Why X posts stall then surge: a fractional physics-informed network splits 24 hours into four windows and lifts the memory order from 0.23 to 0.99
Synopsis
Models of social media spread have either prescribed a fixed growth function or assumed a fixed way in which past activity fades. A fractional physics-informed neural network writes each post's engagement rate as a per-post viral potential times one shared growth law, with a learnable fractional derivative setting how strongly history matters. On 348 training and 86 test posts from X, splitting the 24-hour window into four separately trained segments let the model reconstruct 64% of held-out trajectories within a mean squared error of 0.01, against 44% for Bass diffusion, the strongest of nine classical baselines.
Interpretation
The shape of a spread curve is no longer fixed in advance: the data decide both the form of the growth law and how strongly a post's history drives later engagement. Earlier growth-curve models such as logistic, Gompertz and Bass diffusion commit to a functional form before seeing data and are first-order Markovian, so the future depends only on current engagement; Hawkes-style self-exciting point processes use the full history but require the influence-decay kernel to be specified beforehand. Training used 348 X posts and evaluation used 86 held-out posts, stratified by peak-engagement decile, with each curve max-normalized so only shape is compared; nine classical baselines were fit and evaluated under the same protocol.
The model writes a post's engagement rate as its viral potential multiplied by one growth law shared across all posts, while a fractional derivative carries the memory of history. A fractional derivative generalizes the ordinary derivative by integrating over the whole prior trajectory with a power-law kernel, so a smaller order means stronger memory; here the order is learned from data rather than set in advance. At the deployment resolution of 24 hourly observations, synthetic trajectories with known parameters first confirmed that the order and reaction law can be jointly recovered; under observation noise the pointwise residual drove the order to its lower bound, and a fully whitened residual that propagates observation noise through the fractional operator recovered it.
A single window cannot describe both the fast growth early in a post's life and the decay later, and cutting the 24 hours into four segments with separate parameters raises reconstruction accuracy. The single-window model converged to an order of 0.23, implying strongly memory-dominated dynamics, yet on held-out posts it did worse than a straight line from zero to the peak; the four segments yielded orders of 0.23, 0.74, 0.99 and 0.99. The segmented model reconstructed 64.0% of held-out trajectories within a mean squared error of 0.01, against 44.2% for Bass diffusion and 34.9% for Gompertz; refitting the classical growth curves piecewise on the same four windows raised them by at most 8.1 percentage points and still left them below the segmented model.
Perspective
The conclusions are limited to X posts found through 88 keyword queries spanning technology, AI/ML and startups, collected from November 2025 through January 2026, with 348 training and 86 test posts used; evaluation compares only max-normalized curve shape and does not predict final engagement magnitude.
The authors note the sample is small relative to X's daily volume of new posts and concentrated in technology and startup topics, so other topics may follow different virality dynamics; X's recommendation algorithm differs from other platforms and is frequently updated, so the parameters hold only within the collection window. At the highest noise level the joint fit degrades, which the authors attribute to a flexible absorbing noise structure rather than to the gain problem.