Mean field games efficiently approximate a very large population of strategic agents. While these games can aid the understanding of complex systems, their deployment in real-world settings is challenged by the specification of their parameters: mean field games (MFGs) often involve hidden preferences, constraints, and interactions that can rarely be theoretically derived or directly observed. To address this gap, we present a neural network-based framework for learning parametric, finite-state MFGs from observed population dynamics. To do so, we formulate the parameter calibration as an inverse problem and use implicit differentiation to backpropagate through the games' equilibrium. The resulting approach is fully differentiable and enables us to estimate flexible trajectory-wise parameter paths, including state- and time-dependent specifications without requiring observations of the individual agents' actions or rewards. We provide a proof for the exactness of the gradient computation in a discrete-time formulation. We validate our framework through numerical experiments across four systems of increasing complexity, ranging from synthetic linear-quadratic benchmarks to real-world urban mobility datasets.

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