Finding the initial conditions that led to the current state of the universe is challenging because it involves searching over a vast input space of initial conditions, along with modeling their evolution via tools such as N-body simulations which are computationally expensive. Deep learning has emerged as an alternate modeling tool that can learn the mapping between the linear input of an N-body simulation and the final nonlinear displacements at redshift zero, which can significantly accelerate the forward modeling. However, this does not help reduce the search space for initial conditions. In this paper, we demonstrate for the first time that a deep learning model can be trained for the reverse mapping. We train a V-Net based convolutional neural network, which outputs the linear displacement of an N-body system, given the current time nonlinear displacement and the cosmological parameters of the system. We demonstrate that this neural network accurately recovers the initial linear displacement field over a wide range of scales ($<1$-$2\%$ error up to nearly $k = 1\ \mathrm{Mpc}^{-1}\,h$), despite the ill-defined nature of the inverse problem at smaller scales. Specifically, smaller scales are dominated by nonlinear effects which makes the backward dynamics much more susceptible to numerical and computational errors leading to highly divergent backward trajectories and a one-to-many backward mapping. The results of our method motivate that neural network based models can act as good approximators of the initial linear states and their predictions can serve as good starting points for sampling-based methods to infer the initial states of the universe.

翻译：寻找导致宇宙当前状态的初始条件极具挑战性，因为这涉及对庞大初始条件输入空间的搜索，同时需通过N体模拟等工具（计算成本高昂）对其演化过程进行建模。深度学习作为一种替代建模工具崭露头角，它能够学习N体模拟线性输入与红移为零时的最终非线性位移之间的映射关系，从而显著加速正向建模。然而，这种方法并未帮助缩小初始条件的搜索空间。本文首次证明，深度学习模型可通过训练实现反向映射。我们训练了一个基于V-Net的卷积神经网络，该网络根据当前时刻的非线性位移及系统的宇宙学参数，输出N体系统的线性位移。我们证明，尽管小尺度上逆问题的病态特性显著——非线性效应主导小尺度动力学，导致反向轨迹高度发散且存在一对多反向映射——该神经网络仍能在宽泛尺度范围内精确恢复初始线性位移场（误差<1%~2%，最高至$k = 1\ \mathrm{Mpc}^{-1}\,h$）。我们的方法结果表明，基于神经网络的模型可作为初始线性状态的良好近似器，其预测能为基于采样的方法推断宇宙初始状态提供优质起点。