GraphFM: Graph-Constrained Flow Matching for Dynamics Reconstruction from Temporal Snapshots

Reconstructing continuous population dynamics from sparse, unpaired temporal snapshots is a fundamental yet underdetermined problem in many scientific applications. Many vector fields can reproduce the same finite collection of temporal marginals. Existing flow-matching methods reduce this ambiguity through assumptions on endpoint couplings, interpolation geometry, or temporal smoothness, but typically treat the variables within each state as unstructured. This overlooks known interaction structure in systems such as gene-regulatory and road networks. We introduce Graph-Constrained Flow Matching (GraphFM), which incorporates a variable-interaction graph into both conditional interpolation and vector-field learning. GraphFM parameterizes an endpoint-constrained graph neural interpolant whose message-passing structure restricts each variable to depend only on its graph-defined receptive field. To exploit multiple observed marginals, we train the interpolant with a hierarchical objective combining distributional matching at observed time points, multi-anchor trajectory regularization induced by a multi-marginal coupling, and local temporal smoothness. The derivatives of the learned trajectories then supervise a graph-aware velocity field defined over the same interaction graph. We evaluate GraphFM on synthetic dynamics, singlecell trajectory inference, spatial-transcriptomic interpolation, and traffic-state imputation. GraphFM improves or matches strong multi-marginal flow-matching baselines across these settings, while graph-corruption experiments demonstrate the value of informative structural priors. Our results show that variable-level interaction structure provides a effective inductive bias for reconstructing dynamics from temporal snapshots.