Sampling String Vacua Using Generative Models

TL;DR

This work addresses sampling Type IIB flux vacua in the string landscape by applying two generative-model frameworks to ISD flux configurations within a controlled moduli-space region. Bayesian Flow Networks offer unconditional sampling of flux vectors and show strong interpolation with meaningful extrapolation, while Transformers enable conditional sampling on physics targets like $N_{ extrm{flux}}$ and $|W_0|$. Together, the approaches demonstrate that learned generative models can reproduce the data distribution of valid vacua and generate new, physically plausible configurations, providing a powerful toolkit for exploring the string landscape. The results highlight the potential of combining data-driven generative modeling with traditional string-theory constraints to efficiently search for vacua with desired phenomenological properties, and point to future directions such as balancing training data, extending to more complex geometries, and integrating with existing computational pipelines.

Abstract

We apply generative models to a key problem in the string compactification program, namely construction of type IIB string vacua. To this end, we make use of a Bayesian Flow Network, a generative model capable of handling discrete data, to generate flux vectors that give rise to type IIB vacua. Furthermore, we sample flux vacua that have certain desirable properties by employing a Transformer as a conditional generative model. Both models demonstrate good performance in finding flux vacua and thus prove to be powerful tools in the exploration of the string landscape.

Figures (13)

• Figure 1: Schematic drawing of BFN training for the one-dimensional case with $K=2$. A training step is shown for one data point $x=1$.
• Figure 2: Training evaluation of the BFNs. The validation loss, validity and originality are plotted over various epochs. The values correspond to the mean of five models that have been trained on different subsets of Dataset A and the errors correspond to the standard deviation. The validity corresponds to the percentage of valid fluxes among numerically unique samples. Originality corresponds to the percentage of valid fluxes not contained in the training data.
• Figure 3: Distribution of $N_\textrm{flux}$ computed from samples drawn from all five models at epoch 300 compared to the respective training data. The plotted samples correspond to the valid samples and thus make up $\sim70\%$ of the $10~000$ samples.
• Figure 4: Distribution of $W_0$ computed from samples drawn from five models at epoch 300 compared to the training data (points in each plot correspond to valid samples/training data from all five models).
• Figure 5: Distribution of $z^1, z^2$ and $\tau$ computed from valid samples drawn from five models at epoch 300 (first row) compared to the training data (second row). Points in each plot correspond to samples/training data from all five models. To ensure readability, the sample plots have been cut off resulting to some samples not being shown.