# Flow-Lenia: Towards open-ended evolution in cellular automata through mass conservation and parameter localization

arXiv: https://arxiv.org/abs/2212.07906 · HTML: https://mlsanity.com/paper/2212.07906/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 14 Dec 2022.

## Builds on (most cited first, 5 of 5)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [Lenia - Biology of Artificial Life](https://mlsanity.com/paper/1812.05433/) (2018)
- [evosax: JAX-based Evolution Strategies](https://mlsanity.com/paper/2212.04180/) (2022)
- [Lenia and Expanded Universe](https://mlsanity.com/paper/2005.03742/) (2020)

## Built on by (most cited first, 11 of 11)

- [Automating the Search for Artificial Life with Foundation Models](https://mlsanity.com/paper/2412.17799/) (2024)
- [Biomaker CA: a Biome Maker project using Cellular Automata](https://mlsanity.com/paper/2307.09320/) (2023)
- [Discovering Sensorimotor Agency in Cellular Automata using Diversity Search](https://mlsanity.com/paper/2402.10236/) (2024)
- [Exploring Flow-Lenia Universes with a Curiosity-driven AI Scientist: Discovering Diverse Ecosystem Dynamics](https://mlsanity.com/paper/2505.15998/) (2025)
- [Flow-Lenia: Emergent evolutionary dynamics in mass conservative continuous cellular automata](https://mlsanity.com/paper/2506.08569/) (2025)
- [Expedition & Expansion: Leveraging Semantic Representations for Goal-Directed Exploration in Continuous Cellular Automata](https://mlsanity.com/paper/2509.03863/) (2025)
- [Guiding Evolution of Artificial Life Using Vision-Language Models](https://mlsanity.com/paper/2509.22447/) (2025)
- [Visualizing the Structure of Lenia Parameter Space](https://mlsanity.com/paper/2601.01932/) (2026)
- [Conversable Complexity: Agentic LLM Collectives as Interpretable Substrates](https://mlsanity.com/paper/2607.01047/) (2026)
- [Coralai: Intrinsic Evolution of Embodied Neural Cellular Automata Ecosystems](https://mlsanity.com/paper/2406.09654/) (2024)
- [Evolving Many Worlds: Towards Open-Ended Discovery in Petri Dish NCA via Population-Based Training](https://mlsanity.com/paper/2604.11248/) (2026)

## Abstract (arXiv)

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.