The non-equilibrium dynamics of mesoscale phase transitions are fundamentally shaped by thermal fluctuations, which not only seed instabilities but actively control kinetic pathways, including rare barrier-crossing events such as nucleation that are entirely inaccessible to deterministic models. Machine-learning surrogates for such systems must therefore represent stochasticity explicitly, enforce conservation laws by construction, and expose physically interpretable structure. We develop physics-aware surrogate models for the stochastic Cahn-Hilliard equation in 3D that satisfy all three requirements simultaneously. The key innovation is to parameterize the surrogate at the level of inter-cell fluxes, decomposing each flux into a deterministic mobility-weighted chemical-potential gradient and a learnable noise amplitude. This design guarantees exact mass conservation at every step and adds physical fluctuations to inter-cell mass transport. A learnable free energy functional provides thermodynamic interpretability, validated by independent recovery of the bulk double-well landscape, interfacial excess energy, and curvature-independent interfacial tension. Tests demonstrate accurate reproduction of ensemble statistics and noise-accelerated coarsening, with generalization to spatial domains 64 times larger in volume and temporal horizons 160x longer than those seen during training. Critically, the stochastic surrogate captures thermally activated nucleation in the metastable regime, a qualitative capability that no deterministic surrogate can provide regardless of training, thus establishing flux-level stochasticity as an architectural necessity rather than an optional enhancement.