Neural networks are trained to predict the next state of a CA from the three previous timesteps. Different training examples for a single rule corresponds to different initialization.
The authors study three kinds of generalization:
- Simple generalization: The network is trained on 300 different CA rules and tested on more unseen initial configurations from those 300 rules.
- Level 1 generalization: Same setup as above but the rules have variable sized neighborhoods
- Level 2 generalization: The network is trained on 300 rules with different neighborhoods sizes and tested on new initial configurations and 30 new rules.
- Level 3 generalization: This level has two setups
- Interpolation: train on rules with neighborhoods \(3\times 3\), \(5 \times 5\) and \(9 \times 9\) and evaluate on rules with neighborhood \(7 \times 7\)
- Extrapolation: train on rules with neighborhoods \(3\times 3\), \(5 \times 5\) and \(7 \times 7\) and evaluate on rules with neighborhood \(9 \times 9\)
The neural network used is an encoder-decoder architecture, with convolutions and down-sampling to convert input states into a low dimensional vector and convolutions + up-sampling to convert it back to the next predicted state.
For training and validation, all configurations above reach more than 90% accuracy. For the simple and level 1 generalizations (same CA rules), the test accuracy is also very high (> 90%). The test accuracy drops significantly for Level 2 and 3, where the network probably has trouble inferring the rule from the 4 previous steps only, whereas before it could have memorized those rules during training.
- Aach, Marcel, Jens Henrik Goebbert, and Jenia Jitsev. March 27, 2021. "Generalization over Different Cellular Automata Rules Learned by a Deep Feed-Forward Neural Network". arXiv:2103.14886 [Nlin]. http://arxiv.org/abs/2103.14886.