The authors developed a tensor network-based machine learning model to learn the dynamics of probabilistic cellular automata. Probabilistic cellular automata are systems where the evolution of a sequence depends on multiple local rules, each with a predefined probability of occurrence.
The key highlights of the work are:
The authors use matrix product operators (MPOs), a type of tensor network, to represent the conditional probabilities of obtaining an output sequence given an input sequence. The MPO is trained on pairs of input-output sequences from the probabilistic cellular automata.
The training process involves minimizing a loss function that considers the expected conditional probabilities of the output sequences. The authors use an efficient optimization algorithm that locally updates the MPO tensors.
The trained MPO can accurately predict the output sequences with the correct probabilities, even when the underlying rules are complex (e.g., regular or chaotic) or have vastly different probabilities of occurrence.
The authors find that the performance of the model depends on the bond dimension of the MPO and the number of training samples. Larger bond dimensions and more training samples generally lead to better predictions.
The authors also analyze the effect of the bit-wise distance between the probabilistic rules on the difficulty of learning. Rules that are more similar bit-wise are easier to learn than those that are more different.
The authors demonstrate the generality of their approach by considering cases with two and three probabilistic rules.
Overall, the tensor network-based learning model provides an efficient and accurate way to learn the dynamics of probabilistic cellular automata, with potential applications in modeling realistic noisy or probabilistic systems.
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by Heitor P. Ca... at arxiv.org 04-19-2024
https://arxiv.org/pdf/2404.11768.pdfDeeper Inquiries