PRIME: Poincaré return induced measure for learning partially observed dynamical systems

Learning dynamics from sparse partial observations presents a fundamental dilemma: the pointwise mean squared error (MSE) training objective may induce local contraction and empirically lead to topological collapse, whereas matching the system's invariant measure preserves global geometry but lacks temporal ordering constraints. To resolve this, we introduce the $\textbf{P}$oincaré $\textbf{R}$eturn $\textbf{I}$nduced $\textbf{M}$easur$\textbf{e}$ $(\textbf{PRIME})$ framework. PRIME bridges this gap by adopting a novel perspective of recurrence-level temporal organization, motivated by the return map and roof time structure of suspension flow representations. Instead of relying on unstable pointwise matching, PRIME refines the learned dynamics by leveraging recurrence-level temporal information, specifically by aligning the empirical graph measure of the induced return map over recurrence blocks. We evaluate our framework on the partially observed Lorenz63 system, sensor-observed Kuramoto-Sivashinsky equation, and flow past a cylinder. PRIME effectively reduces topological collapse across our benchmarks, improving temporal consistency while maintaining or enhancing long-term geometric and spectral fidelity.