Flow Matching for Averaged Systems

We extend flow matching to ensembles of linear systems in both deterministic and stochastic settings. Averaging over system parameters induces memory leading to a non-Markovian interpolation problem for the stochastic case. In this setting, a control law that achieves the distributional controllability is characterized as the conditional expectation of a Volterra-type control. This conditional expectation in the stochastic settings motivates an open-loop characterization in the noiseless-deterministic setting. Explicit forms of the conditional expectations are derived for special cases of the given distributions and a practical numerical procedure is presented to approximate the control inputs. A by-product of our analysis is a numerical split between the two regimes. For the stochastic case, history dependence is essential and we implement the conditional expectation with a recurrent network trained using independent sampling. For the deterministic case, the flow is memoryless and a feedforward network learns a time-varying gain that transports the ensemble. We show that to realize the full target distribution in this deterministic setting, one must first establish a deterministic sample pairing (e.g., optimal-transport or Schrodinger-bridge coupling), after which learning reduces to a low-dimensional regression in time.