<p>Nonlinear vibration isolation systems are of fundamental importance in safeguarding equipment, structures, and buildings against harmful vibrational excitations. However, existing intelligent methods are inadequate to fully capture their physical features, which limits the efficient prediction and design. In this study, we propose a novel modular physics-informed neural network (MPINN), enabling dynamic prediction and inverse design of harmonically excited single-degree-of-freedom nonlinear vibration isolators. This framework adopts a modular architecture consisting of a steady-state module informed by the harmonic balance network for predicting steady-state responses, a transient module that predicts decaying components through structured exponential and Fourier layers, and a restoring-force module that learns nonlinear stiffness in a polynomial form. These three modules are integrated through the governing equations and relevant physical quantities, thereby yielding a unified framework. By comparison with analytical solutions under linear conditions, the reliability of the MPINN is validated, and the steady-state and transient modules improve the accuracy by orders of magnitude compared with a specific multilayer perceptron (MLP)-based framework. Incorporating the energy-conservation loss significantly enhances the convergence of the MPINN for dynamic responses of Duffing-type isolators. The MPINN is further adopted to predict the responses of strongly nonlinear isolators, with the residual error reaching only 0.3% of that from the Runge-Kutta method. Finally, the inverse design of vibration isolators is achieved using the MPINN, generating desired results within only 0.52 s. This work proposes a novel modular physics-informed machine learning framework for the prediction and design of vibration isolators, providing new insights into investigating complex dynamic behaviors.</p>