Flow-consistent identification of governing equations from sparsely sampled measurements

Extracting continuous-time governing equations from sparsely sampled measurements remains a fundamental challenge across science and engineering. Under severe temporal sparsity, conventional data-driven approaches often lose physical consistency, leading to distorted dynamics and unreliable qualitative behavior. Here we propose Hybrid Analytic Neural Dynamics Identification (HANDI), a flow-consistent framework that enables accurate identification of governing equations from sparsely sampled time series. Rather than relying on noisy numerical differentiation or integration, HANDI operates directly on the flow map, bridging discrete measurements and continuous-time dynamics through a Koopman-based formulation. By constructing a hybrid observable space that integrates interpretable analytic structures with neural representations, HANDI enables a sampling-robust linearization of the underlying dynamics. Across canonical nonlinear systems (e.g., limit cycles and bistability) and real-world datasets (e.g., inverted-flag flapping and wheel shimmy dynamics), HANDI consistently attains high mechanistic fidelity under extreme temporal sparsity, reliably uncovering the underlying physics and dynamical geometry from limited measurements.