We introduce a generalizable framework for identifying effective Hamiltonians directly from experimental data in solid-state quantum systems. Our unsupervised autoencoder-based approach incorporates the governing physics (here, the S-matrix formalism) directly into the decoder, enabling physically meaningful Hamiltonian inference without labeled Hamiltonian parameters. Through numerical experiments on a triple quantum dot chain, we demonstrate automated characterization of programmable solid-state simulators from transport measurements. The model accurately infers Hamiltonian parameters and generalizes beyond the training domain, while exhibiting finite capacity as the parameter space becomes increasingly broad or diverse. We further demonstrate robustness to noisy measurements and show that the model can be trained on transport data generated outside the Hamiltonian family assumed by the physics decoder, while still identifying meaningful effective parameters.