Training Neural ODEs by Path Matching

The prohibitive training cost of Neural ODEs (NODEs) stems from backpropagation through the ODE solver, whose cost scales with the number of function evaluations (NFE). Flow Matching (FM) removes this backpropagation by reframing training as field regression, but supervising the field on pre-defined trajectories (e.g. straight lines) creates a path-crossing conflict: on paired data the field simply averages conflicting velocities. We present Path Matching (PM), which resolves this conflict by matching the field on-policy: rather than regressing onto a prescribed path, we roll out the model and supervise the field at the states $x_t$ it actually visits, with a bridge target that realigns a drifted rollout back onto a target path (a straight line between endpoints by default). On image classification and a real, irregularly-sampled clinical time-series task, PM reaches accuracy on par with end-to-end NODE training at a fraction of the per-step cost and with no backprop through the solver, an efficiency we characterize with a runtime scaling law. A latent-dimension ablation then explains why the simulation-free alternative is fragile: it fails when a straight coupling cannot avoid path crossings—a limitation PM's curvable, on-policy objective escapes.