In this work, we report the results of applying deep learning based on hybrid convolutional-recurrent and purely recurrent neural network architectures to the dataset of almost one million complete intersection Calabi-Yau four-folds (CICY4) to machine-learn their four Hodge numbers $h^{1,1}, h^{2,1}, h^{3,1}, h^{2,2}$. In particular, we explored and experimented with twelve different neural network models, nine of which are convolutional-recurrent (CNN-RNN) hybrids with the RNN unit being either GRU (Gated Recurrent Unit) or Long Short Term Memory (LSTM). The remaining four models are purely recurrent neural networks based on LSTM. In terms of the $h^{1,1}, h^{2,1}, h^{3,1}, h^{2,2}$ prediction accuracies, at 72% training ratio, our best performing individual model is CNN-LSTM-400, a hybrid CNN-LSTM with the LSTM hidden size of 400, which obtained 99.74%, 98.07%, 95.19%, 81.01%, our second best performing individual model is LSTM-448, an LSTM-based model with the hidden size of 448, which obtained 99.74%, 97.51%, 94.24%, and 78.63%. These results were improved by forming ensembles of the top two, three or even four models. Our best ensemble, consisting of the top four models, achieved the accuracies of 99.84%, 98.71%, 96.26%, 85.03%. At 80% training ratio, the top two performing models LSTM-448 and LSTM-424 are both LSTM-based with the hidden sizes of 448 and 424. Compared with the 72% training ratio, there is a significant improvement of accuracies, which reached 99.85%, 98.66%, 96.26%, 84.77% for the best individual model and 99.90%, 99.03%, 97.97%, 87.34% for the best ensemble.

翻译：本工作中，我们报告了将基于混合卷积-循环及纯循环神经网络架构的深度学习技术，应用于近百万个完全交截卡拉比-丘四重形（CICY4）数据集，以机器学习其四个霍奇数 $h^{1,1}, h^{2,1}, h^{3,1}, h^{2,2}$ 的结果。具体而言，我们探索并实验了十二种不同的神经网络模型，其中九种为卷积-循环（CNN-RNN）混合模型，其循环神经网络单元采用门控循环单元（GRU）或长短期记忆网络（LSTM）。其余四种模型为基于LSTM的纯循环神经网络。在 $h^{1,1}, h^{2,1}, h^{3,1}, h^{2,2}$ 的预测准确率方面，当训练比例为72%时，我们表现最佳的单一模型是CNN-LSTM-400（一种LSTM隐藏层大小为400的混合CNN-LSTM模型），其准确率分别达到99.74%、98.07%、95.19%、81.01%；表现第二佳的单一模型是LSTM-448（隐藏层大小为448的LSTM模型），其准确率为99.74%、97.51%、94.24%和78.63%。通过构建前二、前三乃至前四名模型的集成，这些结果得到了进一步提升。我们最佳的集成模型（由前四名模型组成）达到了99.84%、98.71%、96.26%、85.03%的准确率。在80%训练比例下，表现最佳的两个模型LSTM-448和LSTM-424均为基于LSTM的模型，隐藏层大小分别为448和424。与72%训练比例相比，准确率有显著提升：最佳单一模型达到99.85%、98.66%、96.26%、84.77%，最佳集成模型达到99.90%、99.03%、97.97%、87.34%。