Existing models of social media virality either prescribe the functional form of engagement growth or assume a fixed mechanism by which past activity influences future engagement. We present a fractional physics-informed neural network that learns both the temporal memory and nonlinear growth dynamics of social media engagement, using symbolic regression to translate the learned dynamics into interpretable equations. Synthetic experiments at the deployment resolution of 24 hourly observations demonstrate that the fractional order and reaction law can be jointly recovered from clean data. Under observation noise, however, the standard pointwise physics residual drives the fractional order toward its lower bound through a scale artifact. We address this failure using a fully whitened residual that propagates observation noise through the fractional operator. Applying the framework to 348 posts from X (formerly Twitter), we find that a single-window model estimates strongly memory-dependent dynamics but fails to accurately reconstruct held-out trajectories. Partitioning the 24-hour window into four teacher-forced segments substantially improves performance, yielding segment-specific fractional orders that increase from 0.23 to 0.99 and interpretable growth laws for each phase. The segmented model reconstructs 64% of held-out trajectories within a mean squared error of 0.01, compared with 44% for the strongest of nine classical baselines. Together, these results demonstrate the potential of fractional physics-informed equation discovery to uncover interpretable engagement dynamics while highlighting the importance of noise-aware estimation and temporal segmentation.

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