My understanding is Conway's game of life can be done by simply convolving a 3x3 array of all ones (except a central 0) through every pixel, and then looking at the results of that convolution to see if the pixel is dead or alive, e.g. https://stackoverflow.com/questions/46196346/why-does-my-gam...

If that's the case, would a nice solution be to model it as a recurrent neural network, e.g. pytorch or jax, with a single convolutional layer, with a single kernel, which you don't backprop to, and then try to minimise to loss with reference to the target image, by backpropagating to the input image?

It then becomes highly analogous to Google's deepdream, but instead of optimising the image for the output class, you optimise it to the output image.