The consensus ΛCDM (Lambda-Cold Dark Matter) model of cosmology has shown remarkable explanatory power over a variety of cosmic scales and epochs, and it narrates a reassuring story of a universe currently filled mostly with dark matter and dark energy. Yet, this explanation is not fully satisfactory because the actual nature of the dark components remains a puzzle. Furthermore, cosmologists have recently reported significant anomalies concerning the delicate balance of cosmic expansion and structure growth, without a compelling solution. The main objective of the MAPEX project is to reassess this far-reaching problem from a new perspective, and determine if cosmological tensions can be traced to the most extreme cosmic web environments: deep voids and dense superclusters. This EU-funded action will allow me to access unprecedented new data taken at the Vera Rubin Observatory, solidifying and broadening the Hungarian contributions to the next-generation Legacy Survey of Space and Time (LSST) project based in Chile. To go beyond the state-of-the-art, I will acquire extensive skills on machine learning techniques from expert researchers at Konkoly Observatory to combine with my groundwork results on cosmological data analysis from, above all, the Dark Energy Survey (DES). As a key innovation, I will develop deep learning models to study extreme voids and superclusters. First, I will apply convolutional neural network methods to augment traditional cross-correlations between galaxy density fluctuations and the anisotropies of the Cosmic Microwave Background. Then, I will capture the dependence of their gravitational signals on the physical properties of dark energy and dark matter. The proposed analyses of simulations and early observational LSST data will help resolve whether some as-yet unknown physical effects or systematic biases complicate the picture in cosmology. Either way we will gather fundamentally new knowledge about the Universe on the largest scales.