Collective responses depend on individual differences, contact opportunities, and accumulated experience.
Learning their dynamics from aggregate counts requires connecting a population's response distribution to both current observations and future behavior.
We introduce Differentiable Gaussian Dynamics (DGD)
Which learns this connection through three components:
- a Gaussian mixture representing heterogeneous response propensities,
- differentiable aggregation of contact intensity and behavioral probabilities,
- and feedback recurrence that updates subsequent responses.
Reparameterized integration and temporal recurrence let aggregate prediction errors jointly train the distribution, observation functions, and feedback parameters.
On four windows from KuaiRand-Pure and Online Retail II
DGD achieves lower joint behavioral negative log-likelihood than a DeepAR adaptation with a joint-behavior head.
In Retail 2010, its one-day behavioral-count MAE is 4.71 versus 6.88 for this adaptation.
Learning the distribution reduces behavioral negative log-likelihood by 10.82% relative to a fixed Gaussian in KuaiRand's standard-recommendation window;
Removing feedback dynamics raises joint KL from 0.0340 to 0.2577 in a controlled experiment.
These results establish the value of learning population representations and their feedback process from aggregate observations.
Code is available at https://github.com/OranAi-Ltd/oransim.