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. Its effectiveness is demonstrated on synthetic data from a canine vocal fold model and experimental data from excised pigeon syringes, accurately reconstructing 3D vocal dynamics, aerodynamics, and acoustics from sparse 2D vibration profiles.