Gravity transformed the small initial fluctuations into the complex, non-linear cosmic web. The late-time large-scale structures contain a wealth of information about fundamental physics, but the statistical inference is difficult. I will introduce two research directions to deal with the difficulties. One is simulation-based inference: A machine-learning technique that allows to constrain theory from data in the absence of an explicit likelihood. After a general introduction to the technique, I will argue that multi-fidelity training is a necessary step towards trustworthy results and explain our recent work on it. The second direction is constrained information extraction, exemplified by my recent work on cosmic void finding. I will argue that unconstrained information extraction is unlikely to yield robust constraints. Instead, guardrails should be provided. I will demonstrate this idea using a graph neural network that is trained with a flow-matching objective to find cosmic voids in sparse galaxy redshift surveys.
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