Infinitesimal Operator Discovery for Continuous-Depth Networks

Continuous-depth neural networks replace a finite stack of layers with a learned vector field, creating a natural mathematical view of deep computation but obscuring which infinitesimal transformations are used after training. This work introduces Infinitesimal Operator Discovery (IOD), a diagnostic framework that analyzes the local variational dynamics of a trained continuous-depth model by projecting its Jacobian action onto constrained structural coordinates: diagonal modulation, topology-aware local mixing, low-rank global mixing, sparse nonlocal mixing, and an explicit residual that quantifies unexplained dynamics. The coordinates are deliberately structural rather than full architectural blocks, so descriptive reconstruction is separated from causal evidence. The audit combines residual fit, null baselines, repeated random controls, decision-boundary interventions, dictionary ablations, solver checks, and rank/radius sensitivity. On a controlled structured benchmark spanning 10 task families and 80 task-seed replications, IOD obtains low residual energy (mean 0.0138), operator profiles that are distinct from untrained, label-shuffled, and random-matched controls, and consistent causal support for low-rank global operators. Local operators are also task-relevant in most settings, while sparse nonlocal and residual coordinates remain negligible in the present benchmark. Boundary-focused interventions have larger effects than high-margin interventions, indicating diagnostic value for decision-sensitive examples. These results support a focused contribution: continuous-depth models can be made more transparent by analyzing their infinitesimal operator structure through a conservative, causal, and reproducible diagnostic coordinate system.