Learning to unknot
Learning to unknot
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DOI：
10.1088/2632-2153/abe91f
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发表时间：
2020-10
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影响因子：
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通讯作者：
S. Gukov;James Halverson;Fabian Ruehle;P. Sułkowski
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作者：
S. Gukov;James Halverson;Fabian Ruehle;P. Sułkowski
我们将自然语言处理引入到结理论的研究中，因为结的编织词表示是自然的。我们研究了确定一个给定的结是否是unknot的UNKNOT问题。在描述了一种随机生成N-交叉辫子及其结闭合的算法并讨论了结分布的诱导先验之后，我们将二进制分类应用于UNKNOT决策问题。我们发现，Reformer和共享QK Transformer网络架构的性能优于完全连接的网络，尽管它们的准确率都在95%左右。也许令人惊讶的是，我们发现准确性随着编织词的长度而增加，并且网络学习到预测的置信度与琼斯多项式的次数之间存在直接相关性。最后，我们利用强化学习（RL）来找到马尔可夫移动和编织关系的序列，这些序列简化了结，并可以通过显式给出解开动作的序列来识别解开。信赖域策略优化（TRPO）的性能一直很好，减少了80%的解结，我们测试了多达96个交叉点，并彻底优于其他RL算法和随机游走器。研究这些行动，我们发现辫子关系是更有用的简化到unknot比一个马尔可夫移动。
We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an algorithm to randomly generate N-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform at ≳ 95% accuracy. Perhaps surprisingly, we find that accuracy increases with the length of the braid word, and that the networks learn a direct correlation between the confidence of their predictions and the degree of the Jones polynomial. Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions. Trust region policy optimization (TRPO) performs consistently well, reducing ≳ 80% of the unknots with up to 96 crossings we tested to the empty braid word, and thoroughly outperformed other RL algorithms and random walkers. Studying these actions, we find that braid relations are more useful in simplifying to the unknot than one of the Markov moves.