Phase separation governs microstructural evolution in multiphase materials and strongly influences their macroscopic properties. To date, these relationships have largely been investigated in a forward fashion: given an assumed free-energy functional and associated kinetics, one predicts the resulting microstructural features. The inverse problem—inferring the underlying free energy landscape and kinetic parameters from observed microstructures—is inherently ill-posed, yet critically important for the analysis and design of advanced materials, such as polymer blends, nanoporous alloys and even multi-physics ceramics. As a first step toward addressing this inverse problem, we focus on diffusion-driven phase separation and introduce a framework for identifying the underlying free-energy functional and associated kinetics directly from sparse snapshots of microstructural evolution. Given a microstructure, we first extract salient features using a diverse set of descriptors, including statistical measures (e.g., two-point correlation functions), topological invariants (e.g., Betti numbers), geometric metrics (e.g., interfacial length, curvature, and pore-size distributions), and latent representations obtained from a pre-trained ResNet neural network. These descriptors are then provided as input to a neural network that predicts the free-energy function, represented via a B-spline parameterization, along with additional kinetic parameters. The model is trained on a representative dataset and validated through forward simulations that reproduce microstructures with the target statistical and morphological characteristics. To address the inherent non-uniqueness of the inverse mapping and enable uncertainty quantification, we employ conditional normalizing flows to learn the full posterior distribution of free-energy and kinetic parameters conditioned on the microstructural descriptors. Together, these results establish an uncertainty-aware, descriptor-based learning framework for free-energy and kinetics inference, with direct applicability to the analysis of experimental microstructural images and robust data-driven materials design.