Discovering constitutive models for hyperelastic materials is a challenging inverse problem, particularly when experimental data are extremely scarce. Conventional identification approaches typically require extensive stress–strain measurements or full-field displacement data, which are often difficult to obtain in practical experimental settings. This study proposes a physics-informed neural network (PINN)–based framework that enables the discovery of hyperelastic constitutive models using only sparse displacement measurements and reaction force data from a single material test. The proposed framework integrates PINNs with finite element discretization and a weak-form formulation of the governing equilibrium equations. Spatial gradients are evaluated using finite element shape functions rather than strong-form or automatic differentiation, significantly reducing computational cost while preserving numerical accuracy. The constitutive behavior is represented as a linear combination of predefined strain energy density candidates, formulating the discovery problem as a physics-constrained sparse regression task. To infer both the displacement field and constitutive parameters, a two-stage training strategy is employed. In the first stage, neural network parameters and constitutive coefficients are jointly optimized using the Adam optimizer. In the second stage, the displacement field is refined using the L-BFGS optimizer, while sparse regression with regularization identifies a parsimonious constitutive model. The proposed method is validated using multiple benchmark hyperelastic models under varying noise levels. Numerical results demonstrate accurate reconstruction of full-field displacement and recovery of constitutive laws even with extremely limited and noisy data, highlighting the framework’s potential for experimental scenarios lacking comprehensive measurements.