Learning Individual Dynamics from Sparse Cross-Sectional Snapshots

The paper introduces CADENCE, a probabilistic framework for recovering individualized continuous-time trajectories from extremely sparse, cross-sectional snapshots plus static context, resolving the usual split between dense sequence models like Latent ODEs and population transport methods such as optimal transport and flow matching. Its key novelty is an identifiability theory showing that single-timepoint trajectory inference is possible when the ensemble has contextual structure: a Dynamical Foliation Assumption partitions context space into leaves with shared ODE parameters, Context Observability makes leaf membership readable from static covariates, and a Manifold Regularity Assumption justifies representing dynamics as a convex mixture over a finite expert set via a Soft Mixture-of-Experts router. Spatial ambiguity is removed by a score-based Probability Flow ODE encoder, which collapses latent symmetry from the full diffeomorphism group to Gaussian-preserving isometries. Training is decoupled into a score-matching spatial stage and a low-dimensional temporal stage, with either a semi-longitudinal latent MSE loss or a cross-sectional MMD loss over temporally smoothed subgroup marginals. Across seven benchmarks spanning Lotka–Volterra, Van der Pol, SIR, gene expression, SERGIO, and the LARRY haematopoiesis dataset, CADENCE trained strictly on sparse snapshots matches or exceeds dense-trajectory baselines; on LARRY it reaches 67.8 percent fate prediction accuracy, outperforming scDiffEq at 58.5 percent, and on BM1 it cuts MAE from 27.47 for OT-CFM to 2.05 while also improving routing accuracy to 94 percent.