Learning Collective Dynamics with Differentiable Gaussian Representations

Social-media interactions and market transactions arise from individuals with different response propensities: contact opportunities determine which individuals generate observed events, and accumulated experience changes later responses. Predicting collective dynamics from aggregated event and behavior counts therefore requires learning how individual responses sum to current totals and how feedback shapes later behavior. Because population type composition, contact selection, and experience-induced propensity changes correspond to the components, intensity reweighting, and translation of a Gaussian mixture, we represent response heterogeneity with a trainable Gaussian mixture. Evaluating the intensity of arrivals (recorded contacts or transactions) and conditional behavioral probabilities at the same latent type, the model integrates their product to predict arrival counts, joint behavioral probabilities, and behavioral totals; observed or expected outcomes then update the state. Reparameterized quadrature and cross-day recurrence let predictive losses jointly train the representation, observation functions, and feedback parameters. We also derive conditions for count-likelihood sufficiency and parameter equivalence, and analyze cross-day propagation of quadrature error. On four KuaiRand-Pure and Online Retail II windows, the model has lower joint behavioral negative log-likelihood (NLL) than a DeepAR adaptation with a joint-behavior head, with 1.1232 versus 1.3466 in the KuaiRand-Pure standard-recommendation window; in Retail 2010, one-day count MAE averaged over three basket marks is 4.71 versus 6.88. In the standard-recommendation ablation, learning the distribution reduces behavioral NLL by 10.82% versus a fixed Gaussian; in a controlled dynamics experiment, removing dynamics raises mean joint KL from 0.0340 to 0.2577.