Neural Network Learning and Quantum Gravity

TL;DR

This work proposes that the string theory landscape, while vast, possesses finiteness properties that render many Quantum Gravity learning problems tractable. By combining statistical learning theory (VC and fat-shattering dimensions) with tame geometry via o-minimal structures, the authors argue that low-energy EFTs from quantum gravity are learnable by neural networks because their couplings and interactions are definable in a fixed o-minimal structure. They show that finite shattering dimensions lead to concrete sample-complexity bounds for both classification and regression tasks, and they illustrate this with problems such as reconstructing gauge couplings and assessing slow-roll inflation. The framework lays groundwork for a systematic identification of learnable properties in the string landscape and provides a formal bridge between QG physics and data-driven inference, while noting that decidability remains an open complement to learnability.

Abstract

The landscape of low-energy effective field theories stemming from string theory is too vast for a systematic exploration. However, the meadows of the string landscape may be fertile ground for the application of machine learning techniques. Employing neural network learning may allow for inferring novel, undiscovered properties that consistent theories in the landscape should possess, or checking conjectural statements about alleged characteristics thereof. The aim of this work is to describe to what extent the string landscape can be explored with neural network-based learning. Our analysis is motivated by recent studies that show that the string landscape is characterized by finiteness properties, emerging from its underlying tame, o-minimal structures. Indeed, employing these results, we illustrate that any low-energy effective theory of string theory is endowed with certain statistical learnability properties. Consequently, several learning problems therein formulated, including interpolations and multi-class classification problems, can be concretely addressed with machine learning, delivering results with sufficiently high accuracy.

Figures (8)

• Figure 1: On the left is depicted a data set composed of the points $\{(\varphi_i, g_i)\}$. After applying a machine learning algorithm, one can infer the best model $\hat{g}(\varphi)$ that interpolates the data set, such as the one depicted on the right.
• Figure 2: On the left is a data set composed of points in the parameter space spanned by $c_1$ and $c_2$: the purple dots are those for which slow-roll inflation is not realized, while the light blue squares denote the points at which the slow-roll conditions are satisfied. On the right is an example of a boundary, which can be obtained via machine learning, that separates the regions of the parameter space in which inflation can or cannot be realized.
• Figure 3: On the left is depicted a data set for a two-dimensional input space. The labels $\{0,1\}$ associated with a given point are here encoded in the shape and color of a point: a point with label $1$ is represented with a blue circle, and a point with label $0$ is represented as an orange square. On the right, the example of a function $g(x; \hat{\omega})$ that defines a partition of $\mathbb{R}^2$ via the locus $g(x; \hat{\omega}) = 0$ such that points above have predicted label $1$, and those below predicted value $0$. The function $g(x; \hat{\omega})$ depends on the estimated parameters and, using $g(x; \hat{\omega})$, one can define the actual model \ref{['DL_function_best']}. Notice that, in general, the model does not predict the labels exactly.
• Figure 4: In $\mathbb{R}^2$, a line can always create a partition into two subsets such that three (non-collinear) data points with different labels (here, one label is represented by the data point being a blue circle, the other by the data point being an orange square) fall into the two distinct subsets. However, given a set of four points, a single line cannot always create a partition of $\mathbb{R}^2$ such that four points with different labels fall into the different partition components.
• Figure 5: An example of $\gamma$-shattering for two data points in $\mathbb{R}$. Given $\gamma$, there exist two real numbers, $r_1$ and $r_2$, such that one can introduce four functions $f_{00}$, $f_{10}$, $f_{01}$ and $f_{11}$ obeying \ref{['VC_gamma-shatt_f']}. Here, the subscripts of the functions denote the fictitious labels associated with $x_1$ and $x_2$.