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.

We construct all possible complete intersection Calabi-Yau five-folds in a product of four or less complex projective spaces, with up to four constraints. We obtain $27068$ spaces, which are not related by permutations of rows and columns of the configuration matrix, and determine the Euler number for all of them. Excluding the $3909$ product manifolds among those, we calculate the cohomological data for $12433$ cases, i.e. $53.7 \%$ of the non-product spaces, obtaining $2375$ different Hodge diamonds. The dataset containing all the above information is available at https://www.dropbox.com/scl/fo/z7ii5idt6qxu36e0b8azq/h?rlkey=0qfhx3tykytduobpld510gsfy&dl=0 . The distributions of the invariants are presented, and a comparison with the lower-dimensional analogues is discussed. Supervised machine learning is performed on the cohomological data, via classifier and regressor (both fully connected and convolutional) neural networks. We find that $h^{1,1}$ can be learnt very efficiently, with very high $R^2$ score and an accuracy of $96\%$, i.e. $96 \%$ of the predictions exactly match the correct values. For $h^{1,4},h^{2,3}, \eta$, we also find very high $R^2$ scores, but the accuracy is lower, due to the large ranges of possible values.

Calabi-Yau four-folds may be constructed as hypersurfaces in weighted projective spaces of complex dimension 5 defined via weight systems of 6 weights. In this work, neural networks were implemented to learn the Calabi-Yau Hodge numbers from the weight systems, where gradient saliency and symbolic regression then inspired a truncation of the Landau-Ginzburg model formula for the Hodge numbers of any dimensional Calabi-Yau constructed in this way. The approximation always provides a tight lower bound, is shown to be dramatically quicker to compute (with compute times reduced by up to four orders of magnitude), and gives remarkably accurate results for systems with large weights. Additionally, complementary datasets of weight systems satisfying the necessary but insufficient conditions for transversality were constructed, including considerations of the IP, reflexivity, and intradivisibility properties. Overall producing a classification of this weight system landscape, further confirmed with machine learning methods. Using the knowledge of this classification, and the properties of the presented approximation, a novel dataset of transverse weight systems consisting of 7 weights was generated for a sum of weights $\leq 200$; producing a new database of Calabi-Yau five-folds, with their respective topological properties computed. Further to this an equivalent database of candidate Calabi-Yau six-folds was generated with approximated Hodge numbers.

In recent years, deep learning has become a relevant research theme in physics and mathematics. It is a very efficient method for data processing, and elaboration and exploration of patterns [1]. Though the basic building blocks are not new [2], the increase in computational capabilities and the creation of larger databases lead new deep learning techniques to thrive. Specifically, the understanding of the geometrical structures [3, 4] and the representation learning [5] are of particular interest from a mathematical and theoretical physics points of view [6-9]. We are interested in applications of data science and deep learning techniques for algebraic topology, and especially Hodge numbers, of complete intersection Calabi-Yau (CICY) manifolds [10-12].

The ubiquitous interrelations between algebraic geometry and physics has for centuries flourished fruitful phenomena in both fields. With connections made as far back as Archimedes whose work on conic sections aided development of concepts surrounding the motion under gravity, physical understanding has largely relied upon the mathematical tools available. In the modern era, these two fields are still heavily intertwined, with particular relevance in addressing one of the most significant problems of our time - quantising gravity. String theory as a candidate for this theory of everything, relies heavily on algebraic geometry constructions to define its spacetime and to interpret its matter. However, where new mathematical tools arise their implementation is not always simple.

We revisit the classic database of weighted-P4s which admit Calabi-Yau 3-fold hypersurfaces equipped with a diverse set of tools from the machine-learning toolbox. Unsupervised techniques identify an unanticipated almost linear dependence of the topological data on the weights. This then allows us to identify a previously unnoticed clustering in the Calabi-Yau data. Supervised techniques are successful in predicting the topological parameters of the hypersurface from its weights with an accuracy of R^2 > 95%. Supervised learning also allows us to identify weighted-P4s which admit Calabi-Yau hypersurfaces to 100% accuracy by making use of partitioning supported by the clustering behaviour.

The dataset is considerably larger and richer than the one for CICY threefolds and Topological quantities of manifolds, such as Betti or it consists of about 900000 topological types of manifolds. Hodge numbers, are often non-trivially related to the However, so far, this new dataset has not been data describing the underlying manifold and tend to used for machine learning and the purpose of this letter be difficult to work out. Explicit formulae are usually is to fill this gap. More specifically, we will explore, not known and calculations rely on complicated and frequently within the context of supervised learning, if and to what computationally intense algorithms (see, for example, extent Hodge numbers of CICY fourfolds can be learned the volume [1] and references therein for applications by neural networks.

We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past year, this writing is a very preliminary draft of a book to appear with Springer, by whose kind permission we post to ArXiv for comments and suggestions.

The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi-Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to which the Synthetic Minority Oversampling Technique (SMOTE) is applied to boost performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned.

Theoretical physics now firmly resides within an Age wherein new physics, new mathematics and new data coexist in a symbiosis which transcends interdisciplinary boundaries and wherein concepts and developments in one field are evermore rapidly enriching another. String theory has spearheaded this vision for the past few decades and has, perhaps consequently, become a paragon of the theoretical sciences. That she engenders the cross-fertilization between physics and mathematics is without dispute: interactions on an unprecedented scale have commingled fields as diverse as quantum field theory, general relativity, condensed matter physics, algebraic and differential geometry, number theory, representation theory, category theory, etc. With the advent of increasingly powerful computers, from this fruitful dialogue has also arisen a plethora of data, ripe for mathematical experimentation. This emergence of data in some sense began with the incipience of string phenomenology [1] where compactification of the heterotic string on Calabi-Yau threefolds (CY3) was widely believed to hold the ultimate geometric unification.