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tokens: B ), which can be used not only to capture complicated patterns between variables and to keep a good balance between bias and variance, but also to identify the most important variables that explain model predictions. We address the second problem by using a linear dimension reduction algorithm, the principal component analysis (PCA; see Appendix A ), which can effectively reduce the dimension of the halo AH and the galaxy SFH and yet retain large amounts of information of the histories for the empirical modelling. Finally, we address the third problem by building a deep model that incorporates components of both dimension reduction and ensemble regressor and classifier. Each component in the model is motivated physically and can be optimized separately. This approach makes the model capable of dealing with complex patterns in parameter space, and yet transparent to interpret. The ensemble regressor and linear dimension reduction method has already been used to study the relationship among halo properties in Chen et al. ( 2020 ). Here, we extend it to studying the galaxy–halo connection. As the first in a series, this paper focuses on central galaxies in dark matter haloes. We identify important ingredients that should be included in an empirical model, and demonstrate the limit such a model can reach in describing the stellar masses and SFHs of individual galaxies. Our model is built on the inferences from two hydrodynamic simulations, the Illustris-TNG (e.g. Nelson et al. 2019 ) and EAGLE (e.g. The EAGLE team 2017 ). This paper is organized as follows. In Section 2 , we describe the simulation data we use, and define the halo properties and samples used in our analysis. In Section 3 , we use both the GBDT and PCA to study the relations of galaxy stellar mass and SFR with halo properties for both star-forming galaxies and quenched galaxies. We also identify halo properties that cause a galaxy to quench. In Section 4 , we build an empirical model that predicts the SFH of galaxies in dark matter haloes, testing its performance in several steps. We summarize and discuss our results in Section 5 . 2 THE DATA 2.1 The Illustris-TNG and EAGLE simulations The Illustris-TNG simulation (Marinacci et al. 2018 ; Naiman et al. 2018 ; Nelson et al. 2018 , 2019 ; Springel et al. 2018 ; Pillepich et al. 2018b ) is a suite of cosmological, hydrodynamical simulations implemented with the moving-mesh code Arepo (Springel 2010 ). The cosmological parameters are taken from the Planck 2015 results (Planck Collaboration XIII 2015 ): Hubble constant with h = 0.6774, cosmological constant Ω Λ,0 = 0.6911, matter density Ω M,0 = 0.3089, baryon density Ω B,0 = 0.0486, and initial power spectrum with normalization σ 8 = 0.8159 and index n s = 0.9667. The simulated physical processes for galaxy formation include gas cooling, star formation, stellar feedback, metal enrichment, black hole feedback, and so on. The details can be found in the two method papers, Weinberger