I am trying to understand why they approached this ML problem this way- as a conv net. Conway's game of life is in itself a convolution. The state of a cell is

I am trying to understand why they approached this ML problem this way- as a conv net. Conway's game of life is in itself a convolution. The state of a cell is calculated from the state of the 9 neighboring cells. When these authors made a neural net to learn these 'rules', they were trying to learn a simple convolution kernel from a bunch of macro states. They used a 32x32 grid board and 1 million training examples. It seems from their paper that each example consisted of a pair of states in the 'game of life'-the before state and the after state. But what were the labels? Was the 'after' state being used as a label for the before state? When the problem is stated this way, it seems more akin to to predicting the next word in a sentence based on the present word. Wouldn't an LSTM or perhaps a GPT-3 architecture be better suited? As you can see by the stupidity of my question, I am an ML amateur and neophyte. To be completely naive, how would one approach the problem of designing an ML algorithm to 'learn a convolution kernel' For example, suppose I had a million image pairs generated with a 3x3 'blur' kernel {(.0625,.125,.0625),(.125,.25,.125),(.0625,.125,.0625)} what architecture would be best to feed this data into?