Based on the Onsager variational principle, we reformulate the Cahn-Hilliard equation as a gradient flow of free energy dissipation and establish the variational triple concentration $c$, chemical potential $\mu$, and flux $\mathbf{J}$. A physics-informed neural network jointly outputs these three fields. An energy convex splitting architecture decomposes the free energy into convex and concave components, correspondingly splitting $\mu$ and $\mathbf{J}$ under independent supervision. This decouples interfacial relaxation from phase separation driving forces and inherently suppresses numerical stiffness. Numerical experiments show that the proposed method successfully captures the full coalescence of twin elliptic droplets: rounding, approach, merging, and coarsening, whereas the standard PINN remains trapped in a metastable separated state. This work provides a systematic variational framework for deep learning simulation of phase-field equations.