Continuous limit and parameter scaling for reintegration tracking in Flow-Lenia

Determine the correct scaling relationships among the discretization hyperparameters of the reintegration tracking advection scheme used in Flow-Lenia (including the step-size parameter s and related flow parameters) that yield a well-defined continuous-time-and-space limit of the Flow-Lenia dynamics, and ascertain whether such a continuous limit exists.

Background

The paper contrasts the Mass-Conserving Evolution (MaCE) rule—whose continuous limit is derived explicitly—with Flow-Lenia, which transports mass along the gradient of an affinity field using the reintegration tracking scheme.

The authors note that reintegration tracking introduces additional discretization hyperparameters and that it is unclear how these should scale to achieve a mathematically well-defined continuous limit for Flow-Lenia, or whether such a limit even exists.