Tissue dynamics play critical roles in many physiological functions and provide important metrics for clinical diagnosis. Capturing real-time high-resolution 3D images of tissue dynamics, however, remains a challenge. This study presents a hybrid physics-informed neural network algorithm that infers 3D flow-induced tissue dynamics and other physical quantities from sparse 2D images. The algorithm combines a recurrent neural network model of soft tissue with a differentiable fluid solver, leveraging prior knowledge in solid mechanics to project the governing equation on a discrete eigen space. The algorithm uses a Long-short-term memory-based recurrent encoder-decoder connected with a fully connected neural network to capture the temporal dependence of flow-structure-interaction. The effectiveness and merit of the proposed algorithm is demonstrated on synthetic data from a canine vocal fold model and experimental data from excised pigeon syringes. The results showed that the algorithm accurately reconstructs 3D vocal dynamics, aerodynamics, and acoustics from sparse 2D vibration profiles.