Physical Benchmarking for AI-Generated Cosmic Web

The potential of deep learning based image-to-image translations has recently drawn a lot of attention; one intriguing possibility is that of generating cosmological predictions with a drastic reduction in computational cost. Such an effort requires optimization of neural networks with loss functions beyond low-order statistics like pixel-wise mean square error, and validation of results beyond simple visual comparisons and summary statistics. In order to study learning-based cosmological mappings, we choose a tractable analytical prescription - the Zel'dovich approximation - modeled using U-Net, a convolutional image translation framework. A comprehensive list of metrics is proposed, including higher-order correlation functions, conservation laws, topological indicators, dynamical robustness, and statistical independence of density fields. We find that the U-Net approach does well with some metrics but has difficulties with others. In addition to validating AI approaches using rigorous physical benchmarks, this study motivates advancements in domain-specific optimization schemes for scientific machine learning.