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Although dynamical systems models are a powerful tool for analysing microbial ecosystems, challenges in learning these models from complex microbiome datasets and interpreting their outputs limit use. We introduce the Microbial Dynamical Systems Inference Engine 2 (MDSINE2), a Bayesian method that learns compact and interpretable ecosystems-scale dynamical systems models from microbiome timeseries data. Microbial dynamics are modelled as stochastic processes driven by interaction modules, or groups of microbes with similar interaction structure and responses to perturbations, and additionally, noise characteristics of data are modelled. Our open-source software package provides multiple tools for interpreting learned models, including phylogeny/taxonomy of modules, and stability, interaction topology and keystoneness. To benchmark MDSINE2, we generated microbiome timeseries data from two murine cohorts that received faecal transplants from human donors and were then subjected to dietary and antibiotic perturbations. MDSINE2 outperforms state-of-the-art methods and identifies interaction modules that provide insights into ecosystems-scale interactions in the gut microbiome.