The design of complex self-organising systems producing life-like phenomena, such as the open-ended evolution of virtual creatures, is one of the main goals of artificial life. Lenia, a family of cellular automata (CA) generalizing Conway's Game of Life to continuous space, time and states, has attracted a lot of attention because of the wide diversity of self-organizing patterns it can generate. Among those, some spatially localized patterns (SLPs) resemble life-like artificial creatures and display complex behaviors. However, those creatures are found in only a small subspace of the Lenia parameter space and are not trivial to discover, necessitating advanced search algorithms. Furthermore, each of these creatures exist only in worlds governed by specific update rules and thus cannot interact in the same one. This paper proposes as mass-conservative extension of Lenia, called Flow Lenia, that solve both of these issues. We present experiments demonstrating its effectiveness in generating SLPs with complex behaviors and show that the update rule parameters can be optimized to generate SLPs showing behaviors of interest. Finally, we show that Flow Lenia enables the integration of the parameters of the CA update rules within the CA dynamics, making them dynamic and localized, allowing for multi-species simulations, with locally coherent update rules that define properties of the emerging creatures, and that can be mixed with neighbouring rules. We argue that this paves the way for the intrinsic evolution of self-organized artificial life forms within continuous CAs.

翻译：摘要：设计能够产生类生命现象（如虚拟生物的开放式进化）的复杂自组织系统，是人工生命领域的主要目标之一。作为将康威生命游戏推广至连续空间、时间与状态的元胞自动机家族，Lenia因其能生成丰富多样的自组织模式而备受关注。其中，部分空间局部化模式（SLP）呈现出类似生命的人造生物形态，并展现出复杂行为。然而，这些生物仅存在于Lenia参数空间的一个小子空间中，且发现过程颇具挑战性，需要借助高级搜索算法。更进一步，每个生物仅存在于受特定更新规则支配的世界中，因此无法在同一个世界中互动。本文提出Lenia的质量守恒扩展版本——Flow Lenia，以同时解决上述两个问题。我们通过实验证明其在生成具有复杂行为的SLP方面的有效性，并表明更新规则参数可被优化以产生具有目标行为的SLP。最后，我们展示了Flow Lenia能够将元胞自动机更新规则的参数整合到其动力学过程中，使参数具备动态性与局部化特性，从而支持多物种模拟，其中局部一致的更新规则定义了涌现生物的特性，并可与其邻近规则混合。我们认为，这为连续元胞自动机中自组织人工生命形式的内在进化铺平了道路。