There’s something about visualising ODEs.

The lines connecting the points are solutions to an ODE parameterised by a neural network, better known as a flow model. While flow models are usually trained with a noise distribution such as an independent multivariate normal, another “distribution” has been inserted here for humour. As one of my (former) colleagues put it, “life imitates art”.


The plot below was generated by running Replicator Nash Dynamics on a random initial policy for two players playing rock-paper-scissors. The lines are paraterised by $(x_1(t), y_1(t), z_1(t)) = \left(\pi^1_ \text{rock}(t), \pi^1_ \text{paper}(t), \pi^2_ \text{scissors}(t)\right)$, and $(x_2(t), y_2(t), z_2(t)) = \left(\pi^2_ \text{rock}(t), \pi^2_ \text{paper}(t), \pi^1_ \text{scissors}(t)\right)$ for a more thrilling visual effect (since $\sum_{a} \pi^i_ a(t) = 1$, plotting (x, y, z) as the probabilities of a single policy would make the result lie on a single plane which would look boring). Indeed, the transformation depicted here is the result of solving a time invariant ODE. As a fun aside, I also got to use this as inspiration for a logo for an internal system at a previous firm!