Accurate estimation of treatment effects is crucial for decision-making in various scientific disciplines. However, this task becomes challenging in settings such as social sciences or online marketplaces, where treating one experimental unit can impact the outcomes of other units through direct or indirect interactions. This interaction leads to a dynamic of changing outcomes over time until they reach equilibrium. These interactions also result in biased estimates of the treatment effect that are especially challenging to avoid when the interference structure is unknown. We address this issue by proposing a new class of estimators based on the theory of approximate message-passing, specifically designed for settings with pervasive unknown interference. Our estimator utilizes information from the distribution of unit outcomes and treatments observed over time to estimate the evolution of the system state and ensure efficient use of the observed data. Concretely, we first create time-dependent non-linear basis features from moments of these distributions and learn a function that maps these to the mean and variance of the outcomes in the future periods. This approach enables the estimation of counterfactual quantities related to the treatment effect dynamics, including the total treatment effect and its variance over time. Extensive numerical simulations across multiple domains, using both synthetic and real network data, demonstrate the effectiveness of our approach in estimating treatment effect dynamics throughout the experiment.