Cosmo3DFlow: Wavelet Flow Matching for Spatial-to-Spectral Compression in Reconstructing the Early Universe

Reconstructing the early universe from the evolved present-day universe is a challenging and computationally demanding problem in modern astrophysics. We devise a novel generative framework, Cosmo3DFlow, designed to address dimensionality and sparsity, the critical bottlenecks inherent in current state-of-the-art methods for cosmological inference. By integrating 3D Discrete Wavelet Transform (DWT) with flow matching, we effectively represent high-dimensional cosmological structures. The Wavelet Transform addresses the ``void problem'' by translating spatial emptiness into spectral sparsity. It decouples high-frequency details from low-frequency structures, and wavelet-space velocity fields facilitate stable ordinary differential equation (ODE) solvers with large step sizes. Using large-scale cosmological $N$-body simulations at $128^3$ resolution, we achieve up to $46\times$ faster sampling than diffusion models. Our results enable initial conditions to be sampled in seconds, compared to minutes for previous methods.