia is a continuous generalization of Conway's Game of Life. Bert Wang-Chak Chan has discovered and published many seemingly organic dynamics in his Lenia simulations since 2019. These simulations follow the Euler curve algorithm starting from function space initial conditions. The Picard-Lindelöf Theorem for the existence of integral curves to Lipschitz vector fields on Banach spaces fails to guarantee solutions, because the vector field associated with the integro-differential equation defining Lenia is discontinuous. However, we demonstrate the dynamic Chan is using to generate simulations is actually an arc field and not the traditional Euler method for the vector field derived from the integro-differential equation. Using arc field theory we prove the Euler curves converge to a unique flow which solves the original integro-differential equation. Extensions are explored and the modeling of entropy is discussed. Keywords: arc fields; discontinuous vector fields; integro-differential equations; entropy models
