Neural population dynamics are jointly shaped by numerous factors at the cellular, synaptic, and network levels. Mechanistic models, especially spiking neural networks, are crucial for understanding how these interacting variables influence neural activity. However, the process of adjusting model parameters to match experimental recordings, i.e., inverse modeling, is challenging, and often requires a trade-off between model complexity and tractability. Consequently, models often use fixed or manually tuned parameters, limiting their dynamic range and explanatory power while making cross-study comparisons difficult. Here, we introduce Automated Model Inference from Neural Dynamics (AutoMIND), a simulation-based inference framework for identifying candidate circuit model configurations from neural population recordings. The forward model is a network of clustered adaptive integrate-and-fire neurons, which exhibits diverse population dynamics endowed by its many free parameters. For inverse modeling, deep generative models are trained solely on network simulations, which can then return many configurations consistent with experimental data. We apply AutoMIND to several datasets, obtaining models of synchronous network bursting in human brain organoids during early development, as well as models consistent with complex, broadband frequency profiles from Neuropixels recordings in mouse cortex and hippocampus. In each case, multiple valid model configurations are identified for a single observation—even for diverse synthetic recordings with known parameters—which define a parameter subspace where dynamics remain invariant. The geometry of invariances is critical, but surprisingly, depends on the specific target dynamic, rather than being a global property of the model parameterization. Thus, AutoMIND facilitates efficient inverse modeling, while shedding light on state-dependent invariance of circuit mechanisms underlying neural population dynamics.