Observational data relevant for astroparticle physics and astrophysical searches for dark matter becomes increasingly complex and detailed. We are in a situation where often what we can learn from new observations is limited not by the amount of data, but by the sophistication of our analysis tools and the quality and detail of our physical models. Classical statistical techniques, like Markov Chain Monte Carlo, severely limit model realism and complexity, due to their high simulation requirements and limitation on the number of free parameters. Neural simulation-based inference algorithms have the capability to break through these barriers in surprising ways. However, using these new classes of algorithms without compromising the precision and accuracy of statistical inference results remains challenging. I will present examples from strong gravitational lensing and cosmology, and discuss successes and typical pitfalls.