DEMO :

grid.mp4

Flow Lenia model

The affinity map U is defined the same way the growth is in the original Lenia. For a detailed decription of Lenia see [1, 2].

The matter in Flow Lenia tends to flow towards higher affinity regions by following the gradient in space of U. The affinity gradient is combined with the negative concentration gradient to avoid getting all the matter in a single location. Alpha weights the importance of term such that the concentration gradient dominates in high concentrations regions.

Matter can then be moved according to the flow F using reintegration tracking algorithm [3]. Each cell projects its matter along the flow onto a square distribution of side length 2s. The proportion of matter from cell p' going into cell p is then given by the integral of this square distribution on the cell domain of p.

parameter embedding

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. Formally, this comes to define a parameter map P mapping cells to vector of parameters and locally modifying how the affinity map is computed :

A question is how to mix parameters arriving in the same cell. We propose two methods namely weighted average and softmax sampling

Weighted Average

Weighted average is going to average incoming parameters with respect to incoming quantities of matter. Intuitively, the more represented set of parameters will have greater impact on resulting parameters.

Softmax Sampling

Softmax sampling on the other hand samples one of the incoming set of parameters w.r.t the softmax distribution given by incoming quantities of matter. Intuitively, the more represented set of parameters has greater probability of being selected.

References

[1] Chan, Bert Wang-Chak. “Lenia - Biology of Artificial Life.” Complex Systems 28, no. 3 (October 15, 2019): 251–86. https://doi.org/10.25088/ComplexSystems.28.3.251.

[2] Chan, Bert Wang-Chak. “Lenia and Expanded Universe.” In The 2020 Conference on Artificial Life, 221–29, 2020. https://doi.org/10.1162/isal_a_00297.

[3] Moroz, M. (s. d.). Reintegration tracking. https://michaelmoroz.github.io/Reintegration-Tracking/

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