Predicting the Initial Conditions of the Universe using a Deterministic Neural Network

Finding the initial conditions that led to the current state of the universe is challenging because it involves searching over an intractable input space of initial conditions, along with modeling their evolution via tools such as N-body simulations which are computationally expensive. Recently, deep learning has emerged as a surrogate for N-body simulations by directly learning the mapping between the linear input of an N-body simulation and the final nonlinear output from the simulation, significantly accelerating the forward modeling. However, this still does not reduce the search space for initial conditions. In this work, we pioneer the use of a deterministic convolutional neural network for learning the reverse mapping and show that it accurately recovers the initial linear displacement field over a wide range of scales ($<1$-$2\%$ error up to nearly $k\simeq0.8$-$0.9 \text{ Mpc}^{-1}h$), despite the one-to-many mapping of the inverse problem (due to the divergent backward trajectories at smaller scales). Specifically, we train a V-Net architecture, which outputs the linear displacement of an N-body simulation, given the nonlinear displacement at redshift $z=0$ and the cosmological parameters. The results of our method suggest that a simple deterministic neural network is sufficient for accurately approximating the initial linear states, potentially obviating the need for the more complex and computationally demanding backward modeling methods that were recently proposed.