Machine Learning

The tools discussed in the previous sections can be used to implement a number of different machine learning approaches to approximating the Calabi-Yau metric. Some methods for a particular approach to this using Flax can be found in the cyjax.ml submodule. This includes functions for initializing and working with the Cholesky decomposition of the Hermitian matrix and a batched sampler class.

ml.cholesky_decode(diag, upper)	Construct hermitian matrix from Cholesky decomposition.
ml.cholesky_from_param(h_par)	Construct hermitian matrix from Cholesky decomposition parameters.
ml.hermitian_param_init(key, basis_size[, ...])	Initialize parametrization to yield identity for Hermitian matrix.
ml.BatchSampler(seed, variety, params_sampler)

One can use multiple different losses which effectively measure the Ricci-flatness of the approximated metric. In particular, we use here the so called \(\sigma\) accuracy and a Monge-Ampere loss, which rely on the property that the Ricci flat metric \(g\) gives rise to a volume form which must be proportional to the one given by the holomorphic top form (Headrick & Nassar, 2013). If \(\Omega\) is the holomorphic top form, we can define the ratio \(\eta = \frac{\det g}{\Omega \wedge \Omega}\). The \(\sigma\) accuracy measures the deviation from \(\eta\) being constant as the integral

\[ \sigma = \int_X |\eta - 1| d\mathrm{vol}_{\Omega} \,. \]

For training, we use the related variance-like Monge Ampere loss

\[ \mathcal{L}_{MA} = \sum_{z \sim X} (\eta - 1)^2 w(z)\,, \]

which approximates an integral with respect to the volume form \(d\mathrm{vol}_{\Omega}\) using Monte Carlo weights \(w(z)\). The latter “undo” the bias introduced by the sampling scheme used to sample points \(z\) on the manifold, as discussed in a previous section.

ml.variance_eta_loss(h, sample, algebraic_metric)

Compute variance-based eta loss.

Lastly, there is a configurable MLP-like network for learning the moduli dependence of the \(H\) matrix.

ml.HNetMLP(basis_size, layer_sizes[, ...])

Dense network for learning moduli dependence of the H matrix.

A schematic overview of the MLP-like network is given below.