Artificial Neural Network in Cosmic Landscape

TL;DR

The paper tackles the challenge of generating high-dimensional inflationary landscapes by proposing artificial neural networks as a global, efficient generator for the potential $V(\varphi)$. It leverages the universal approximation theorem to construct random functions with polynomial complexity and validates the approach through a toy multi-field inflation model, analyzing landscape properties such as volume, rotational symmetry, and Taylor-coefficient statistics. The cosmological application demonstrates a 20-field inflation scenario with $V(\varphi)=V_0(\varphi_1)+V_{rand}(\varphi)$ and $V_0(\varphi_1)=\tfrac{1}{2} m^2 \varphi_1^2$, showing slow-roll dynamics and activation-dependent ruggedness. This framework offers a scalable method for exploring high-dimensional cosmological landscapes and points toward richer networks and broader models in future work.

Abstract

In this paper we propose that artificial neural network, the basis of machine learning, is useful to generate the inflationary landscape from a cosmological point of view. Traditional numerical simulations of a global cosmic landscape typically need an exponential complexity when the number of fields is large. However, a basic application of artificial neural network could solve the problem based on the universal approximation theorem of the multilayer perceptron. A toy model in inflation with multiple light fields is investigated numerically as an example of such an application.

Figures (6)

• Figure 1: Potential surface with null, small and large randomness. We choose $\Delta_x\varphi=0.05$, $m=10^{-6}$, $\varphi_{\max}=30$, and $\mathcal{A}=0,0.1,1$ respectively for the left, middle and right figure. The distribution is uniform for random numbers in the interpolation function, and the interpolation is for $C^1$ functions.
• Figure 2: Slices of two 100-dimensional random functions. We generate 100-dimensional random functions with parameters $\alpha=50$, $\beta=50$, $\omega=50$, $h=1000$ for sigmoid (left) and sine (right) activation functions respectively. We plot the function $A_h(0,0,\ldots,x,y)$ (The first 98 components of the variable vector are all zero and the last two components are chosen to be $x$ and $y$.)
• Figure 3: The probability distribution of the Taylor series coefficients $a_{10}$ and $a_{11}$ for neural network in two dimension. We consider the sigmoid activation (upper) and the sine activation (lower) respectively. The data is constructed via $x_0=y_0=1$, $\alpha=100$, $\beta=100$, $w=1/h$ and $h=100$, and we realize the network for 1000 times.
• Figure 4: The absolute value of (linear) correlations between Taylor series coefficients. We consider the sigmoid activation (upper) and the sine activation (lower) respectively. The data is constructed via $x_0=y_0=1$, $\alpha=100$, $\beta=100$, $w=1/h$ and $h$ changes from 100 to 1000, and we realize the network for 1000 times.
• Figure 5: The evolution of trajectories for dimension 20. In this case we choose the sigmoid activation function and we set $\alpha=\beta=50$, $\omega=1$, $\mathcal{A}=10^{-14}$ and $h=100$. And also we set $V_\text{rand}=A_h(\varphi_1-\varphi_\text{max}/2,(\varphi_{i\ne 1}-\varphi_\text{max}/2)\times 10)$. We show the trajectories $\varphi_1(k)$ (left) as a function of time step $k$ for inflaton, and $\varphi_\text{another}(k)$ (left) for another light field.