Least-Action Causal Discovery via SDEs for Stochastic Dynamical Systems

We introduce the Principle of Least-Action Causal Identification of Dynamics (PLACID), a controlled-SDE testbed for learning latent dynamics and examining sparse causal drift structure from irregularly sampled time series. Under explicit regularity assumptions, Girsanov’s theorem relates path-space relative entropy to a quadratic steering cost, motivating a decomposition into reusable structural drift and data-dependent control. Theoretically, we give a conditional drift-consistency analysis and sufficient coverage, smoothness, and signal-margin conditions for oracle recovery, a counterexample shows that vanishing action and uniformly accurate drift can still retain a false edge. Empirically, within this controlled-SDE setting, Lorenz96 and Glycolysis experiments show drift-ranking signal, including for 50-variable Lorenz models, although fixed-threshold graph recovery remains poor in that high-dimensional setting. Checkpoint diagnostics show that similar prediction error can accompany different drift rankings. A matched Lorenz D10 comparison finds no resolved ranking benefit from control or stochastic diffusion over deterministic drift, a released-code NGM-GL comparator gives similar ranking with substantial checkpoint sensitivity. Ablations show that bounded control prevents constant-drift collapse in a zero-action-penalty stress test, while proximal sparsity improves Lorenz graph parsimony. Together, these results position PLACID as a testbed for examining how control regularization and sparse drift parameterization shape structure learning in stochastic dynamical systems.